Help me solve the following questions.

1 Calculate E(X |Y =10) for the joint distribution: Y 10 20 30 0.2 0.2 0.1 X D.2 0.3 0 2 The random variable V has a Poisson distribution with mean 5. For a given value of V, the style random variable U is distributed as follows: U|(V =v) ~U(0,v) Obtain the mean and variance of the marginal distribution of U . [4] 3 Given that X and Y are continuous random variables, prove from first principles that: E(Y) = E [E(Y1X)] [3] The random variable X has a gamma distribution with parameters o =3 and 1=2. Y is a related variable with conditional mean and variance of: E(Y | X = x) =3x+1 var(Y | X =x) =2x-+5 Calculate the unconditional mean and standard deviation of Y . [5] [Total 8] 4 Suppose that a random variable X has a standard normal distribution, and the conditional style distribution of a Poisson random variable Y , given the value of X = x , has expectation g(x) = x- +1. Determine E(Y) and var(y). [5].5 The table below shows the bivariate probability distribution for two discrete random variables X and Y : X=0 X =1 X -2 Y =1 0.15 0.20 0.25 Y =2 0.05 0.15 0.20 Calculate the value of E(X | Y = 2)