Gamma分布 メモ

gamma分布.ipynb

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      "source": " # Table of Contents\n<div class=\"toc\" style=\"margin-top: 1em;\"><ul class=\"toc-item\" id=\"toc-level0\"><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Prob_Stat/ProbabilityAndStatics/gamma%E5%88%86%E5%B8%83.ipynb#ガンマ分布\" data-toc-modified-id=\"ガンマ分布-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>ガンマ分布</a></span><ul class=\"toc-item\"><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Prob_Stat/ProbabilityAndStatics/gamma%E5%88%86%E5%B8%83.ipynb#意味\" data-toc-modified-id=\"意味-1.1\"><span class=\"toc-item-num\">1.1&nbsp;&nbsp;</span>意味</a></span></li><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Prob_Stat/ProbabilityAndStatics/gamma%E5%88%86%E5%B8%83.ipynb#確率密度関数\" data-toc-modified-id=\"確率密度関数-1.2\"><span class=\"toc-item-num\">1.2&nbsp;&nbsp;</span>確率密度関数</a></span><ul class=\"toc-item\"><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Prob_Stat/ProbabilityAndStatics/gamma%E5%88%86%E5%B8%83.ipynb#累積密度関数\" data-toc-modified-id=\"累積密度関数-1.2.1\"><span class=\"toc-item-num\">1.2.1&nbsp;&nbsp;</span>累積密度関数</a></span></li><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Prob_Stat/ProbabilityAndStatics/gamma%E5%88%86%E5%B8%83.ipynb#期待値\" data-toc-modified-id=\"期待値-1.2.2\"><span class=\"toc-item-num\">1.2.2&nbsp;&nbsp;</span>期待値</a></span></li><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Prob_Stat/ProbabilityAndStatics/gamma%E5%88%86%E5%B8%83.ipynb#分散\" data-toc-modified-id=\"分散-1.2.3\"><span class=\"toc-item-num\">1.2.3&nbsp;&nbsp;</span>分散</a></span></li><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Prob_Stat/ProbabilityAndStatics/gamma%E5%88%86%E5%B8%83.ipynb#ガンマ分布のモーメント母関数\" data-toc-modified-id=\"ガンマ分布のモーメント母関数-1.2.4\"><span class=\"toc-item-num\">1.2.4&nbsp;&nbsp;</span>ガンマ分布のモーメント母関数</a></span></li></ul></li><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Prob_Stat/ProbabilityAndStatics/gamma%E5%88%86%E5%B8%83.ipynb#ガンマ分布と指数分布の比較\" data-toc-modified-id=\"ガンマ分布と指数分布の比較-1.3\"><span class=\"toc-item-num\">1.3&nbsp;&nbsp;</span>ガンマ分布と指数分布の比較</a></span></li><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Prob_Stat/ProbabilityAndStatics/gamma%E5%88%86%E5%B8%83.ipynb#他分布との関係\" data-toc-modified-id=\"他分布との関係-1.4\"><span class=\"toc-item-num\">1.4&nbsp;&nbsp;</span>他分布との関係</a></span><ul class=\"toc-item\"><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Prob_Stat/ProbabilityAndStatics/gamma%E5%88%86%E5%B8%83.ipynb#$\\chi^2$分布\" data-toc-modified-id=\"$\\chi^2$分布-1.4.1\"><span class=\"toc-item-num\">1.4.1&nbsp;&nbsp;</span>$\\chi^2$分布</a></span></li></ul></li><li><span><a href=\"http://localhost:8888/notebooks/GoogleDrive/JupyterNote/Prob_Stat/ProbabilityAndStatics/gamma%E5%88%86%E5%B8%83.ipynb#参考資料\" data-toc-modified-id=\"参考資料-1.5\"><span class=\"toc-item-num\">1.5&nbsp;&nbsp;</span>参考資料</a></span></li></ul></li></ul></div>"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "# ガンマ分布"
    },
    {
      "metadata": {
        "trusted": true,
        "ExecuteTime": {
          "start_time": "2017-12-12T11:45:55.535883Z",
          "end_time": "2017-12-12T11:46:02.145565Z"
        },
        "collapsed": true
      },
      "cell_type": "code",
      "source": "import scipy as sp\nfrom scipy import special\nfrom scipy import stats\nimport matplotlib.pyplot as plt",
      "execution_count": 1,
      "outputs": []
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "- [gist](https://gist.github.com/e6da7070126b1d733bd3b03fad01fd9f)\n- [nbviewer](http://nbviewer.jupyter.org/gist/Cartman0/e6da7070126b1d733bd3b03fad01fd9f)"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "## 意味"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "期間$\\mu$ ごとに1回起こるランダムな事象が$n$回起こるまでの時間の分布を表す．\n\n例：「10年に1度の割合でランダムに起こるイベントが3回起こるまでに何年かかるか」という問題は，期待値は30年，\n\nパラメータは，\n- 尺度母数$\\mu = 10$\n- 形状母数$n = 3$\n\nのガンマ分布に対応する．"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "## 確率密度関数"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "\\begin{eqnarray}\nf(x) = \\frac{1}{\\Gamma(n)\\mu^n}x^{n−1}e^{−\\frac{x}{\\mu}}, (x≥0)\n\\end{eqnarray}\n\n- パラメータ: $\\mu, n > 0$"
    },
    {
      "metadata": {
        "trusted": true,
        "ExecuteTime": {
          "start_time": "2017-12-11T15:49:40.739460Z",
          "end_time": "2017-12-11T15:49:40.754471Z"
        }
      },
      "cell_type": "code",
      "source": "help(stats.gamma)",
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "stream",
