Due 11/11

Complete the following exercises:

1.

Suppose that R and S are independent random variables with gamma densities with $\beta = 1$. Find the density of R/S.

2.

The t density with 1 degree of freedom (k=2) is called the Cauchy density. Graph the Cauchy density and and find a simple expression for this density.

3.

For which degrees of freedom do t densities fail to have means? When the mean exists, what is its value?

4.

Calculus practice: Find a closed formula for the distribution function of a random variable with the t distribution with 1, 2 and 3 degrees of freedom.

5.

Using the tables in the back of MSA, find a value x so that if V is a random sample of size 10 from a normal distribution with mean $\mu$,

$$\Pr\left(\frac{\vert\mu_{MLE}(V) - \mu\vert}{\sigma_{MLE}(V)} \gt x\right) = 0.05$$

6.

Confidence intervals. Let x and V be as in the preceding problem. Explain why

$$\Pr(-x\sigma_MLE(V) + \mu_{MLE}(V) \leq \mu \leq x\sigma_MLE(V) + \mu_{MLE}(V)) = 0.95.$$

On the basis of this computation, the random interval

$$[-x\sigma_MLE(V) + \mu_{MLE}(V), x\sigma_MLE(V) + \mu_{MLE}(V)]$$

is called a 95% confidence interval for $\mu$. Find the 95% confidence interval for $\mu$ if the V = (1,2,-2,3,5,7,-10,11,0,4).

7.

Graph the t densities for 5, 10, 15, 20, 25 and 30 degrees of freedom, along with the standard normal density. How does what you see explain the range of values for the degrees of freedom in the table of the t density in MSA?