NOTE: This homework is due on Tuesday, March 3. Please give the problems a try tomorrow and come on Thursday ready to ask questions.


  1. In a month with 24 working days, I decide to come to work on the first 12 working days by bike and the next 12 working days by walking. If I bike to work, the probability of me running into my friend JoVi is 1/3, independent across days. If I walk to work, the probability of me running into my friend JoVi is 1/2 again independent across days.

    1. What is the chance that I run into JoVi exactly once while biking?
    2. What is the chance that I run into JoVi exactly twice while walking?
    3. What is the chance that the total number of times I run into JoVi in the month while going to work is less than or equal to 2 times?�) 2 times?



2. Suppose that \(X\) is a random variable with cumulative distribution function

\[F(x) = \left\{ \begin{array}{ll} 0 & \text{if}\ x<-1 \\ \frac{1}{12} & \text{if}\ -1\leq x<0\\ \frac{5}{12} & \text{if}\ 0\leq x<1\\ \frac{11}{12} & \text{if} \ 1\leq x<3.5 \\ 1 & \text{if} \ x\geq 3.5 \end{array}\right.\] What is the probability mass function (pmf) of \(X\)? In other words, make a table illustrating the values the random variable \(X\) takes and the corresponding probabilities.


3. Caroline is applying for jobs this summer. Let \(X\) be the number of interviews she gets. Caroline thinks that this random variable has a pmf given by the following table:

  1. What is the chance that Caroline gets at least 2 interviews?
  2. Find \(E(X)\). The interpretation of this value is as follows: If we had a huge number of candidates identical to Brad (so with the same distribution of interviews as above), then some of these candidates would get 0 interviews this summer, some would get 1, some would get 2 and some would get 3 and if we calculated the average number of interviews over these candidates, this would be close to the number you found.
  3. Find \(E(X^2)\).
  4. Find \(Var(X)\).
  5. Caroline estimates that for each interview that she gets it takes 10 hours to prepare. She also estimates that she has taken about 20 hours applying for the interviews, uploading her resume to various job portals and so on. If Y denotes the total amount of time Caroline spends on her job application process, find \(E(Y)\). (Hint: Here \(Y = 10X + 20\)).
  6. Find \(Var(Y)\).


4. Suppose you roll two fair 6-side dice. Define the random variable \(X\) to be the number of dice that show an even number. Thus for example if the outcome was (1,4) for the two dice, then \(X=1\) Find the probability mass function of the random variable \(X\).


5. An urn contains 5 balls numbered 1 to 5. We draw 3 at random without replacement.

  1. Let \(X\) be the largest number showing on the 3 balls drawn. Find the pmf of \(X\).
  2. Let \(Y\) be the smallest number showing on the 3 balls drawn. Find the pmf of \(Y\).


6. Suppose you select two numbers at random with replacement from the set {1, 2, 3} and then you form a rectangle with the first number as the length of the rectangle and the second number as the height of the rectangle. Let X be the area of the rectangle so formed. Find the probability mass function of X.


7. Shankar either walks or bikes the 3 miles from his home to office. If the weather is nice (which happens with probability 0.7) Shankar walks to work. Shankar’s speed for walking is 4 miles per hour. If the weather is bad (which happens with probability 0.3), Shankar bikes to work. His speed for biking is 15 miles per hour. Let \(X\) denote the amount of time (measured in minutes) it takes Shankar to get to work. Find \(E(X)\).


8. Caroline really likes driving to the Outer Banks. The only problem is Caroline occasionally speeds. Assume that the chance that Caroline gets a speeding ticket on her way to the outer banks is 1/4 (you may assume that on each trip she gets either zero speeding tickets or one speeding ticket; the implication being if he gets one speeding ticket she slows down for the rest of the trip). You may also assume that outcomes over different trips are independent. Caroline will go to the Outer Banks 2 times this year. Let \(X\) denote the number of speeding tickets she gets.

  1. Make a table demonstrating the pmf of \(X\)
  2. Find \(E(X)\).


9. Let \(X\) be a random variable having expected value \(\mu\) and variance \(\sigma^2\). Find the expected value and variance of \[Y = \frac{X-\mu}{\sigma}.\]


10. Let \(X\) be a binoial random variable \(B(n,p)\). What value of \(p\) maximizes \(P[X=k]\) for \(0\leq k \leq n\)? (Prove your answer.)

This is an example of a statistical method used to estimate \(p\) when a binomial \(B(n,p)\) random variable is observed to equal \(k\). If we assume that \(n\) is known, then we estimate \(p\) by choosing the value of \(p\) that maximizes \(P[X=k]\). This is known as the method of maximum likelihood estimation.


11. Let \(X\) be the number of successes that result from \(2n\) independent drials, when each trial is a success with probability \(p\). Show that \(P[X=n]\) is a decreasing function of \(n\).