# Probability - Poisson Random Variable?

1. 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 ? 2. 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 α = 0.16 potholes/mile x represents 0, 1, 2, ... , 30 is this correct? Expected value of X= α = 0.16 potholes/mile Variance of X = expected value of X = α = 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 ## Answers and Replies vela Staff Emeritus Science Advisor Homework Helper Education Advisor 1. 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 ?

2. 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 α = 0.16 potholes/mile

x represents 0, 1, 2, ... , 30 is this correct?
No. The random variable X denotes "the number of potholes the county will have to repair." Why would that number be limited to 30? ##\alpha## is the expected value of X, so it should be a number, not a number per mile.

You may be overcomplicating the problem a bit :)

Look at it this way: if, historically, the city averages about 1.6 potholes/ 10 miles, how many would you average in 30 miles?

Now to find the average expense of Y, there is a way we can look at it. Y = 5000*λ where lambda is the number of pot holes. So we can expect to pay say 5000*1.6 = \$8000 for 10 miles of road. So how much would that be for 30 miles?

Now, let's talk about variance. Remember the definition of expected value and variance for Poisson? They are both λ .

So if you need to find Var(Y) = Var(5000*λ), what do we do with constant terms in variance? Hint: It's a large number, but that OK because variance isn't as helpful to know as standard deviation.