Stat 401: Applied Statistical Methods II Fall 2016 Homework #3 Due Date: October 7 (in your lab) 1. An exam has two questions. Let (X, Y) be the outcomes for a randomly selected student. X = 1 if the student answers question 1 correctly; and X = 0 if the student does not. Similarly, Y denotes the corresponding outcome for question 2. The joint pmf of (X, Y) is given below. From the marginal distribution, you can get the following: E(X) = 0.5 and Var(X) = 0.25. y x 0 1 Marginal distribution of Y 0 1 0.4 0.1 0.1 0.4 Marginal distribution of X 0.5 0.5 0.5 0.5 1.0 a) What is correlation between X and Y? b) The exam scores are determined as follows: the student gets 50 points for answering a question correctly and zero otherwise. There are no partial credits. Let S be the total score that the student will get in the exam. What are the mean and variance of S? (Do not compute this from the distribution of S. Use your answers in parts (a) and (b) and the other information you are given to answer this question.) c) Write down the pmf of S. 2. Joe Smith plays a betting game with his friends. He puts in one dollar to join the game. If he wins the game, he gets $2. So his gain is $1. If he loses, he gets nothing, so his gain is -$1. Let V be the random variable that denotes the gain (amount of money in dollars) made by Joe Smith. (Remember that a loss is counted as negative gain). The pmf of V is given below. v p(v) a) b) c) d) -1 0.6 1 0.4 Consider the random variable V2 given by V times V. Write down the joint pmf of (V, V2). Are V and V2 independent? Why or why not? Consider the random variable V3 given by V times V times V. What is the joint pmf of (V, V3)? What is the correlation between V and V3? [Hint: You can answer this question without computing the covariance of V and V3 but you must explain clearly.] 1 3. Consider again the game in Question 2. Joe Smith is planning to play the game 100 times. The 100 outcomes will be independent of each other. a) What is his expected total gain if he plays 100 times? b) What is the variance of his total gain? c) Use the Central Limit Theorem to compute the probability that his total gain will be positive. 4. The distribution of piston diameters is normally distributed with mean inches and standard deviation 5 inches. Engineer Dan takes a sample of 25 pistons from the manufacturing assembly line. Let X1, X2, ..., X25 be denote these observations which are iid. a. What is the probability that Xbar, the sample mean of Dan's 25 observations, is within 1 inch of the population mean . b. Suppose engineer Jane decides to take a sample of 30 observations. Let Y1, Y2, ..., Y30 denote these observations which are also iid. Jane computes the probability that her sample mean Ybar is within 1 inch of the population mean . Without doing any calculations, do you think this probability will be smaller or bigger than that for Dan's sample mean? Explain your answer clearly. 5. Consider two random variables X and Y with variances 4 and 9 respectively. a. What is the variance of (X - Y) when cor (X, Y) = 0? b. What is the variance of (X - Y) when cor(X, Y) = 0.5? c. What is the variance of (X - Y) when cor(X, Y) = - 0.5? 6. An appliance dealer sells three different models of freezers with advertised capacity of 13, 16 and 20 cubic feet. Let X = the amount of storage space of the freezer purchased by the next customer. Suppose X has the pmf given below: x p(x) 13 0.2 16 0.5 20 0.3 a. The price of a freezer having cubic capacity X is given by 25X - 20. What are the expected value and variance of the price paid by the next customer? b. Suppose that although the advertised capacity of a freezer is X, the actual capacity is h(X) = X - 0.01 X2. What are the expected value and variance of h(X), the actual capacity of the freezer bought by the next customer? 2