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Final Examination

## STAT 3083

10 December, 1997
Time: 3 Hours. Total Marks: 67 (values are indicated at the beginning of each question). One 8 by 11 cheat sheet is allowed. Answer all questions on the question paper.   Show all work.
1. (21 points, 3 points each)
The joint distribution of the random variables X and Y is given by
fXY = c (x² + y²)    for x=0,1;   y=1,2,3.
(a) Find c.
(b) Find the marginal distribution of X.
(c) Find the marginal distribution of Y.
(d) Are X, Y independent? Explain.
(e) Find the conditional distribution of Y given X=-1.
(f) Find E(XY).
(g) Find E(X | X <0).

2. (18 points, 2 points each) Let X, Y, and Z be random variables such that E(X) = -3, Var(X)=5, E(Y)=4, Var(Y)=4, E(Z)=1, Var(Z)=7, Cov(X,Y)=3, Cov(X,Z)=-2, and Y and Z are independent. Using the information given find, if possible, the values of the following expressions. (If it is not possible to find the value of a given expression, just write "not possible.")
 (a) Var(X + 3) (b) E(7X + 5Z) (c) E(1 / X²) (d) E(Y2/3) (e) Var(Y+7Z) (f) E(Y Z) (g) E(X Y) (h) E(Y / Z) (i) E(Y(3+2Z))
3. (6 points, 3 points each)
Let X1,X2, X3, . . . X40 be a random sample from a distribution with mean 8 and variance 5. Let Xav be the average of the 40 X's.

(a) Approximate P(Xav <8).

(b) Let Y1,Y2, Y3, . . . Y50 be a random sample from a distribution with mean 0 and variance 1. Let Yav be the average of the 50 Y's. What is the approximate distribution of Xav + Yav ?

4. (12 points, 4 points each)
Do any three of the following five problems. To get partial marks start out by stating the appropriate distribution and the values of the associated parameters.
 (a) What is the probability that a Revenue Canada auditor will catch only two income tax returns with illegitimate deductions if she randomly selects five returns from among 15 returns of which 9 contain illegitimate deductions? (b) In the inspection of a fabric produced in continuous rolls the number of imperfections per ten yards is a random variable having a Poisson distribution with lambda=2.8. Find the probability that ten yards of the fabric will have at most three imperfections. (c) If 40% of the mice used in an experiment will become very aggressive within one minute after having been administered an experimental drug, find the probability that exactly six of the fifteen mice which have been administered the drug will become very aggressive within one minute. (d) The probability that a newborn child will be male is about 52%. If a large hospital has 1290 live births in a given year, what is the probability that 700 or more of the babies born during this year at this hospital will be male? (e) The probability that a salmon caught in the Ottawa River will be of a suitable size to be kept, according to game regulations, is 0.08. What is the probability that an individual will have to catch more than 9 fish to be able to keep one?
5. (5 points) It is reported that 50% of all computer chips produced are defective. Inspection ensures than only 5% of the chips legally marketed are defective. Unfortunately, some chips are stolen before inspection. If 1% of all chips on the market are stolen, find the probability that a given chip is stolen before it is defective.
6. (5 points) Let X be a binomial random variable with parameters n and p. Show that the moment generating function for X is given by
mX(t) = ((1-p) + p et) n