6.54 Let Y1, Y2,..., Y, be independent Poisson random variables with means 1, 2,..., An respectively. Find the a probability function of Y. b conditional probability function of Y1, given that Y = m. Y1 = m. c conditional probability function of Y1+Y2, given that 6.55 Customers arrive at a department store checkout counter according to a Poisson distribution with a mean of 7 per hour. In a given two-hour period, what is the probability that 20 or more customers will arrive at the counter? 6.56 The length of time necessary to tune up a car is exponentially distributed with a mean of .5 hour. If two cars are waiting for a tune-up and the service times are independent, what is the probability that the total time for the two tune-ups will exceed 1.5 hours? [Hint: Recall the result of Example 6.12.] 6.57 Let Y, Y2,..., Y,, be independent random variables such that each Y, has a gamma distribution with parameters a, and B. That is, the distributions of the Y's might have different a's, but all have the same value for ẞ. Prove that U Y1+Y2++Y,, has a gamma distribution with parameters a1+α2++α, and B. 6.58 We saw in Exercise 5.159 that the negative binomial random variable Y can be written as Y=W, where W1, W2,..., W, are independent geometric random variables with parameter p. 349/939 6.6 Multivariable Transformations Using Jacobians (Optional) a Use this fact to derive the moment-generating function for Y. b Use the moment-generating function to show that E(Y) = r/p and V(Y) = r(1 - p)/p2. c Find the conditional probability function for W1, given that Y = W1+ W2+...+W, = m. 6.59 Show that if Y, has a x2 distribution with v1 degrees of freedom and Y2 has a x2 distribution with v2 degrees of freedom, then U = Y1+ Y2 has a x2 distribution with v1 + v2 degrees of freedom, provided that Y, and Y2 are independent. 6.60 Suppose that W = Y1 + Y2 where Y1 and Y2 are independent. If W has a x2 distribution with v degrees of freedom and W1 has a x2 distribution with v1

A manufacturer has determined that a machine averages one faulty unit for every 500 it produces. What is the probability that an order of 300 units will have one or more faulty units?

Sick leave probability that a given worker at Dyno Nutrition Will call in sick on a Monday is 004. The packaging department has eight workers. What is the probability that two or more ackaging workers Will call in sick next Monday ?

If a binomial experiment has probability p success, then the probability of failure is ____________________. The probability of getting exactly r successes in n trials of this experiment is C(_________, _________)p (1p)