Due 10/07/98

1.

Derive the formulae for the mean, variance and standard deviation for the uniform distribution, the exponential distribution, the gamma-type distribution, the chi-square distribution with v degrees of freedom, the normal distribution and the beta distribution. You will find it helpful to read the proofs of Theorems 4.6, 4.8, 4.9, 4.10 and 4.11.

2.

Graph the probability density function and the distribution function for a random variable with a normal distribution with expected value of 3 and variance of 2.

3.

Graph the probability density function and the distribution function for a random variable with a gamma-type distribution with expected value 4 and variance 8.

4.

Graph the probability density function and the distribution function for a random variable with a chi-square distribution with 4 degrees of freedom.

5.

Graph the probability density function and the distribution function for a random variable with a beta distribution with $\alpha = 2$ and $\beta = 5$.

6.

Suppose that the random variable X has a exponential distribution and expected value 1. Suppose that H(z) = e^tz. Compute E[H(X)]. Be careful to describe for which values of t E[H(X)] exists.

7.

Suppose that the random variable X has a gamma-type distribution with $\beta = 1$. Suppose that H(z) = e^tz. Compute E[H(X)]. Be careful to describe for which values of t E[H(X)] exists. How does your answer compare with the answer to the previous question?

Extra Credit Problems

1.

Verify that $\Gamma(\alpha+\beta)B(\alpha,\beta) = \Gamma(\alpha)\Gamma(\beta)$. Hint: Show that

$$\Gamma(t) = 2\int_0^\infty z^{2t-1}\exp(-z^2)dz$$

Then write the righthand side of the identity as a double integral and evaluate it with polar coordinates.

2.

Show $B(1/2,1/2) = \pi$ directly from its definition as an integral. Use this in combination with the last problem to show that $\Gamma(1/2) = \sqrt{\pi}$.

3.

It is impossible to find a simple algebraic formula for the distribution function for a random variable with a normal distribution. Suppose that we X is a random variable with a normal distribution with E[X] = 0 and V[X] = 1. Let f_X(t) denote its density and F_X(t) denote its distribution function. Show

$$\lim_{n\rightarrow\infty}\frac{1 - F_X(t)}{t^{-1}f_X(t)} = 1$$

and

$$1 - F_X(t) \leq t^{-1}f_X(t)\;\;{\rm for\;} t \gt 0.$$