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

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

Its x-12.5 not expone

Moment generating function is given: Mx(t)= 1/(4-t)^2 Find the mean and the vairance for the random variable x.

α = 0.01 for a left-tailed-test,  Find the critical t value if n =   23

4) Considey molecules cnd 100 10 cells of equal energy. Find log. energy most probabile and ii the wfor j the distribution least probable distribution.

Suppose you are fitting a survival model and the outcome of interest is time to the diagnosis of cancer (T). A subject entered the study at t=0 and the diagnosis of cancer did not occur by the end of the study period, what kind of event is that? It is an interval-censored observation because an event happened between time of entry and the exit. It is a right-censored observation because no event will be observed during the study period. It is a left-censored observation where the event of interest happened before the study starts. It is a right-censored observation because the event has already occurred.

5

The Volatility X for the S&P stock index on a given day is a normal random variable with mean = 10 and standard deviation = 2 The volatilities recorded over a 100-day period on the S&P500 are Y1, Y2, ... Y100. Assume that these Yi's are independent and identically distributed, uniform on the interval [5,15]. Let V = (Y1 + Y2 + ... + Y100)/100. What approximately is P[9.5 < V < 10.5]?