Hi there, could you help me with these 6 questions? Thank you!

Overdispersion and Smoothing Homework Questions Our fundamental smoothing (or conditioning) identities for the mean and variance of a random variable Y that depends on another variable X are: mn=Ewwmn and Var(Y) = E[Var(Y|X)] + Var(E[Y|X]). Consider an example in which the variable X has a stande exponen- tial distribution. The mean and variance of this distribution are both one. Given X , the variable Y is drawn from a Poisson distribution with mean A equal to 3X. (1) Find the expected value of Y. (2) Find the variance of Y. (3) If Y had a Poisson distribution, then the mean and variance of Y would be the same. Are they the same? Which is larger? (4) Write code in R to generate 10,000 (X,Y) pairs. Hints: The function rexp(n) will generate n standard exponential variables. The function rpois(n,lambda) will generate n Poisson variables. Also, the argument lambda to this function can be a vector. (5) Construct a scatter-plot of the (X , Y) pairs and a histogram of the Y values. (6) Find the sample mean and sample variance of the 10,000 Y values in the simulation. Do the values agree with the exact values computed in the rst questions