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MTH 524 Section D (Fall 2018) Homework 3

September 7, 2018

1. Two teams, A and B, play a series of games. If team A has probability p of winning each game, for what values of p is it to its advantage for team A to play ta single game over the best two out of three? Assume the outcomes of successive games are independent.

2. Two loaded six-sided dice are to be rolled alternately till a six is observed. Assume that the die to be rolled first has probability 0.2 of yielding a six while the second die has probability 0.1 of yielding a six.

(a) obtain the frequency function for the number of rolls needed.

(b) What is the probability that an even number of rolls is needed?

3. In a sequence of independent trials with probability p of success, what is the frequency function for the number of successes before the rth failure, where r is a fixed positive integer?

4. Consider the binomial distribution with n trials and probability p of success on each trial. For what value of k is P(X = k) maximized? This value is called the mode of the distribution. (Hint: Consider the ratio of successive terms.)

5. Phone calls are received at a certain residence as a Poisson process with parameter λ = 2 per hour.

a. If Diane takes a 10-min. shower, what is the probability that the phone rings during that time?

b. How long can her shower be if she wishes the probability of receiving no phone calls to be at most .5?

6. Suppose the cdf F of a random variable X is defined by

F(x) =

 

0, −∞ < x < −0.5, 0.5 + x, −0.5 ≤ x < 0.25, 0.9, 0.25 ≤ x < 0.75, 1, 0.75 ≤ x < ∞.

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(a) Determine all x for which P(X = x) > 0, and for each such case determine the value of that probability.

(b) Determine P(X ∈ J) for all all possible intervals of real numbers with with endpoints 0.25 and 0.75.

7. Suppose X is a random variable whose c.d.f. F satisfies

F(x) = √ x, 0 < x < 1.

Calculate the c.d.f. of Y = √ X.

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