Stochastic Calculus, Fall 2002 (http://www.math.nyu.edu/faculty/goodman/teaching/StochCalc/)

Assignment 1.

Given September 5, due September 12. Last revised, September 4.

Objective: Review of Basic Probability.

1. We have a container with 300 red balls and 600 blue balls. We mix the balls well and choose one at random, with each ball being equally likely to be chosen. After each choice, we return the chosen ball to the container and mix again.

a. What is the probability that the first n balls chosen are all blue?

b. Let N be the number of blue balls chosen before the first red one. What is the P (N = n)? What are the mean and variance of N. Explain your answers using the formulae

∑_{n=0}^{∞} x^{n} = \frac{1}{1 − x}  for |x| < 1
∑_{n=0}^{∞} nx^{n} = x\frac{d}{dx} \frac{1}{1 − x} for |x| < 1

etc.

c. What is the probability that N = 0 given that N ≤ 2?

d. What is the probability that N is an even number? Count 0 as an even number.

2. A tourist decides between two plays, called “Good” (G) and “Bad” (B). The probability of the tourist choosing Good is P (G) = 10%. A tourist choosing Good likes it (L) with 70% probability (P (L | G) = .7) while a tourist choosing Bad dislikes it with 80% probability (P (D | B) = .8).

a. Draw a probability decision tree diagram to illustrate the choices.

b. Calculate P (L), the probability that the tourist liked the play he or she saw.

c. If the tourist liked the play he or she chose, what is the probability that he or she chose Good?

3. A “triangular” random variable, X, has probability density function (PDF) f(x) given by

f(x) = \{ \begin{matrix}2(1 − x) & if 0 ≤ x ≤ 1, \\ 0 & otherwise. \end{matrix}

a. Calculate the mean and variance of X. b. Suppose X_{1} and X_{2} are independent samples (copies) of X and Y = X_{1} +X_{2}. That

is to say that X_{1} and X_{2} are independent random variables and each has the same density f. Find the PDF for Y.

c. Calculate the mean and variance of Y without using the formula for its PDF. d. Find P (Y > 1). e. Suppose X_{1}, X_{2}, …, X_{100} are independent samples of X. Estimate P r(X_{1} + ⋯ +

X_{100} > 34) using the central limit theorem. You will need access to standard normal probabilities either through a table or a calculator or computer program.

4. Suppose X and Y have a joint PDF

f(x, y) = \frac{1}{8π} \{ \begin{matrix}4 − x^{2} − y^{2} & if x^{2} + y^{2} ≤ 4, \\ 0 & otherwise. \end{matrix}

a. Calculate P (X^{2} + Y^{2} ≤ 1). b. Calculate the marginal PDF for X alone. c. What is the covariance between X and Y? d. Find an event depending on X alone whose probability depends on Y. Use this to

show that X is not independent of Y. e. Write the joint PDF for U = X^{2} and V = Y^{2}. f. Calculate the covariance between X^{2} and Y^{2}. It may be easier to do this without

using part e. Use this to show, again, that X and Y are not independent.