probability

Math 464 - Fall 13 - Homework 10

1. We flip a fair coin until we get 100 heads. Use the central limit theorem to find (approximately) the probabilities that it will take at least 200,210, 220 and 300 flips. Hint: Let X be the number of flips to get the 100 heads. Write X as a sum of 100 i.i.d. random variables.

2. Let Z1, Z2, · · · Zn be independent, each having the standard normal dis- tribution. Let

Qn = n

∑

i=1

Z 2

i

The distribution of Qn is called the chi-squared distribution with n degrees of freedom. It is an important distribution in statistics. When n is large we can use the central limit theorem to approximate it. (a) Use the central limit theorem to find (approximately)

P(80 ≤ Q100 ≤ 120) (1)

(b) In a previous homework you showed that the distribution of Q1 = Z 2

1 is

a gamma distribution. Use this to show that the distribution of Qn is also a gamma distribution and find the parameters of the gamma distribution.

(c) (Optional) Use software like R and your answer to (b) to compute that probability in (a) exactly.

3. Let X and Y be continuous random variables with joint pdf

f(x, y) = {

e−y, if 0 < x < y 0 otherwise

(2)

Find E[X|Y = y] and E[Y |X = x].

4. Our computer has two random number generators. One will return a random number that is uniformly distributed on [0, 1]. The other will return a random number with the exponential distribution with parameter λ. (You specify λ when you call the generator.) We generate a random number X by the following two stage process. First, we call the uniform generator and let λ be the number it returns. Then we call the exponential generator using λ as the parameter. X is the number returned by the exponential generator.

(a) Find E[X].

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(b) Find the pdf of X.

5. A statistician want to estimate the mean age of the population in the US. Let X be the age of a randomly selected person in the US. (a) Make a crude estimate of the variance of X using just your common sense. (b) She wants to estimate the mean of X within one year with a 99.9% confidence interval. In other words, she wants the 99.9% confidence interval to have a width of 2 years. How many people does she need to sample? (Of course, your answer will depend on your estimate in part (a).)

6. We roll a six-sided die 100 times. Let X be the sum of the rolls. (So X ranges from 100 to 600.) The central limit theorem says that X is approxi- mately normal. The goal of this problem is to see this in a simulation. (a) Find analytically the mean and variance of X. (b) Write a program (in R or whatever software you want) to generate a large number of samples of X and plot a histogram of them. On the same graph plot the pdf of a normally distributed random variable with the mean and variance you found in part (a). Note that to generate one sample of X requires rolling the die 100 times. (The computer will be happy to do that for you.)

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