task_2

{"metadata": {"kernelspec": {"display_name": "Python 3", "name": "python3", "language": "python"}, "language_info": {"file_extension": ".py", "nbconvert_exporter": "python", "codemirror_mode": {"name": "ipython", "version": 3}, "name": "python", "pygments_lexer": "ipython3", "mimetype": "text/x-python", "version": "3.4.3"}}, "cells": [{"metadata": {"trusted": true, "collapsed": false}, "cell_type": "code", "source": "%pylab inline\nimport numpy as np\nfrom functools import reduce\nimport scipy.stats as st\nSAMPLE_SIZE = 100\nBS_SIZE = 5000\nMU = 5", "outputs": [{"text": "Populating the interactive namespace from numpy and matplotlib\n", "name": "stdout", "output_type": "stream"}], "execution_count": 151}, {"metadata": {}, "cell_type": "markdown", "source": "#\u21163\n\u041f\u0443\u043d\u043a\u0442 \u0430): \u0421\u0433\u0435\u043d\u0435\u0440\u0438\u043c \u0432\u044b\u0431\u043e\u0440\u043a\u0443 \u0438 \u043f\u043e\u0441\u0442\u0440\u043e\u0438\u043c \u0434\u043e\u0432\u0435\u0440\u0438\u0442\u0435\u043b\u044c\u043d\u044b\u0435 \u0438\u043d\u0442\u0435\u0440\u0432\u0430\u043b\u044b \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u0431\u0443\u0442\u0441\u0442\u0440\u0435\u043f\u043d\u044b\u0445 \u043c\u0435\u0442\u043e\u0434\u043e\u0432 \u0438 \u0434\u0435\u043b\u044c\u0442\u0430-\u043c\u0435\u0442\u043e\u0434\u0430(se \u0434\u043b\u044f \u043d\u0435\u0433\u043e \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u0435\u0442\u0441\u044f, \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u0437\u043d\u0430\u043d\u0438\u044f \u043e \u0442\u043e\u043c, \u0447\u0442\u043e se \u043e\u043c\u043f \u043e\u0446\u0435\u043d\u043a\u0438 \u0441\u0440\u0435\u0434\u043d\u0435\u0433\u043e \u0434\u043b\u044f \u043d\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0440\u0430\u0432\u043d\u043e $(sigma**2)/n$):"}, {"metadata": {"trusted": true, "collapsed": true}, "cell_type": "code", "source": "samples = np.random.normal(MU, 1, size=SAMPLE_SIZE)", "outputs": [], "execution_count": 174}, {"metadata": {"trusted": true, "collapsed": false}, "cell_type": "code", "source": "def estimate_theta(samples):\n    return np.exp(sum(samples) / len(samples))\n\ndef estimate_mu(samples):\n    return sum(samples) / len(samples)\n\ndef estimate_sigma(samples):\n    mu_est = estimate_mu(samples)\n    return np.sqrt(reduce(lambda x, y: x + (y-mu_est)**2, samples) / len(samples))\n\ndef estimate_sigma_bsparam(samples):\n    mu_est = estimate_mu(samples)\n    bs = [estimate_theta(np.random.normal(mu_est, 1, size=SAMPLE_SIZE)) for _ in range(BS_SIZE)]\n    return estimate_sigma(bs)\n\ndef estimate_sigma_bsnonparam(samples):\n    bs = [estimate_theta(np.random.choice(samples, size=SAMPLE_SIZE)) for _ in range(BS_SIZE)]\n    return estimate_sigma(bs)\n\ntheta = estimate_theta(samples)\nsigma_param, sigma_nonparam = np.sqrt(estimate_sigma_bsparam(samples)), np.sqrt(estimate_sigma_bsnonparam(samples))\nsigma_delta = np.sqrt((theta**2) / SAMPLE_SIZE)\nz = st.norm.ppf(0.05/2)\n\nprint((theta + sigma_param*z, theta - sigma_param*z))\nprint((theta + sigma_nonparam*z, theta - sigma_nonparam*z))\nprint((theta + sigma_delta*z, theta - sigma_delta*z))\nprint(np.exp(5))", "outputs": [{"text": "(150.65524019997767, 166.16922493984859)\n(151.32246771974445, 165.50199742008181)\n(127.36400551515186, 189.4604596246744)\n148.413159103\n", "name": "stdout", "output_type": "stream"}], "execution_count": 175}, {"metadata": {}, "cell_type": "markdown", "source": "*\u0414\u0430, \u043d\u0430 \u0441\u043e\u0442\u043d\u0435 \u0441\u044d\u043c\u043f\u043b\u043e\u0432 \u0447\u0442\u043e \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0439, \u0447\u0442\u043e \u043d\u0435\u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0431\u0443\u0442\u0441\u0442\u0440\u0435\u043f\u044b \u0440\u0430\u0431\u043e\u0442\u0430\u044e\u0442 \u0438\u0437 \u0440\u0443\u043a \u0432\u043e\u043d \u043f\u043b\u043e\u0445\u043e. \u0421 \u0434\u0435\u043b\u044c\u0442\u0430-\u043c\u0435\u0442\u043e\u0434\u043e\u043c \u0432\u0441\u0451 \u0442\u043e\u0436\u0435 **\u043d\u0435\u0433\u043b\u0430\u0434\u043a\u043e**.*"}, {"metadata": {}, "cell_type": "markdown", "source": "\u041f\u0443\u043d\u043a\u0442 \u0431): \u043e\u0446\u0435\u043d\u0438\u043c \u043f\u043b\u043e\u0442\u043d\u043e\u0441\u0442\u0438 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u043e\u0446\u0435\u043d\u043a\u0438 \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u0433\u0438\u0441\u0442\u043e\u0433\u0440\u0430\u043c\u043c"}, {"metadata": {"trusted": true, "collapsed": false}, "cell_type": "code", "source": "samples = np.random.normal(MU, 1, size=SAMPLE_SIZE)\n\nmu_est = estimate_mu(samples)\ntheta_est = estimate_theta(samples)\nbsparam_samples = [estimate_theta(np.random.normal(mu_est, 1, size=SAMPLE_SIZE)) for _ in range(BS_SIZE)]\nbsnonparam_samples = [estimate_theta(np.random.choice(samples, size=SAMPLE_SIZE)) for _ in range(BS_SIZE)]\nmtcarlo_samples = [estimate_theta(np.random.normal(MU, 1, size=SAMPLE_SIZE)) for _ in range(BS_SIZE)]\ndelta_samples = np.random.normal(theta_est, np.sqrt((theta**2)/SAMPLE_SIZE), size=BS_SIZE)\n\n\n\nfigure(figsize=(15,6))\n\nplt.subplot(131)\n_, bins, patches_mntcarlo = hist(mtcarlo_samples, alpha=0.3)\n_, _, patches_param = hist(bsparam_samples, bins=bins, alpha=0.3)\nlegend([patches_param[0], patches_mntcarlo[0]], ['bs-param', 'monte-carlo'])\n\nplt.subplot(132)\n_, bins, patches_mntcarlo = hist(mtcarlo_samples, alpha=0.3)\n_, _, patches_nonparam = hist(bsnonparam_samples, bins=bins, alpha=0.3)\nlegend([patches_nonparam[0], patches_mntcarlo[0]], ['bs-nonparam', 'monte-carlo'])\n\nplt.subplot(133)\n_, bins, patches_mntcarlo = hist(mtcarlo_samples, alpha=0.3)\n_, _, patches_delta = hist(delta_samples, bins=bins, alpha=0.3)\nlegend([patches_delta[0], patches_mntcarlo[0]], ['delta', 'monte-carlo'])", "outputs": [{"metadata": {}, "output_type": "execute_result", "data": {"text/plain": "<matplotlib.legend.Legend at 0x7f4e0e429710>"}, "execution_count": 181}, {"metadata": {}, "output_type": "display_data", "data": {"text/plain": "<matplotlib.figure.Figure at 0x7f4e0e5485c0>", "image/png": 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oSjJoPmaT9Ajm1hhza2/f3L2H3teWe0hyLPCL9HWmSU4Gzqa37+p64MK+LvcvgFdV1YnA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\nSZKkDrC5kyRJkqQ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