Due 10/21

Complete the following exercises:

1.

Show that if R and S are independent random variables with Poisson distributions with means a and b respectively, then R + S has a Poisson distribution with mean a + b.

2.

Suppose that V is a random sample of size N from a Poisson distribution with mean 1. Put $R_N = V_1 + \cdots + V_N$, and

$$R^*_N = \frac{R_N - E[R_N]}{\sqrt{{\rm Var}[R_N]}}$$

Make accurate graphs of the probability mass function of R^*_N for N = 10, 20, 30, 40, 50, 60. How do these compare with the graphs you got for binomial random variables on the last homework?

3.

Suppose that V is a random sample of size N from an exponential distribution with mean 1. Put $R_N = V_1 + \cdots + V_N$, and

$$R^*_N = \frac{R_N - E[R_N]}{\sqrt{{\rm Var}[R_N]}}$$

Make accurate graphs of the density function of R^*_N for N = 10, 20, 30, 40, 50, 60. How do these compare with the graph of the standard normal density?

4.

Show that if R and S are independent random variables having normal distributions with mean 0 and variances 1 and b² respectively, then R + S has a normal distribution with variance 1 +b².

5.

By scaling and shifting, use the previous exercise to show that if R and S are independent random variables having normal distributions then R + S has a normal distribution.

6.

Find the maximumm likelihood estimate of the variance of a normal random population with mean 0, based on N observations.

7.

Find the density of the square of normal random variable with mean 0. Use this to give the density for the MLE estimate you derived in the previous problem.

8.

Suppose that V is a random sample of size N from a population with distribution function F. Let $m(V) = \min(V_1,\dots,V_N)$. Show that the distribution function of m(V) is given by

F_m(V)(t) = 1 - (1-F(t))^N.

Hint: Try to find $1 - F_{m(V)}(t) \equiv \Pr(m(V) \gt t)$.