Probability question

1.On average, 80% of a bank’s customers are males, and 20% are females. If the sexes of arriving customers are independent trials find the probability that, starting at a given hour,

(a) three male customers arrive, followed by two females

(b) 3 males and 2 females are the first to arrive, in unspecified order

(c) the first female arrives as the 5th customer

2.Let X be uniformly distributed on the set{1,2,3,4}. Find:

(a) E(X), the expected value of X (b) E(2^X), using LOTUS

3.Suppose the probability mass function of a random variable is given by

p(x) = c(1/3)x, x = 0,1,...

Find the value of c so that p is a bona fide pmf.

4.There are n eggs, each of which hatches a chick with probability p (independently). Each of these chicks survives with probability r, independently. Give the name and pmf formula for the distribution of the number of chicks that survive.

5.The number of fish in a certain lake is Pois(λ). Worried that there might be no fish at all, a statistician adds one fish to the lake. Let Y be the resulting number of fish (so Y is one plus a Pois(λ) random variable.) Find E(Y 2).

6.Let X be a random variable having mean µ = −2 and variance σ2 = 4. Find the mean and variance of the random variable Y = 1−3X.

7.Each class period the professor picks 3 students at random from his class of 15 students to present problems at the board.

(a.) What is the expected number of times Jim (one of the students) will be picked in a week (5 classes)?

(b.) What is the expected number of classes between ones in which Jim is picked?

8.One hundred year floods are floods of such a size as to happen only once every hundred years on average. Assume the number of such floods occuring in a given time period has a Poisson distribution. Find the probability that 2 or more hundred year floods will occur in a given 150 year time period.

9.Let

f(x) = cx/(1+x^2)

, 0 < x < 4.

Find the value of c so that f is a bona fide probability density function.

10.Find E|Z| if Z has a standard normal distribution. (Here |Z| denotes the absolute value of Z.)

11.If X has a N(2,16) distribution, find P(−1 < X ≤ 2) in terms of Φ, the standard normal cdf.