NOTE: More problems may be added on Thursday (This homework is due Tuesday, April 21, at 11:59 pm).


  1. 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. Find \(E[X]\) and \(E[X^2]\). Write an interpretation of what \(E[X]\) represents.
  2. Let \(M_X(t) = E[e^{tX}]\) denote the moment generation function of \(X\). Give a formula for \(M_X(t)\).
  3. Give a formula for the first derivative of the Moment Generating Function. \[\frac{d}{dt}M_X(t) = M_X'(t).\]
  4. Evaluate \(M'(0)\) and check that it matches \(E[X]\).
  5. Calculate \(M''(t)\) and check that \(M''(0) = E[X^2]\).



2. Shankar really likes playing Badminton. On a particular day at Fetzer gym he plays two games (whose outcomes are independent of each other). He has .3 probability of winning the first game and he has .9 probability of winning the second game. There are no ties and as stated above the outcomes of each game are independent of each other. Let \(X\) denote the number of games Shankar wins.

  1. Calculate \(E[X]\) and \(E[X^2]\).
  2. Find a formula for \(M(t)\), the moment generating function for \(X\).
  3. Find a formula for \(M'(t)\) and check that \(M'(0) = E[X]\).
  4. Find a forumla for \(M''(t)\) and use it to find \(Var[X]\).


3. \(X_1\) has a Poisson distribution with parameter 3, \(X_2\) has a Poisson distributino with parameter 0.5 and \(X_3\) has a Poisson distribution with parameter 2. \(X_1, X_2, X_3\) are independent. Let \[Y= X_1 + X_2 +X_3\] Calculate \(P[Y=6]\).


4. An engineer working in a cellphone company factory sees that the distribution of lifetimes of batteries (time from full charge to discharge during regular use) has exponential distribution with parameter \(\lambda = 1/8\) (in hours). He takes a sample of 30 cellphones to test their battery lifetime. Let \(X_i, (1 \leq i \leq 30)\) be the lifetime of the battery from cellphone \(i\) and assume these are independent with exponential distribution with parameter \(\lambda = 1/8\). Suppose the engineer is interested in the distribution of the average battery lifetime in his sample i.e. \[\overline{Y} = \frac{\sum_{i=1}^{30} X_i}{30}.\]

  1. Calculate the MGF of \(Y = X_1 + X_2 + ... + X_{30}\). Use it to identify the distribution of \(Y\).
  2. (Challenge) Use the techniques of Section 5.7 to find the distribution of \(\overline{Y}\) from the distribution of \(Y\). You will need the PDF of \(Y\), which can be found on the MGF tables. (This question is meant to be hard. You should be pleased if you can do it, but not stress overly if the solution escapes you.)
    Comment: Taken from an old final.



5. Luke takes a loan of 400 dollars to buy 20 units of stock of Company A and 30 units of stock of Company B. After 1 year he will sell these stocks. Suppose after a year his selling price (dictated by the market at that time) per unit for Stock A, \(X_A\) has \(N(10,σ^2 = 1)\) (i.e. normal) distribution and the sale price \(X_B\) per unit of Stock B has \(N(11,σ^2 = 1.2)\) distribution. Further \(X_A\) and \(X_B\) are independent. Thus the total amount he gets back is \[Y = 20X_A + 30X_B.\] He needs to get back 480 to repay the bank. What is \(P (Y > 480)\)?
Hint: Calculate the MGF of \(Y\) and use this to understand the distribution of \(Y\).


  1. \(X_1, X_2, X_3\) are three independent normally distributed random variables with \[X_1: N(2, \sigma = 3), \ \ \ \ \ X_2: N(5, \sigma = 7), \ \ \ \ \ X_3 ~ N(4, \sigma = 8).\] Suppose \[Y = 2X_1 -3X_2 - 5 X_3.\]
  1. Use moment generating functions to find the distribution of \(Y\).
  2. Use the distribution identified in (a) to calculate \(P[Y>3]\).


  1. (Taken from an old final.) Let \(X, Y, Z\) be independent Normal random variables with expectation \(-2, 3, 1\) and variances \(4, 9, 36\), respectively. Let \[f(x,y,z) = 2 + yz - 3x^2y + 6x^2 y^2 z^2.\]
  1. Find \(E[f(X,Y,Z)]\).
  2. Find \(P(2X + Y \geq Z-Y)\).


8. Let \(X_1\) be binomially distributed with parameters \(n_1, p\) and let \(X_2\) be \(B(n_2, p)\). Let \(X_1\) and \(X_2\) be independent and let \(Y=X_1 + X_2\). It turns out \(Y\) has a familiar distribution. Use MGFs to find the distribution of \(Y\). Please include a proof along with a description of the distribution of \(Y\).

  1. Concerning the final: Please think for 5 minutes and give me three sentences describing why the UNC Honor Code is important to you personally and a statement that you will abide by it during the final exam. Please write your full name and PID after this statement.