The joint probability mass function of X and Y is given by Compute the following probabilities: P(X + Y >4) P(XY = 2) = P(> 2) = p(1, 1) = 0.4 p(2, 1) 0.1 p(3, 1) = 0.1 p(1,2) = 0.1 p(1, 3) = 0.1 p(2,2) = 0.05 p(2,3) = 0.1 p(3,2) = 0.05 p(3, 3) = 0

If X and Y have the joint probability distributionf(−1, 0) = 0, f(−1, 1) = 1 4 , f(0, 0) = 16 , f(0, 1) = 0, f(1, 0) = 112 , and f(1, 1) = 12 , show that (a) cov(X, Y) = 0;(b) the two random variables are not independent.

X is a discrete random variable and takes the values 0,1 and 2 with probabilities of 1/6, 1/3 and 1/2, respectively. What is the moment generator function M(t) of X?

If y is random variable has a probability mass function defined as follows find 1. and 2.

(b) Let Z be a discrete random variable with E(Z) = 0. Does it necessarily follow that E(Z³) = 0? If yes, give a proof; if no, give a counterexample.

Find the moment-generating function of the contin-uous random variable X whose probability density is given by f(x) =1 for 0 < x < 10 elsewhere and use it to find μ1,μ2, and σ2.

The cumulative distribution function F(x) of a discrete random variable X is given by F(0) = .30, F(1) = .70, F(2) = .90, and F(3) = 1.0, then the value of the probability mass function p(x) at x = 1 is what?

If Y is a discrete random variable with possible values of 1, 2, and 4, and the probability mass function is given by P(Y = 1) = 0.2, P(Y = 2) = 0.5, and P(Y = 4) = 0.3, what is the variance of Y?

Suppose a random variable X has range {−3, −1, 1, 2}. Its proba- bility distribution (probability mass function) is given by P (X = −3) = 0.25 P (X = −1) = 0.25 P (X = 1) = 0.45 P (X = 2) = 0.05 a) Find E(X). b) Find E(X^2). b) Find V (X). b) Find V (X^2).

Lana arrives at a bus stop at a uniformly random time. When Lana arrives at the bus stop, either of two independent bus lines (both of which can take her school) may come by. Company A’s bus arrival times are exactly 10 minutes apart, whereas the time from one Company B bus to the next is distributed as Expo (1/10​) . What is the probability that Company B’s bus arrives first solve by double integration.