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          "text": "Help on gamma_gen in module scipy.stats._continuous_distns object:\n\nclass gamma_gen(scipy.stats._distn_infrastructure.rv_continuous)\n |  A gamma continuous random variable.\n |  \n |  %(before_notes)s\n |  \n |  See Also\n |  --------\n |  erlang, expon\n |  \n |  Notes\n |  -----\n |  The probability density function for `gamma` is::\n |  \n |      gamma.pdf(x, a) = x**(a-1) * exp(-x) / gamma(a)\n |  \n |  for ``x >= 0``, ``a > 0``. Here ``gamma(a)`` refers to the gamma function.\n |  \n |  `gamma` has a shape parameter `a` which needs to be set explicitly.\n |  \n |  When ``a`` is an integer, `gamma` reduces to the Erlang\n |  distribution, and when ``a=1`` to the exponential distribution.\n |  \n |  %(after_notes)s\n |  \n |  %(example)s\n |  \n |  Method resolution order:\n |      gamma_gen\n |      scipy.stats._distn_infrastructure.rv_continuous\n |      scipy.stats._distn_infrastructure.rv_generic\n |      builtins.object\n |  \n |  Methods defined here:\n |  \n |  fit(self, data, *args, **kwds)\n |      Return MLEs for shape (if applicable), location, and scale\n |      parameters from data.\n |      \n |      MLE stands for Maximum Likelihood Estimate.  Starting estimates for\n |      the fit are given by input arguments; for any arguments not provided\n |      with starting estimates, ``self._fitstart(data)`` is called to generate\n |      such.\n |      \n |      One can hold some parameters fixed to specific values by passing in\n |      keyword arguments ``f0``, ``f1``, ..., ``fn`` (for shape parameters)\n |      and ``floc`` and ``fscale`` (for location and scale parameters,\n |      respectively).\n |      \n |      Parameters\n |      ----------\n |      data : array_like\n |          Data to use in calculating the MLEs.\n |      args : floats, optional\n |          Starting value(s) for any shape-characterizing arguments (those not\n |          provided will be determined by a call to ``_fitstart(data)``).\n |          No default value.\n |      kwds : floats, optional\n |          Starting values for the location and scale parameters; no default.\n |          Special keyword arguments are recognized as holding certain\n |          parameters fixed:\n |      \n |          - f0...fn : hold respective shape parameters fixed.\n |            Alternatively, shape parameters to fix can be specified by name.\n |            For example, if ``self.shapes == \"a, b\"``, ``fa``and ``fix_a``\n |            are equivalent to ``f0``, and ``fb`` and ``fix_b`` are\n |            equivalent to ``f1``.\n |      \n |          - floc : hold location parameter fixed to specified value.\n |      \n |          - fscale : hold scale parameter fixed to specified value.\n |      \n |          - optimizer : The optimizer to use.  The optimizer must take ``func``,\n |            and starting position as the first two arguments,\n |            plus ``args`` (for extra arguments to pass to the\n |            function to be optimized) and ``disp=0`` to suppress\n |            output as keyword arguments.\n |      \n |      Returns\n |      -------\n |      mle_tuple : tuple of floats\n |          MLEs for any shape parameters (if applicable), followed by those\n |          for location and scale. For most random variables, shape statistics\n |          will be returned, but there are exceptions (e.g. ``norm``).\n |      \n |      Notes\n |      -----\n |      This fit is computed by maximizing a log-likelihood function, with\n |      penalty applied for samples outside of range of the distribution. The\n |      returned answer is not guaranteed to be the globally optimal MLE, it\n |      may only be locally optimal, or the optimization may fail altogether.\n |      \n |      \n |      Examples\n |      --------\n |      \n |      Generate some data to fit: draw random variates from the `beta`\n |      distribution\n |      \n |      >>> from scipy.stats import beta\n |      >>> a, b = 1., 2.\n |      >>> x = beta.rvs(a, b, size=1000)\n |      \n |      Now we can fit all four parameters (``a``, ``b``, ``loc`` and ``scale``):\n |      \n |      >>> a1, b1, loc1, scale1 = beta.fit(x)\n |      \n |      We can also use some prior knowledge about the dataset: let's keep\n |      ``loc`` and ``scale`` fixed:\n |      \n |      >>> a1, b1, loc1, scale1 = beta.fit(x, floc=0, fscale=1)\n |      >>> loc1, scale1\n |      (0, 1)\n |      \n |      We can also keep shape parameters fixed by using ``f``-keywords. To\n |      keep the zero-th shape parameter ``a`` equal 1, use ``f0=1`` or,\n |      equivalently, ``fa=1``:\n |      \n |      >>> a1, b1, loc1, scale1 = beta.fit(x, fa=1, floc=0, fscale=1)\n |      >>> a1\n |      1\n |      \n |      Not all distributions return estimates for the shape parameters.\n |      ``norm`` for example just returns estimates for location and scale:\n |      \n |      >>> from scipy.stats import norm\n |      >>> x = norm.rvs(a, b, size=1000, random_state=123)\n |      >>> loc1, scale1 = norm.fit(x)\n |      >>> loc1, scale1\n |      (0.92087172783841631, 2.0015750750324668)\n |  \n |  ----------------------------------------------------------------------\n |  Methods inherited from scipy.stats._distn_infrastructure.rv_continuous:\n |  \n |  __init__(self, momtype=1, a=None, b=None, xtol=1e-14, badvalue=None, name=None, longname=None, shapes=None, extradoc=None, seed=None)\n |      Initialize self.  See help(type(self)) for accurate signature.\n |  \n |  cdf(self, x, *args, **kwds)\n |      Cumulative distribution function of the given RV.