# Probability - Random variables

• tjackson

## Homework Statement

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 $5000. If Y denotes the county's pothole repair expense for next winter,find the mean value and variance of Y ? ## Homework Equations and Attempt at a solution 1.) Pretty sure this is a Poisson random variable 2.) P = ($\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 Hello! You are correct, the random variable X is a Poisson random variable with a rate parameter of 0.16 potholes per mile. The expected value of X would be 4.8 potholes (0.16 * 30 miles) and the variance would also be 4.8 potholes. For the second part, Y would follow a normal distribution with a mean of 5000 * 4.8 =$24,000 and a variance of (5000^2) * 4.8 = $576,000,000. This means that the county can expect to spend an average of$24,000 on pothole repairs next winter, with a variance of \$576,000,000.

Hope this helps! Let me know if you have any other questions.