# Probability - Random variables

1. Mar 8, 2012

### tjackson

1. The problem statement, all variables and given/known data

During a typical Pennsylvania winter, I80 averages 1.6 potholes per 10 miles. A certain county is responsible for repairing potholes in a 30 mile stretch of the interstate. Let X denote the number of potholes the county will have to repair at the end of next winter.
1. The random variable X is

(i) binomial (ii) hypergeometric (iii) negative binomial (iv) Poisson

2. Give the expected value and variance of X.

3. The cost of repairing a pothole is \$ 5000. If Y denotes the county's pothole repair expense for next winter,find the mean value and variance of Y ?

2. Relevant equations and Attempt at a solution

1.) Pretty sure this is a Poisson random variable

2.) P = ($\alpha$x * e-$\alpha$ )/ x!

In this case $\alpha$ = 0.16 potholes/mile

x represents 0, 1, 2, ... , 30 is this correct?

Expected value of X= $\alpha$ = 0.16 potholes/mile
Variance of X = expected value of X = $\alpha$ = 0.16 potholes/mile

Y = aX + b

X = potholes that need to be fixed
a = 5000 (cost to fix each pothole)
b = 0

Expected value of Y = a * Expected value of X

Variance of Y = a2 * Variance of X