1) Given the following table representing the joint probability distribution of the discrete Random variable (X, Y) X 1 2 3 1/12 0 2 1/6 1/9 1/18 1/4 3 0 1/5 2/15 a) Evaluate the marginal distribution of X and Y. b) Find the conditional distribution of X given Y = 2. c) Find the conditional distribution of Y given X= 3.

12. i. State Central limit theorem and tell what is the main theme of it. ii. Show that if the random variables (X, Y) follow the probability density of a bivariate normal distribution and if the correlation p = 0, X and Y are independent. (Show that f(x, y) = g(x) h(y))

Customers arrive at a shop according to a Poisson process at a mean rate of 2 customers every ten minutes. The shop opens at 9am. (c) Let Y be the time (in hours) until the 4th customer arrives at the shop. State the distribution of Y and give the value(s) of its parameter(s). (d) Show that the probability of the time until the 4th customer arrives being greater than 15 minutes is 13e-3. (e) Let M be the number of customers who arrive at the shop between 9am and 12 noon. State the distribution of M and give the value(s) of its parameter(s). (f) Calculate P(M = 30) using a normal approximation with the continuity correction.

Suppose a random variable x is best described by a uniform probability distribution with range 0 to 5. Find the value of a that makes the following probability statements true.