Due 10/14

1.

Describe a reasonable probability model (sample space, sigma algebra, probability measure) for the following experiment: A red and a green die are rolled and the number of dots on the top of each die is recorded. The dice are assumed to be symmetrical and not to influence each other. Explain your choice of the probability measure in terms of the assumptions about the dice.

2.

A classical problem going back to Cardano (1501-1576). Which is more probable: to get at least one six when rolling a die four times or to get at least one twelve when rolling a pair of dice 24 times? This is known as de Mére's paradox.

3.

Suppose that R has an exponential distribution with mean a. Define the random variable G as the value of R rounded down to the nearest integer. Show that G has a geometric distribution.

4.

Suppose that R has a gamma distribution with $\alpha = 2$. Define the random variable N as the value of R rounded down to the nearest integer. Find a formula for the probability mass function of N.

5.

Suppose that R_N is the number of heads obtained in N tosses of a fair coin. Graph the probability mass function of the discrete random variable

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

for N = 10, N=20, N=30, N=40, N=50, N=60. Use some sort of computer/calculator to draw these graphs, as it is important that they be accurate.

6.

A random sample of size 20 is drawn from a population which is assumed to be normally distributed. The following data were recorded:

$$\begin{array} {ccccc} 20.2 & 65.6 & 11.0 & 54.2 & 48.3 \ 12... ....0 & 12.7 & 92.7 \ 47.9 & 03.7 & 72.9 & 40.9 & 87.5\end{array}$$

What are the maximum likelihood estimates of the mean and the variance of this normal distribution based on this data?

7.

A density function f is defined by the rule f(x) = A(1+cos(x)) for $x \in [-\pi, \pi]$ and 0 otherwise. Determine the value of A, graph this density, and determine the mean and variance of a random variable with this density.

8.

Let A and B be events, and let I_C denote the indicator any event C. Show that $E[I_C] = \Pr(C)$. Now show

$$I_A + I_B = I_{A\cup B} + I_{A\cap B},$$

and use this to show that $\Pr(A\cup B) = \Pr(A) + \Pr(B) - \Pr(A\cap B)$.

9.

Suppose that A, B, and C are any events. Derive a formula for $\Pr(A\cup B \cup C)$ in terms of the probabilities of A, B, C and their intersections.

10.

Suppose that V = (R, S) is a vector valued random variable with distribution function $F_V(x,y) = \Pr(R \leq x, S \leq y)$. Express $\Pr(a < R \leq b, c < S \leq d)$ in terms of F_V.