Problem Set 2

Problem Set Due Wednesday October 9th at 7pm on Canvas.

You will hand in a .Rmd file and a knitted html output.

I have provided the raw RMD of this problem set you can use as a template for answering.

Question 1

Solve the following questions about the specified Binomial (\(B\)), Uniform (\(U\)) and Normal (\(N\)) random variables:

\(Y \sim B(n=500, \pi=.45)\)

  1. \(E[Y]\)

  2. \(V[Y]\)

  3. \(P(Y=150)=\)?

  4. \(P(Y \leq 200)=\)?

  5. \(P(200<Y<223)=\)?

  6. \(P(200\leq Y \leq 223)=\)?

\(Q \sim U(min=10, max=200)\)

  1. \(E[Q]\)

  2. \(V[Q]\)

  3. \(P(Q=12)=\)?

  4. \(P(12<Q<20)=\)?

  5. \(P(12\leq Q \leq20)=\)?

\(X \sim N(\mu=26, \sigma^2=49)\)

  1. \(p(X>30) =\) ?

  2. \(p(24 < X < 30) =\) ?

  3. Solve for \(i\) where \(p(X<i)=.75\)

  4. Solve for \(i,j\) where \(p(i<X<j)=.85\)

  5. Solve for \(c\) such that \(p(X < 45) = p(Z < c)\), where \(Z \sim N(0, 1)\)

Question 2

Women’s height is normally distributed with mean of 64 and a standard deviation of 3. Using this linked table, and without using R (you can use it as a calculator but can’t use the norm functions): what is the approximate probability a woman has a height between 63 and 66 inches? In a few sentence describe how you got your answer.

Question 3

Consider the following probability mass function of a random variable, \(K\):

\[ f(K) = p(K=k) = \begin{cases} \frac{1}{6} \text{ if } 1\\ \frac{1}{3} \text{ if } 2\\ \frac{1}{3} \text{ if } 4\\ \frac{1}{6} \text{ if } 10\\ \end{cases} \]

  1. What is the CDF of \(k\)?

  2. What is the expected value of K?

  3. What is the variance of K?

  4. Use R to draw 10,000 samples of K. Confirm that the expected value and variance you calculated above is roughly correct.

  5. Plot the PMF and CDF of K, comparing the simulated and calculated values. For the CDF try out the cumsum() function.