W13 Homework B

Due date: Tuesday 4/14, 11:59pm

Sample Mean, Tails, Law of Large Numbers

01

09

Chebyshev

Suppose $X$ is an RV with $E[X]=17$ and $E[X^2]=298$.

Use Chebyshev’s inequality to find:

(a) A lower bound for $P[10<X<24]$.

(b) An upper bound for $P[|X-17|\ge 16]$.

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02

08

Normal v. Chebyshev - Jewelry insurance

A jewelry insurance provider has 2500 customers. The expected payout to a customer each year is $1000 with a standard deviation of $900.

What premium should be charged to each customer to ensure that the premiums will cover the claims, with probability at least 99.9%?

(a) Solve the problem using a normal approximation.

(b) Solve the problem using Chebyshev’s inequality.

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03

05

Train arrivals

The time between train arrivals at a station is exponentially distributed with a mean of 5 minutes. Therefore, the arrival time of the $n^{\text{th}}$ train, $T_n$, can be represented as $X_1+X_2+\cdots +X_n$ where each $X_i$ is independent of the others and is exponentially distributed with a mean of 5 minutes. Suppose there are enough customers waiting to fill 3 trains.

(a) Find the mean and variance of $T_3$, the time elapsed when the third train arrives.

(b) Use Chebyshev’s inequality to find an upper bound for the probability that more than 35 minutes will pass before all customers can board a train.

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Solution

Solutions---5290-05

04

04

Random walk forward

You play a game where you roll a die, and if the outcome is 1 or 2 you take a step forward, otherwise you take two steps forward. Let $X_n$ be your position (measured in steps forward) after playing the game $n$ times.

(a) Estimate $P[X_{90}\ge 160]$ using a normal approximation for a certain relevant binomial distribution.

(b) Find ${\lim }_{n\to \infty }P[X_n>1.6n]$ and ${\lim }_{n\to \infty }P[X_n>1.7n]$.

Hint: Rewrite the conditions into a form where you can apply the Law of Large Numbers.

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Review

05

10

Body weights

Assume that body weights of men are Normally distributed with a mean of 170 pounds and a standard deviation of 30 pounds.

What is the body weight threshold separating the lightest 90% from the heaviest 10%?

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Solution

Solutions---5280-10

06

07

Normal approximation of the binomial - Double ones

Roll a pair of dice 10,000 times. Estimate the odds that the number of “snake eyes” (double ones) obtained is in the range $[\mathrm{290,300}]$.

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