INTRODUCTION TO PROBABILITY

AND STOCHASTIC PROCESSES

MATH. 3610

SECOND MIDTERM EXAMINATION

WINTER

1997

NAME:........................................................................

ID #:............................................................................

THE EXAM WAS ...........EASY ..........FAIR ..........DIFFICULT

FOR INSTRUCTORS USE 1. a........./6 b........./6 c......./6 ........../18 2. a........./8 b........./8 c......./8 d........./8 e........./8 ........../40 3. a........./9 b........./8 c......./8 ........../25 4. a........./8 b........./9 ........../17 TOTAL ........../100

1. a. If a student selects answers randomly and independently on a true-false examination, determine the probability that the first correct answer is that of question 3. b. A pediatrician wishes to recruit 5 couples, each of whom is expecting their first child, to participate in a new natural childbirth regimen. Suppose that the probability of selecting a couple who agrees to participate is 0.2. What is the probability that 8 couples must be asked before 5 are found who agree to participate? c. A doctor has five patients with migraine headaches. She prescribes for all five a drug that relieves the headaches of 82% of such patients. What is the probability that the medicine will not relieve the headaches of two of these patients?

2. Let X be a continuous random variable with the following probability density function a. What is the cumulative distribution function of X? b. Find P(X>0.1) c. Find E[X] d. Find Var[X] e. If X and Y are independent and identically distributed random variables with the common probability density function given above, find E[X+2Y] and Var[X+2Y].

3. A restaurant serves three fixed-price dinners costing $7, $9, and $10. For a randomly selected couple dining at this restaurant, let X=the cost of the man's dinner and Y=the cost of the woman's dinner. The joint probability mass function of X and Y is given in the following table:

a. Compute the marginal probability mass function of X and probability mass function of Y. b. Calculate the probabilities associated with the total cost of the dinner for the two people. That is find the probability mass function of X+Y. c. Are X and Y independent? Please justify your answer.

4. (Use the moment generating function to answer the following questions) a. Suppose that X has an exponential distribution with l=6. What is the distribution of Y=3X? b. Suppose that X and Y are independent normal random variables with parameters , respectively. What is the distribution of X+Y?