1. Let X be a random variable with mean 2 and variance 1. Find the mean and variance of Y where Y = 4X + 10. 2. Let X1,X2, X3,X4 be iid random variables with mean 10 and variance 16. Find the mean and variance of Y=X1+X2+X3+X4. Find the mean and variance of Z=(X1+X2+X3+X4)/4. Solution 1)E(Y)=E(4X+10)=4E(X)+10=18 V(Y)=V(4X+10)=16V(X)+0=16 2)E(Y)=E(X1)+E(X2)+E(X3)+E(X4)=40 V(Y)=V(X1)+V(X2)+V(X3)+V(X4)=64 E(Z)=E(Y)/4=10 V(Z)=V(Y)/16=16.

1. 1. Let X1,X2,....,Xn be a random sample from a Poisson distribution with parameter theta .
a. Find a maximum likelihood estimator for theta.
b.Find a maximum likelihood estimator for P(X = 0).
Solution
a ) theta = (1/n) [ X1 + X2 + ..... +Xn ]
b ) P(X=0) = e^(-theta)