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Chapter 6: Q.6.22 (page 272)
The joint density function of X and Y is
(a) Are X and Y independent?
(b) Find the density function of X.
(c) Find
a. X and Y are not independent.
b. The density function of X is
c. The value of
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A model proposed for NBA basketball supposes that when two teams with roughly the same record play each other, the number of points scored in a quarter by the home team minus the number scored by the visiting team is approximately a normal random variable with mean 1.5 and variance 6. In addition, the model supposes that the point differentials for the four quarters are independent. Assume that this model is correct.
(a) What is the probability that the home team wins?
(b) What is the conditional probability that the home team wins, given that it is behind by 5 points at halftime?
(c) What is the conditional probability that the home team wins, given that it is ahead by 5 points at the end of the first quarter?
Let X1, ... , Xn be independent uniform (0, 1) random variables. Let R = X(n) 鈭� X(1) denote the range and M = [X(n) + X(1)]/2 the midrange of X1, ..., Xn. Compute the joint density function of R and M.
Let U denote a random variable uniformly distributed over (0, 1). Compute the conditional distribution of U given that
(a) U > a;
(b) U < a; where 0 < a < 1.
(a) If X has a gamma distribution with parameterswhat is the distribution of
(b) Show that has a gamma distribution with parameters when n is a positive integer and is a chi-squared random variable with degrees of freedom
Each throw of an unfair die lands on each of the odd numbers with probability C and on each of the even numbers with probability .
(a) Find C.
(b) Suppose that the die is tossed. Let X equal if the result is an even number, and let it be otherwise. Also, let Y equal if the result is a number greater than three and let it be otherwise. Find the joint probability mass function of X and Y. Suppose now that independent tosses of the die are made.
(c) Find the probability that each of the six outcomes occurs exactly twice.
(d) Find the probability that of the outcomes are either one or two, are either three or four, and are either five or six.
(e) Find the probability that at least of the tosses land on even numbers.
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