\n |      \n |      Parameters\n |      ----------\n |      x : array_like\n |          quantiles\n |      arg1, arg2, arg3,... : array_like\n |          The shape parameter(s) for the distribution (see docstring of the\n |          instance object for more information)\n |      loc : array_like, optional\n |          location parameter (default=0)\n |      scale : array_like, optional\n |          scale parameter (default=1)\n |      \n |      Returns\n |      -------\n |      cdf : ndarray\n |          Cumulative distribution function evaluated at `x`\n |  \n |  expect(self, func=None, args=(), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds)\n |      Calculate expected value of a function with respect to the\n |      distribution.\n |      \n |      The expected value of a function ``f(x)`` with respect to a\n |      distribution ``dist`` is defined as::\n |      \n |                  ubound\n |          E[x] = Integral(f(x) * dist.pdf(x))\n |                  lbound\n |      \n |      Parameters\n |      ----------\n |      func : callable, optional\n |          Function for which integral is calculated. Takes only one argument.\n |          The default is the identity mapping f(x) = x.\n |      args : tuple, optional\n |          Shape parameters of the distribution.\n |      loc : float, optional\n |          Location parameter (default=0).\n |      scale : float, optional\n |          Scale parameter (default=1).\n |      lb, ub : scalar, optional\n |          Lower and upper bound for integration. Default is set to the\n |          support of the distribution.\n |      conditional : bool, optional\n |          If True, the integral is corrected by the conditional probability\n |          of the integration interval.  The return value is the expectation\n |          of the function, conditional on being in the given interval.\n |          Default is False.\n |      \n |      Additional keyword arguments are passed to the integration routine.\n |      \n |      Returns\n |      -------\n |      expect : float\n |          The calculated expected value.\n |      \n |      Notes\n |      -----\n |      The integration behavior of this function is inherited from\n |      `integrate.quad`.\n |  \n |  fit_loc_scale(self, data, *args)\n |      Estimate loc and scale parameters from data using 1st and 2nd moments.\n |      \n |      Parameters\n |      ----------\n |      data : array_like\n |          Data to fit.\n |      arg1, arg2, arg3,... : array_like\n |          The shape parameter(s) for the distribution (see docstring of the\n |          instance object for more information).\n |      \n |      Returns\n |      -------\n |      Lhat : float\n |          Estimated location parameter for the data.\n |      Shat : float\n |          Estimated scale parameter for the data.\n |  \n |  isf(self, q, *args, **kwds)\n |      Inverse survival function (inverse of `sf`) at q of the given RV.\n |      \n |      Parameters\n |      ----------\n |      q : array_like\n |          upper tail probability\n |      arg1, arg2, arg3,... : array_like\n |          The shape parameter(s) for the distribution (see docstring of the\n |          instance object for more information)\n |      loc : array_like, optional\n |          location parameter (default=0)\n |      scale : array_like, optional\n |          scale parameter (default=1)\n |      \n |      Returns\n |      -------\n |      x : ndarray or scalar\n |          Quantile corresponding to the upper tail probability q.\n |  \n |  logcdf(self, x, *args, **kwds)\n |      Log of the cumulative distribution function at x of the given RV.\n |      \n |      Parameters\n |      ----------\n |      x : array_like\n |          quantiles\n |      arg1, arg2, arg3,... : array_like\n |          The shape parameter(s) for the distribution (see docstring of the\n |          instance object for more information)\n |      loc : array_like, optional\n |          location parameter (default=0)\n |      scale : array_like, optional\n |          scale parameter (default=1)\n |      \n |      Returns\n |      -------\n |      logcdf : array_like\n |          Log of the cumulative distribution function evaluated at x\n |  \n |  logpdf(self, x, *args, **kwds)\n |      Log of the probability density function at x of the given RV.\n |      \n |      This uses a more numerically accurate calculation if available.\n |      \n |      Parameters\n |      ----------\n |      x : array_like\n |          quantiles\n |      arg1, arg2, arg3,... : array_like\n |          The shape parameter(s) for the distribution (see docstring of the\n |          instance object for more information)\n |      loc : array_like, optional\n |          location parameter (default=0)\n |      scale : array_like, optional\n |          scale parameter (default=1)\n |      \n |      Returns\n |      -------\n |      logpdf : array_like\n |          Log of the probability density function evaluated at x\n |  \n |  logsf(self, x, *args, **kwds)\n |      Log of the survival function of the given RV.\n |      \n |      Returns the log of the \"survival function,\" defined as (1 - `cdf`),\n |      evaluated at `x`.