# Homework 10¶

• Type your name and email in the “Student details” section below.

• Develop the code and generate the figures you need to solve the problems using this notebook.

• For the answers that require a mathematical proof or derivation you can either:

• Type the answer using the built-in latex capabilities. In this case, simply export the notebook as a pdf and upload it on gradescope; or

• You can print the notebook (after you are done with all the code), write your answers by hand, scan, turn your response to a single pdf, and upload on gradescope.

• The total homework points are 100. Please note that the problems are not weighed equally.

Note

• This is due before the beginning of the next lecture.

• Please match all the pages corresponding to each of the questions when you submit on gradescope.

## Student details¶

• First Name:

• Last Name:

• Email:

import matplotlib.pyplot as plt
%matplotlib inline
import seaborn as sns
sns.set(rc={"figure.dpi":100, 'savefig.dpi':300})
sns.set_context('notebook')
sns.set_style("ticks")
from IPython.display import set_matplotlib_formats
set_matplotlib_formats('retina', 'svg')
import numpy as np
import scipy.stats as st


## Problem 1 - An alternative way to construct the generalized uniform distribution $$U([a,b])$$¶

Let $$Z$$ be a standard uniform random variable:

$Z\sim U([0,1]).$

Define the random variable:

$X = a + (b-a) Z.$
• Show that:

$X\sim U([a,b]).$

Hint: Prove that the CDF of $$X$$ is $$F_X(x) = p(X\le x) = \frac{x-a}{b-a}$$. This is one line.

• The function numpy.random.rand gives you uniform random samples in $$[0, 1]$$. Take 1,000 such samples and transform them to uniform samples in $$[-1, 5]$$. Hint: Fill in the missing code below.

a = -1
b = 5
z = np.random.rand(1000)
x = # Your code here


Test if you are getting the right answer by doing the histogram of your samples (it should be almost flat between -1 and 5):

fig, ax = plt.subplots()
ax.hist(x, density=True, alpha=0.5)
ax.set_xlabel('$x$')
ax.set_ylabel('$p(x)$');


# Problem 2 - The Exponential distribution¶

The Exponential distribution models the probability distribution of the time between events which occur continuously and independently at a constant rate. Examples of such a situation are:

• The time between phone calls in a help center.

• The time between the arrival of cars at a toll station.

• The time between the arrival of customers.

• The time between two earthquakes.

• The time between two micrometeoroid impacts on an Moon research base.

• The time between faults in a mechanical system. However, this is a gross approximation because the rate of faults in a mechanical system increases with time, it is not constant.

We write:

$T \sim \text{Exp}(\lambda),$

$$T$$ follows an Exponential distribution with rate parameter $$\lambda$$.

The rate parameter $$\lambda$$ is positive and has units of inverse time. You can think of $$\lambda$$ as the number of events per unit of time.

The CDF of the Exponential is:

$\begin{split} F(t) = \begin{cases} 0,& t < 0,\\ 1 - e^{-\lambda t},& t \ge 0. \end{cases} \end{split}$
• Prove mathematically that the PDF of the random variable $$T$$ is:

$p(t) = \lambda e^{-\lambda t}.$

Hint: Use one of the properties of the PDF.

• Micrometeoroids make space exploration very challenging. Even though the mass of these projectiles is very small (less than 1 gram), they move with a very high-velocity (of the order of 10 km per second) and thus they are will be gradually degrading the protective layers of deep space habitats. For a Moon base with an area of 1000 squared meters, the rate of micrometeoroid impacts is about:

$\lambda = 3\times 10^{-6}\;\text{s}^{-1}.$

Read the scipy.stats.expon documentation and make an Exponential random variable $$T$$ with this rate:

# You cannot use lambda because it is a reserved word.
# Call the rate lam instead:
lam = 3e-6 # in inverse seconds
T = st.expon(scale= # PICK THE RIGHT VALUE HERE
)

• Take 1000 samples from the random variable you just constructed and draw their histogram.

# Your code here

• Plot the CDF of $$T$$:

# Your code here

• Find the probability that we have a micrometeoroid impact within a day. Hint: Remember that the units of $$T$$ are seconds.

# Your code here

• Plot the PDF of T.

# Your code here