Problem #$1$

Write a Python function named CentralLimitDemoExp that takes as input arguments:

n $-$ the number of samples to be taken from an exponential distribution,

sample$\_$sizes $-$ a $1$ x $6$ numpy array holding the size of each of the n samples for that value of n different size of each sample for a fixed,

mu $-$ the mean of the exponential random variable.

And returns as output:

fig $-$ a figure displaying six histograms, one for each value of sample$\_$size specified for the given value of n$,$

sample$\_$means $-$ a numpy array of size $1$ x n holding the n sample$\_$means,

sample$\_$means$\_$stdevs $-$ a numpy array of size $1$ x $6$ holding the standard deviation of the list of sample$\_$means.

Test your function by setting sample$\_$sizes $=$ np$.$array$([4,50,100,500,1000,2000])$ and running the command fig$1,$ means$1,$ devs$1=$ CentralLimitDemoExp$(1000,$ sample$\_$sizes, $170).$

import numpy as np

import matplotlib.pyplot as plt

import random

import statistics

from numpy.random import seed

def CentralLimitDemoExp$($n$,$ sample$\_$sizes, mu$)$:

# This line will re$-$initialize the random seed everytime the function is run and produce the same

# samples every time, you can comment it out

seed$(1)$

# Create a figure consisting of $6$ subplots arranged into two rows and three columns

fig, axs $=$ plt$.$subplots$(2,3,$ figsize $=(12,8))$

fig.subplots$\_$adjust$($hspace $=0\mathrm{.}4)$

fig.subplots$\_$adjust$($wspace $=0\mathrm{.}4)$

# Figure indeces

axs$\_$row $=0$

axs$\_$col $=0$

# Create a numpy array to hold the standard deviation of the list of sample means $(1$ x $6)$

sample$\_$means$\_$stdev $=$ # Fill in code here

# Loop to generate n samples of sizes sample$\_$sizes$[0],...,$ sample$\_$sizes$[5]$

# keep track of how many of the listed sample$\_$sizes have been completed

k $=0$

for sample$\_$size in sample$\_$sizes:

# Create an np$.$array to hold all n means

sample$\_$means $=$ # Fill in code here

# Loop to generate the n samples of size sample$\_$size

for j in range$(0,$n$)$:

# Create a sample by drawing sample$\_$size elements from an exponential distribution

sample $=$ # Fill in code here

# Find its mean

sample$\_$mean $=$ # Fill in code here

# And add it to the list of sample$\_$means

sample$\_$means$[$j$]=$ sample$\_$mean

# Find the standard deviation of each of the six lists of n sample means and store them in sample$\_$means$\_$stdev

sample$\_$means$\_$stdev$[$k$]=$ np$.$std$($sample$\_$means$)$

# Increase the count of completed sample$\_$sizes

k $+=1$

# Generate the histogram of the n sample$\_$means of size sample$\_$size

# Histogram

axs$[$axs$\_$row, axs$\_$col$].$hist$($sample$\_$means, color $=$ "green"$)$

# Plot and axis labels

axs$[$axs$\_$row, axs$\_$col$].$set$\_$title$("$sample$\_$size $=\%$s$"\%$ sample$\_$size$)$

axs$[$axs$\_$row, axs$\_$col$].$set$\_$ylabel$("$Frequency$")$

axs$[$axs$\_$row, axs$\_$col$].$set$\_$xlabel$("$Observed Sample Mean"$)$

# Figure title

fig.suptitle$(\text{'}$Sample Mean Distribution for different values of sample$\_$size'$)$

# Advance figure row and column counters

if axs$\_$col $<2$:

axs$\_$col $+=1$

else:

axs$\_$col $=0$

axs$\_$row $+=1$

return fig, sample$\_$means, sample$\_$means$\_$stdev