\n |      \n |      Parameters\n |      ----------\n |      x : array_like\n |          quantiles\n |      arg1, arg2, arg3,... : array_like\n |          The shape parameter(s) for the distribution (see docstring of the\n |          instance object for more information)\n |      loc : array_like, optional\n |          location parameter (default=0)\n |      scale : array_like, optional\n |          scale parameter (default=1)\n |      \n |      Returns\n |      -------\n |      logsf : ndarray\n |          Log of the survival function evaluated at `x`.\n |  \n |  nnlf(self, theta, x)\n |      Return negative loglikelihood function.\n |      \n |      Notes\n |      -----\n |      This is ``-sum(log pdf(x, theta), axis=0)`` where `theta` are the\n |      parameters (including loc and scale).\n |  \n |  pdf(self, x, *args, **kwds)\n |      Probability density function at x of the given RV.\n |      \n |      Parameters\n |      ----------\n |      x : array_like\n |          quantiles\n |      arg1, arg2, arg3,... : array_like\n |          The shape parameter(s) for the distribution (see docstring of the\n |          instance object for more information)\n |      loc : array_like, optional\n |          location parameter (default=0)\n |      scale : array_like, optional\n |          scale parameter (default=1)\n |      \n |      Returns\n |      -------\n |      pdf : ndarray\n |          Probability density function evaluated at x\n |  \n |  ppf(self, q, *args, **kwds)\n |      Percent point function (inverse of `cdf`) at q of the given RV.\n |      \n |      Parameters\n |      ----------\n |      q : array_like\n |          lower tail probability\n |      arg1, arg2, arg3,... : array_like\n |          The shape parameter(s) for the distribution (see docstring of the\n |          instance object for more information)\n |      loc : array_like, optional\n |          location parameter (default=0)\n |      scale : array_like, optional\n |          scale parameter (default=1)\n |      \n |      Returns\n |      -------\n |      x : array_like\n |          quantile corresponding to the lower tail probability q.\n |  \n |  sf(self, x, *args, **kwds)\n |      Survival function (1 - `cdf`) at x of the given RV.\n |      \n |      Parameters\n |      ----------\n |      x : array_like\n |          quantiles\n |      arg1, arg2, arg3,... : array_like\n |          The shape parameter(s) for the distribution (see docstring of the\n |          instance object for more information)\n |      loc : array_like, optional\n |          location parameter (default=0)\n |      scale : array_like, optional\n |          scale parameter (default=1)\n |      \n |      Returns\n |      -------\n |      sf : array_like\n |          Survival function evaluated at x\n |  \n |  ----------------------------------------------------------------------\n |  Methods inherited from scipy.stats._distn_infrastructure.rv_generic:\n |  \n |  __call__(self, *args, **kwds)\n |      Freeze the distribution for the given arguments.\n |      \n |      Parameters\n |      ----------\n |      arg1, arg2, arg3,... : array_like\n |          The shape parameter(s) for the distribution.  Should include all\n |          the non-optional arguments, may include ``loc`` and ``scale``.\n |      \n |      Returns\n |      -------\n |      rv_frozen : rv_frozen instance\n |          The frozen distribution.\n |  \n |  __getstate__(self)\n |  \n |  __setstate__(self, state)\n |  \n |  entropy(self, *args, **kwds)\n |      Differential entropy of the RV.\n |      \n |      Parameters\n |      ----------\n |      arg1, arg2, arg3,... : array_like\n |          The shape parameter(s) for the distribution (see docstring of the\n |          instance object for more information).\n |      loc : array_like, optional\n |          Location parameter (default=0).\n |      scale : array_like, optional  (continuous distributions only).\n |          Scale parameter (default=1).\n |      \n |      Notes\n |      -----\n |      Entropy is defined base `e`:\n |      \n |      >>> drv = rv_discrete(values=((0, 1), (0.5, 0.5)))\n |      >>> np.allclose(drv.entropy(), np.log(2.0))\n |      True\n |  \n |  freeze(self, *args, **kwds)\n |      Freeze the distribution for the given arguments.\n |      \n |      Parameters\n |      ----------\n |      arg1, arg2, arg3,... : array_like\n |          The shape parameter(s) for the distribution.  Should include all\n |          the non-optional arguments, may include ``loc`` and ``scale``.\n |      \n |      Returns\n |      -------\n |      rv_frozen : rv_frozen instance\n |          The frozen distribution.\n |  \n |  interval(self, alpha, *args, **kwds)\n |      Confidence interval with equal areas around the median.\n |      \n |      Parameters\n |      ----------\n |      alpha : array_like of float\n |          Probability that an rv will be drawn from the returned range.\n |          Each value should be in the range [0, 1].