Q4(33 marks)You are given a joint probability distribution for two random variables,X and Y.X can take thevalues 2,4,6,while Y can take the values 1,2,3.YP(X=xnY=y)12320.150.20.05X40.05B0.1060.050.100.25a)The probability,P(X =4 n Y =2),has not been provided.Calculate the value of B that would makethe above a well-defined joint probability distribution.b)Calculate the marginal distribution of X and the marginal distribution of Y.c)From these,calculate the expectation and variance of X and Y,E(X),E(Y),var(X),var(Y).d)Calculate the conditional expectation of X for when Y=1,for when Y =2 and when Y=3.Are X and Yindependent variables?Explain your answer.e)Using the joint distributions above,construct a new variable Z=X×Y.(So,Z takes on the values2,4,6,8,12,18).What is the probability distribution of Z?f)Calculate the expected value of Z(also known as E[Z]).g)Calculate the covariance between X and Y.h)Calculate the correlation between two random variables (px.y).
