|Final Examination||1995 December 15||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.|
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).|
|(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].|
Find E(X) and Var(X) for the following densities:
(d) f(x) is the density function of a chi-square random variable with 15 degrees of freedom.
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?