\n |      arg1, arg2, ... : array_like\n |          The shape parameter(s) for the distribution (see docstring of the\n |          instance object for more information).\n |      loc : array_like, optional\n |          location parameter, Default is 0.\n |      scale : array_like, optional\n |          scale parameter, Default is 1.\n |      \n |      Returns\n |      -------\n |      a, b : ndarray of float\n |          end-points of range that contain ``100 * alpha %`` of the rv's\n |          possible values.\n |  \n |  mean(self, *args, **kwds)\n |      Mean of the distribution.\n |      \n |      Parameters\n |      ----------\n |      arg1, arg2, arg3,... : array_like\n |          The shape parameter(s) for the distribution (see docstring of the\n |          instance object for more information)\n |      loc : array_like, optional\n |          location parameter (default=0)\n |      scale : array_like, optional\n |          scale parameter (default=1)\n |      \n |      Returns\n |      -------\n |      mean : float\n |          the mean of the distribution\n |  \n |  median(self, *args, **kwds)\n |      Median of the distribution.\n |      \n |      Parameters\n |      ----------\n |      arg1, arg2, arg3,... : array_like\n |          The shape parameter(s) for the distribution (see docstring of the\n |          instance object for more information)\n |      loc : array_like, optional\n |          Location parameter, Default is 0.\n |      scale : array_like, optional\n |          Scale parameter, Default is 1.\n |      \n |      Returns\n |      -------\n |      median : float\n |          The median of the distribution.\n |      \n |      See Also\n |      --------\n |      stats.distributions.rv_discrete.ppf\n |          Inverse of the CDF\n |  \n |  moment(self, n, *args, **kwds)\n |      n-th order non-central moment of distribution.\n |      \n |      Parameters\n |      ----------\n |      n : int, n >= 1\n |          Order of moment.\n |      arg1, arg2, arg3,... : float\n |          The shape parameter(s) for the distribution (see docstring of the\n |          instance object for more information).\n |      loc : array_like, optional\n |          location parameter (default=0)\n |      scale : array_like, optional\n |          scale parameter (default=1)\n |  \n |  rvs(self, *args, **kwds)\n |      Random variates of given type.\n |      \n |      Parameters\n |      ----------\n |      arg1, arg2, arg3,... : array_like\n |          The shape parameter(s) for the distribution (see docstring of the\n |          instance object for more information).\n |      loc : array_like, optional\n |          Location parameter (default=0).\n |      scale : array_like, optional\n |          Scale parameter (default=1).\n |      size : int or tuple of ints, optional\n |          Defining number of random variates (default is 1).\n |      random_state : None or int or ``np.random.RandomState`` instance, optional\n |          If int or RandomState, use it for drawing the random variates.\n |          If None, rely on ``self.random_state``.\n |          Default is None.\n |      \n |      Returns\n |      -------\n |      rvs : ndarray or scalar\n |          Random variates of given `size`.\n |  \n |  stats(self, *args, **kwds)\n |      Some statistics of the given RV.\n |      \n |      Parameters\n |      ----------\n |      arg1, arg2, arg3,... : array_like\n |          The shape parameter(s) for the distribution (see docstring of the\n |          instance object for more information)\n |      loc : array_like, optional\n |          location parameter (default=0)\n |      scale : array_like, optional (continuous RVs only)\n |          scale parameter (default=1)\n |      moments : str, optional\n |          composed of letters ['mvsk'] defining which moments to compute:\n |          'm' = mean,\n |          'v' = variance,\n |          's' = (Fisher's) skew,\n |          'k' = (Fisher's) kurtosis.\n |          (default is 'mv')\n |      \n |      Returns\n |      -------\n |      stats : sequence\n |          of requested moments.\n |  \n |  std(self, *args, **kwds)\n |      Standard deviation of the distribution.\n |      \n |      Parameters\n |      ----------\n |      arg1, arg2, arg3,... : array_like\n |          The shape parameter(s) for the distribution (see docstring of the\n |          instance object for more information)\n |      loc : array_like, optional\n |          location parameter (default=0)\n |      scale : array_like, optional\n |          scale parameter (default=1)\n |      \n |      Returns\n |      -------\n |      std : float\n |          standard deviation of the distribution\n |  \n |  var(self, *args, **kwds)\n |      Variance of the distribution.