### DEPARTMENT OF MATHEMATICS & STATISTICS STAT 3083

Closed book --- A table of the normal distribution was supplied
 1995 December 15 Final Examination Time: 2 1/2 Hours
1. Consider a binomial experiment where the random variable, X, denotes a person's score at the end of n trials. Let p denote the hit rate of this person. Calculate, with appropriate approximate method, the following:
(4)   (a) P[X < 2]     with n = 150,     p = 0.03,
(4)   (b) P[X > 80]     with n = 300,     p = 0.25.

2. Let the random variable X have the following density function:

(4)   (i) Find the cumulative function , F(x).
(2)   (ii) Find P[ X < 50 ].
(4)   (iii) Find E(X | X < 150).

3. Consider an urn containing 30 red and 50 green chips.
(2)   (a) Suppose the chips are drawn with replacement and we draw 10 chips. What is the probability that all ten are of the same color.
(3)   (b)Suppose chips are drawn one at a time, with replacement, until the fifth draw of a red chip. What is the probability that the 5th chip is drawn on the 12th attempt.
(4)   (c) Suppose chips are drawn wthout replacement and we draw 16 chips. Let X be the random variable denoting the number of red chips drawn. Find P[X = 6],   E(X).

(10)   4.
Let X and Y be jointly distributed random variables with the following joint distribution:
 0 2 4 X Y 0.12 0.15 0.09 0.14 0.16 0.10 0.10 0.09 0.04
(a) Find Cov(X,Y).
(b) Are X, Y independent?
(c) Are X, Y identically distributed?
(d) Find   P[ X+Y = 2].
(e) Find E( XY | XY > 2 ].

(8)   5. Amount of rainfall during a one-year period, X, may be assumed to be a normal random variable with a mean of 200 cm and a standard deviation of 20 cm. Let Y = (X1+X2+...+X25)/25 denote the average rainfall over a 25-year period. Assume X1, X2 ... are independent. Find E(Y), Var(Y), and P[ Y < 195].

(12)   6. Find E(X) and Var(X) for the following densities:

(a)

(b)

(c)

(d)  f(x) is the density function of a chi-square random variable with 15 degrees of freedom.

(4)   7. The probability of functioning of the i-th relay in the circuits shown is given by   p1, p2, p3, p4, p5.   Assuming independence, what is the probability that the current flows?

(a)

(b)