Solve the questions in the attachment.

1 Let X and Y have joint density function given by: f(x, y)=c(x+ 3y) 0 0.5). .2 The continuous random variables X, Y have the bivariate PDF: style f(x, y) =2 xty0, y>0 (i) Derive the marginal PDF of Y . [2] Use the result from part (i) to derive the conditional PDF of X given Y =y. [1] [Total 3] .3 Show that, for the joint random variables M, N, where P(M = m, N = n) = m 35 x20-2 7 , form =1, 2, 3, 4 and n =1, 2, 3 the conditional probability functions for M given N=n and for N given M=m are equal to the corresponding marginal distributions. 4 Let X and Y have joint density function: i style fx, y (x,y)=(3x2 +xy)0 1, / >0, and k is a constant. Derive an expression for k in terms of a and B. Show using convolutions that if X and Y are independent random variables and X has a X7 distribution and Y has a 2% distribution, then X +Y has a Xm+, distribution. LO Let X be a random variable with mean 3 and standard deviation 2, and let Y be a random tyle variable with mean 4 and standard deviation 1. X and Y have a correlation coefficient of -0.3. Let Z =X+Y Calculate: (i) cov(X, Z) [2] var(2). [2] [Total 4] 11 X has a Poisson distribution with mean 5 and Y has a Poisson distribution with mean 10. If cov(X, Y) = -12 , calculate the variance of Z where Z = X-27 + 3. [2] tyle 12 Show that if X has a negative binomial distribution with parameters k andp, and Y has a negative binomial distribution with parameters m and p, and X and Y are independent, then X +Y also has a negative binomial distribution, and specify its parameters. 13 For a certain company, claim sizes on car policies are normally distributed about a mean of E1,800 tyle and with standard deviation $300, whereas claim sizes on home policies are normally distributed about a mean of E1,200 and with standard deviation $500. Assuming independence among all claim sizes, calculate the probability that a car claim is at least twice the size of a home claim. [4]