\n |      \n |      Parameters\n |      ----------\n |      arg1, arg2, arg3,... : array_like\n |          The shape parameter(s) for the distribution (see docstring of the\n |          instance object for more information)\n |      loc : array_like, optional\n |          location parameter (default=0)\n |      scale : array_like, optional\n |          scale parameter (default=1)\n |      \n |      Returns\n |      -------\n |      var : float\n |          the variance of the distribution\n |  \n |  ----------------------------------------------------------------------\n |  Data descriptors inherited from scipy.stats._distn_infrastructure.rv_generic:\n |  \n |  __dict__\n |      dictionary for instance variables (if defined)\n |  \n |  __weakref__\n |      list of weak references to the object (if defined)\n |  \n |  random_state\n |      Get or set the RandomState object for generating random variates.\n |      \n |      This can be either None or an existing RandomState object.\n |      \n |      If None (or np.random), use the RandomState singleton used by np.random.\n |      If already a RandomState instance, use it.\n |      If an int, use a new RandomState instance seeded with seed.\n\n"
        }
      ]
    },
    {
      "metadata": {
        "trusted": true,
        "ExecuteTime": {
          "start_time": "2017-12-11T16:02:35.677903Z",
          "end_time": "2017-12-11T16:02:35.685910Z"
        }
      },
      "cell_type": "code",
      "source": "special.gamma(1), special.gamma(2)",
      "execution_count": 15,
      "outputs": [
        {
          "output_type": "execute_result",
          "metadata": {},
          "data": {
            "text/plain": "(1.0, 1.0)"
          },
          "execution_count": 15
        }
      ]
    },
    {
      "metadata": {
        "trusted": true,
        "ExecuteTime": {
          "start_time": "2017-12-11T16:03:19.044808Z",
          "end_time": "2017-12-11T16:03:19.098650Z"
        }
      },
      "cell_type": "code",
      "source": "special.gamma(.5), sp.sqrt(sp.pi)",
      "execution_count": 17,
      "outputs": [
        {
          "output_type": "execute_result",
          "metadata": {},
          "data": {
            "text/plain": "(1.7724538509055159, 1.7724538509055159)"
          },
          "execution_count": 17
        }
      ]
    },
    {
      "metadata": {
        "trusted": true,
        "ExecuteTime": {
          "start_time": "2017-12-11T16:07:48.715857Z",
          "end_time": "2017-12-11T16:07:48.727864Z"
        }
      },
      "cell_type": "code",
      "source": "print(special.gamma(3. / 2))  # gamma(1+1/2)=1/2gamma(1/2)\n0.5 * sp.sqrt(sp.pi)",
      "execution_count": 18,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": "0.886226925453\n"
        },
        {
          "output_type": "execute_result",
          "metadata": {},
          "data": {
            "text/plain": "0.88622692545275794"
          },
          "execution_count": 18
        }
      ]
    },
    {
      "metadata": {
        "trusted": true,
        "ExecuteTime": {
          "start_time": "2017-12-11T16:13:36.138093Z",
          "end_time": "2017-12-11T16:13:36.198137Z"
        }
      },
      "cell_type": "code",
      "source": "def gamma_pdf(mu, n, x):\n    x = sp.array(x)\n    return 1. / (special.gamma(n) * mu**n) * x**(n - 1) * sp.exp(-x / mu)\n\ngamma_pdf(mu=1, n=1, x=[0,.5,1.])",
      "execution_count": 25,
      "outputs": [
        {
          "output_type": "execute_result",
          "metadata": {},
          "data": {
            "text/plain": "array([ 1.        ,  0.60653066,  0.36787944])"
          },
          "execution_count": 25
        }
      ]
    },
    {
      "metadata": {
        "trusted": true,
        "ExecuteTime": {
          "start_time": "2017-12-11T16:13:51.447904Z",
          "end_time": "2017-12-11T16:13:51.506561Z"
        }
      },
      "cell_type": "code",
      "source": "stats.gamma(a=1, scale=1).pdf([0, .5, 1])",
      "execution_count": 26,
      "outputs": [
        {
          "output_type": "execute_result",
          "metadata": {},
          "data": {
            "text/plain": "array([ 1.        ,  0.60653066,  0.36787944])"
          },
          "execution_count": 26
        }
      ]
    },
    {
      "metadata": {
        "trusted": true,
        "ExecuteTime": {
          "start_time": "2017-12-11T16:18:00.047359Z",
          "end_time": "2017-12-11T16:18:00.061367Z"
        }
      },
      "cell_type": "code",
      "source": "gamma_pdf(mu=10, n=3, x=[0,.5,1.]), stats.gamma(a=3, scale=10).pdf([0, .5, 1])",
      "execution_count": 28,
      "outputs": [
        {
          "output_type": "execute_result",
          "metadata": {},
          "data": {
            "text/plain": "(array([ 0.        ,  0.0001189 ,  0.00045242]),\n array([ 0.        ,  0.0001189 ,  0.00045242]))"
          },
          "execution_count": 28
        }
      ]
    },
    {
      "metadata": {
        "trusted": true,
        "ExecuteTime": {
          "start_time": "2017-12-11T17:10:30.669822Z",
          "end_time": "2017-12-11T17:10:32.061146Z"
        }
      },
      "cell_type": "code",
      "source": "%matplotlib inline\n\nx = sp.linspace(0, 10, 200)\nmu_s = sp.arange(0.5, 3+.5, .5)\nn = 2\nfor mu in mu_s:\n    plt.plot(x, stats.gamma(a=n, scale=mu).pdf(x), \n             label=\"$\\mu={}, n={}$\".format(mu, n))\n\nplt.legend()\nplt.grid()\nplt.show()\n\n\nn_s = sp.arange(0.5, 3+.5, .5)\nmu = 2\nfor n in n_s:\n    plt.plot(x, stats.gamma(a=n, scale=mu).pdf(x), \n             label=\"$\\mu={}, n={}$\".format(mu, n))\n\nplt.legend()\nplt.grid()\nplt.show()",
