Exponential and uniform distributions in Python

Uniform and Exponential Distribution.py

```python
#Import libraries
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.mlab as mlab
from __future__ import division
%matplotlib inline
```

Question 1

Part A


```python
#CDF of the uniform distribution
def CDFUniform( a,b,x ):
    if x>=a and x<=b:
        cdf=(x-a)/(b-a)
    elif x>=b:
        cdf=1
    else:
        cdf=0
    return cdf
```


```python
#CDF for x=3/4, a=0 and b=1
print CDFUniform(0,1,3/4)
```

Part B


```python
#Mean of the uniform distribution
def MeanUniform( a,b ):
    mean=(b+a)/2
    return mean
#Variance of the uniform distribution
def VarianceUniform( a,b ):
    Var=((b-a)**2)/12
    return Var
```


```python
print MeanUniform( 0,1 )
print VarianceUniform( 0,1 )
```

Question 2

Part A


```python
#Generating a random sample of size 1000 from a standard uniform distribution
U=np.random.uniform(0,1,1000)
print(np.mean(U))#Display the sample mean
print(np.var(U))#Display the sample variance
```

Part B


```python
#Plot uniform histogram
plt.hist(U,facecolor='blue')
plt.title("Uniform Histogram")
plt.xlabel("Value")
plt.ylabel("Frequency")
plt.show()
```

Part C


```python
#Plot of the PDF
# the histogram of the data
n, bins, patches = plt.hist(U, 50, normed=1,
    facecolor='green', alpha=0.75)
#Add best fit line
from scipy.stats import uniform
rv = uniform()
l = plt.plot(bins, rv.pdf(bins), 'r--', linewidth=1)

plt.xlabel('Value')
plt.ylabel('Probability')
plt.title("Uniform Histogram and its PDF")
plt.axis([0, 1, 0, 1.5])
plt.grid(True)

plt.show()
```

Question 3

Part A

#Function for the CDF of the exponential distribution
def CDFExponential(lamb,x): #lamb = lambda
    if x<=0:
        cdf=0
    else:
        cdf=1-np.exp(-lamb*x)
    return cdf
#Function to compute the mean of the exponential distribution
def MeanExponential(lamb):
    return 1/lamb;
def VarianceExponential(lamb):
    return (1/lamb)**2;

Part B


```python
#CDF of the exponential distribution
#when lambda=1/5 and x=3
print CDFExponential(1/5,3)
print MeanExponential(1/5)
print VarianceExponential(1/5)
```

Question 4

Part A


```python
#Generate 1000 random random samples
#from a standard uniform distribution
U=np.random.uniform(0,1,1000)
lamb=1/5
X=-np.log(1-U)/lamb
```

Part B


```python
print(np.mean(X))#Display the sample mean
print(np.var(X))#Display the sample variance
```

Question 5

Part A


```python
#Exponential CDF plot
plt.hist(X)
plt.title("Exponential Histogram")
plt.xlabel("Value")
plt.ylabel("Frequency")
plt.show()
```

Part B


```python
#Exponential pdf histogram
n, bins, patches = plt.hist(X, 50, normed=1,
                            facecolor='green', alpha=0.75)
#Add best fit line
from scipy.stats import expon
rv = expon()
lamb = plt.plot(bins, lamb*rv.pdf(bins*lamb), 'r--', linewidth=1)

plt.xlabel('Value')
plt.ylabel('Probability')
plt.title("Uniform Histogram and its PDF")
plt.axis([0, 50, 0, 0.3])
plt.grid(True)
plt.show()
```