Let X be the number of trials until the r -th success in a sequence of independent Bernoulli variables with probability of success p Give the limiting moment generating distribution of Y = pX as p → 0 evaluated at t = 0,3 and r = 6

Exercise 4. Failure times (in years) for certain model of appliance follow a Gamma(2,4) distribution. Let X be the failure time for the appliance you purchased, so X ~ Gamma(2,4). (b) What is the expected lifetime of your appliance, in months? (d) Use a Gamma calculator to compute the exact value of P(|Y – E(Y)| > 70). (e) Compute the 90th percentile of [the distribution of] Y. (f) True or False: 24 ~x²(4). Explain.

In the daily production of a certain kind of rope, the number of defects per foot given by Y is assumed to have a Poisson distribution with mean  ? = 4.  The profit per foot when the rope is sold is given by X, where  X = 70 − 3Y − Y2.  Find the expected profit per foot.

The geometric distribution gives the probability that the first success occurs at the xth trial with success probability p. f(x)=(1-p)x-1p, x=1,2,3,... Show that E(X)=1/p