      "execution_count": 36,
      "outputs": [
        {
          "metadata": {},
          "output_type": "display_data",
          "data": {
            "image/png": 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8wNReqazu0GoEk6L8OKhG7oriVr1dZg8ce6tHRERQWFjIvHnzGDt2LLNnz27VTpXZ62Ol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\nqEzbVC2CykWQP3lExR4g35zPsoplLKtY1lfni/vY79vPPt8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RxV1FRUVlDKKKu4qKisoY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\nGM+Vn4EbAAAAAElFTkSuQmCC\n",
            "text/plain": "<matplotlib.figure.Figure at 0x2793c0b41d0>"
          }
        },
        {
          "metadata": {},
          "output_type": "display_data",
          "data": {
            "image/png": 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K6l5L+CMrZudT19bJrjrp1imESB2mBNKoAbadfon8B+BFrXVQKfU14Fngkwl+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\nAwUCJRj7pYqzAAAAAElFTkSuQmCC\n",
            "text/plain": "<matplotlib.figure.Figure at 0x2793be4e1d0>"
          }
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### 累積密度関数"
    },
    {
      "metadata": {
        "trusted": true,
        "ExecuteTime": {
          "start_time": "2017-12-11T17:33:07.373467Z",
          "end_time": "2017-12-11T17:33:07.405489Z"
        }
      },
      "cell_type": "code",
      "source": "help(stats.gamma.cdf)",
      "execution_count": 37,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": "Help on method cdf in module scipy.stats._distn_infrastructure:\n\ncdf(x, *args, **kwds) method of scipy.stats._continuous_distns.gamma_gen instance\n    Cumulative distribution function of the given RV.\n    \n    Parameters\n    ----------\n    x : array_like\n        quantiles\n    arg1, arg2, arg3,... : array_like\n        The shape parameter(s) for the distribution (see docstring of the\n        instance object for more information)\n    loc : array_like, optional\n        location parameter (default=0)\n    scale : array_like, optional\n        scale parameter (default=1)\n    \n    Returns\n    -------\n    cdf : ndarray\n        Cumulative distribution function evaluated at `x`\n\n"
        }
      ]
    },
    {
      "metadata": {
        "trusted": true,
        "ExecuteTime": {
          "start_time": "2017-12-11T17:38:27.832637Z",
          "end_time": "2017-12-11T17:38:28.258423Z"
        }
      },
      "cell_type": "code",
      "source": "%matplotlib inline\n\nx = sp.linspace(0, 15, 200)\nmu = 2\nn = 2\nplt.plot(x, stats.gamma(a=n, scale=mu).cdf(x), \n         label=\"$\\mu={}, n={}$\".format(mu, n))\n\nplt.legend()\nplt.grid()\nplt.show()",
      "execution_count": 41,
      "outputs": [
        {
          "metadata": {},
          "output_type": "display_data",
          "data": {
            "image/png": 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Y80H4M4Yt3+TJk+e5+7hGB7p7gxfgYuCxOtuXA787xNivUbPnntzY82ZlZXlz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\nRK5CYII=\n",
            "text/plain": "<matplotlib.figure.Figure at 0x2793d62c5c0>"
          }
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### 期待値"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "参考：[確率論 (Probability Theory) 第12週 ポアソン分布と指数分布とガンマ分布, http://stat.inf.uec.ac.jp](http://stat.inf.uec.ac.jp/lib/exe/fetch.php?media=prob:prob-c-note-and-quizzes-20130711.pdf)\n\n$$\nE[X] = \\int_0^{\\infty} xf(x)dx \\\\\n= \\int_0^{\\infty}x \\frac{1}{\\Gamma(n)\\mu^n} x^{n-1} e^{-\\frac{x}{\\mu}}dx \\\\\n= \\frac{1}{\\Gamma(n)\\mu^n} \\int_0^{\\infty}x^{n} e^{-\\frac{x}{\\mu}}dx\n$$\n\nここで，$\\frac{x}{\\mu} = u, \\frac{du}{dx} = \\frac{1}{\\mu}$ に変数変換\n\n\\begin{eqnarray*}\n&=& \\frac{1}{\\Gamma(n)\\mu^n} \\int_0^{\\infty} (\\mu u)^{n} e^{-u} \\mu du\\\\\n&=& \\frac{\\mu^{n+1}}{\\Gamma(n)\\mu^n} \\int_0^{\\infty} u^{n} e^{-u} du\\\\\n&=& \\frac{\\mu^{n+1}}{\\Gamma(n)\\mu^n} \\Gamma(n+1)\\\\\n&=& \\frac{\\mu^{n+1}}{\\mu^n} \\frac{\\Gamma(n+1)}{\\Gamma(n)}\\\\\n&=& \\mu \\frac{\\Gamma(n+1)}{\\Gamma(n)}\\\\\n&=& \\mu n\\\\\n\\end{eqnarray*}"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### 分散"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "$$\nE[X^2] = \\int_0^{\\infty} x^2 f(x)dx \\\\\n= \\int_0^{\\infty}x^2 \\frac{1}{\\Gamma(n)\\mu^n} x^{n-1} e^{-\\frac{x}{\\mu}}dx \\\\\n= \\frac{1}{\\Gamma(n)\\mu^n} \\int_0^{\\infty}x^{n+1} e^{-\\frac{x}{\\mu}}dx\n$$\n\nここで，$\\frac{x}{\\mu} = u, \\frac{du}{dx} = \\frac{1}{\\mu}$ に変数変換\n\n\\begin{eqnarray*}\n&=& \\frac{1}{\\Gamma(n)\\mu^n} \\int_0^{\\infty} (\\mu u)^{n+1} e^{-u} \\mu du\\\\\n&=& \\frac{\\mu^{n+2}}{\\Gamma(n)\\mu^n} \\int_0^{\\infty} u^{n+1} e^{-u} du\\\\\n&=& \\frac{\\mu^{n+2}}{\\Gamma(n)\\mu^n} \\Gamma(n+2)\\\\\n&=& \\frac{\\mu^{n+2}}{\\mu^n} \\frac{\\Gamma(n+2)}{\\Gamma(n)}\\\\\n&=& \\mu^2 (n+1)n\\\\\n\\end{eqnarray*}\n\nよって，\n\n$$\nV[X] = E[X^2] - (E[X])^2\\\\\n= \\mu^2 (n+1)n - (\\mu n)^2\\\\\n= \\mu^2 n\n$$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### ガンマ分布のモーメント母関数"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "$$\nM_{\\Gamma{(\\mu, n)}}(t) = E_{\\Gamma(\\mu, n)}[\\exp{(tX)}]\\\\\n= \\int_0^{\\infty}e^{tx}\\frac{1}{\\Gamma(n)\\mu^n} x^{n-1} e^{-\\frac{x}{\\mu}}dx\\\\\n= \\frac{1}{\\Gamma(n)\\mu^n} \\int_0^{\\infty}e^{tx} x^{n-1} e^{-\\frac{x}{\\mu}}dx\\\\\n= \\frac{1}{\\Gamma(n)\\mu^n} \\int_0^{\\infty}\nx^{n-1} e^{-(\\frac{1}{\\mu}-t)x}dx\n$$\n\n$u = (\\frac{1}{\\mu}-t)x \\to x = \\frac{1}{(\\frac{1}{\\mu}-t)}u$と変数変換する．\n$du = (\\frac{1}{\\mu}-t)dx$\n\n$$\n= \\frac{1}{\\Gamma(n)\\mu^n} \\int_0^{\\infty}\n(\\frac{u}{\\frac{1}{\\mu}-t})^{n-1} e^{-u} \\frac{1}{\\frac{1}{\\mu}-t}du\\\\\n= \\frac{1}{\\Gamma(n)\\mu^n} (\\frac{1}{\\frac{1}{\\mu}-t})^{n} \\int_0^{\\infty} u^{n-1} e^{-u}du\\\\\n= \\frac{1}{\\Gamma(n)\\mu^n} (\\frac{1}{\\frac{1}{\\mu}-t})^{n} \\Gamma(n)\\\\\n= \\frac{1}{\\mu^n} (\\frac{1}{\\frac{1}{\\mu}-t})^{n} \\\\\n= \\frac{1}{(1-\\mu t)^{n}} \\\\\n$$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "## ガンマ分布と指数分布の比較"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "- 指数分布：期間$\\mu$ごとに1回起こるまでの時間の分布\n- $X_1, X_2, \\cdots , X_n$ が互いに独立に平均$\\mu$ の指数分布に従うとき，$Y = X_1 + X_2 + \\cdots + X_n$はパラメータ$\\mu, n$のガンマ分布に従う．\n\n指数分布のモーメント母関数\n\n$$\nf(x) = \\frac{1}{\\mu} e^{-\\frac{x}{\\mu}}\\\\\nM_x(t) = E[e^{tx}] = \\int_0^{\\infty} e^{tx} \\frac{1}{\\mu} e^{-\\frac{x}{\\mu}} dx\\\\\n= \\frac{1}{\\mu} \\int_0^{\\infty} e^{(t-\\frac{1}{\\mu})x} dx\\\\\n$$\n\n- $t \\geq \\frac{1}{\\mu}$は発散\n- $t < \\frac{1}{\\mu}$\n\n\\begin{eqnarray*}\nM_x(t) \n&=& \\frac{1}{\\mu} \\int_0^{\\infty} e^{-(\\frac{1}{\\mu}-t)x} dx\\\\\n&=&\\frac{1}{\\mu}[-\\frac{1}{\\frac{1}{\\mu}-t}e^{-(\\frac{1}{\\mu}-t)x}]_0^{\\infty}\\\\\n&=&\\frac{1}{\\mu}[0 + \\frac{1}{\\frac{1}{\\mu}-t}]\\\\\n&=&\\frac{1}{\\mu}\\frac{1}{\\frac{1}{\\mu}-t}\\\\\n&=&\\frac{1}{1-\\mu t}\\\\\n\\end{eqnarray*}\n\n$X_1+X_2 + \\cdots + X_n = Y_n, (X_i \\sim Exp(\\mu))$ のモーメント母関数は，\n\n$$\nM_{Y_n}(t) = E[e^{tY_n}] \n= E[e^{t \\sum{X_i}}]\n= E[\\Pi_i e^{t X_i}]\n= \\Pi_i E[e^{t X_i}]\n= \\frac{1}{(1-\\mu t)^n}\n$$"
    },
    {
      "metadata": {
        "trusted": true,
        "ExecuteTime": {
          "start_time": "2017-12-11T18:14:45.271577Z",
          "end_time": "2017-12-11T18:14:46.100194Z"
        }
      },
      "cell_type": "code",
      "source": "n = 2\nmu = 3\nx = stats.expon(scale=mu).rvs(size=(10000, n))\nY = x.sum(axis=1)\nplt.hist(Y, 100, normed=True,\n         label=\"$X_1+X_2, X_i \\sim Exp({})$\".format(mu))\n\nx = sp.linspace(0, 40, 100)\nplt.plot(x, stats.gamma(a=n, scale=mu).pdf(x),\n        label=\"$Ga({}, {})$\".format(mu, n))\n\nplt.legend()\nplt.grid()\nplt.show()",
      "execution_count": 62,
      "outputs": [
        {
          "metadata": {},
          "output_type": "display_data",
          "data": {
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pp57yuV7lSGILFy48axSx6v3uhGpkMC36AEcyIaEtB8v8eAq3PpfeAn//PWxbrEVfRY2O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\nKopo0VdKqSjy/wHVwUommiXDQQAAAABJRU5ErkJggg==\n",
            "text/plain": "<matplotlib.figure.Figure at 0x2793be07128>"
          }
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "## 他分布との関係"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "|分類|1回起こる（形状パラメータ$k=1$）|n回起こる（形状パラメータ$k=n>1$）|\n|:--:|:------------------------------:|:--------------------------------:|\n|離散|幾何分布                        |負の二項分布|\n|連続|指数分布                        |ガンマ分布|\n"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### $\\chi^2$分布"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "- 尺度母数$\\mu=2, 形状母数n=\\frac{n}{2}$ のとき，自由度$n$の$\\chi^2$分布に従う．\n\n確率密度関数を求めてみると，\n\n$$\nf(x;\\mu=2, n=\\frac{n}{2}) \n= \\frac{x^{\\frac{n}{2}-1}e^{-\\frac{x}{2}}}{\\Gamma(\\frac{n}{2}) 2^{\\frac{n}{2}}}\n$$\n\nこれは$\\chi^2$分布の確率密度関数と一致する．"
    },
    {
      "metadata": {
        "ExecuteTime": {
          "start_time": "2017-12-12T12:33:34.436279Z",
          "end_time": "2017-12-12T12:33:35.290346Z"
        },
        "trusted": true,
        "scrolled": true
      },
      "cell_type": "code",
      "source": "# 自由度n=3のchi^2分布\nn = 3\nx = sp.linspace(0, 20, 100)\nchi2 = stats.chi2(df=n)\nplt.plot(x, chi2.pdf(x), label=\"$\\chi^2(n={})$\".format(n))\n\n# Ga(mu=2, n=3/2)\nmu = 2\nrvs = stats.gamma(a=n / 2., scale=mu).rvs(size=10000)\nplt.hist(rvs, 100, normed=True, label=\"$Ga(\\mu={}, n={})$\".format(mu, n/2.))\n\nplt.legend()\nplt.grid()\nplt.show()",
      "execution_count": 20,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "- ベイズ統計では\n\nガンマ分布は，正規分布の分散パラメータ$\\sigma^2$の逆数である精度の共役事前分布になる．\n\nよって，\n\n$$\n\\text{事後分布：ガンマ分布}=\\text{尤度関数：正規分布}\\cdot \\text{事前分布：ガンマ分布}\n$$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "## 参考資料"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "- [ガンマ分布の意味と期待値、分散](https://mathtrain.jp/gammadist)\n- [確率論 (Probability Theory) 第12週 ポアソン分布と指数分布とガンマ分布\n, stat.inf.uec.ac.jp](http://stat.inf.uec.ac.jp/lib/exe/fetch.php?media=prob:prob-c-note-and-quizzes-20130711.pdf)"
    }
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