Probability - Poisson Random Variable?

In summary, The random variable X is Poisson with an expected value of 0.16 potholes/mile. The expected value and variance of X are both 0.16 potholes/mile. The cost of repairing a pothole is $5000. If Y denotes the county's pothole repair expense for next winter, the mean value of Y is $8000 for 30 miles of road and the variance of Y is $40000 for 30 miles of road.
  • #1
tjackson
5
0
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 =( [itex]\alpha[/itex]x * e -[itex]\alpha[/itex] ) / 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
 
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  • #2
tjackson said:
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 =( [itex]\alpha[/itex]x * e -[itex]\alpha[/itex] ) / 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.
 
  • #3
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.
 

FAQ: Probability - Poisson Random Variable?

1. What is a Poisson random variable?

A Poisson random variable is a discrete random variable that represents the number of events that occur in a given time interval or space. It is used to model situations where the events occur randomly and independently of one another.

2. How is a Poisson random variable different from a normal random variable?

A Poisson random variable only takes on non-negative integer values, while a normal random variable can take on any real value. Additionally, a Poisson random variable is used to model count data, while a normal random variable is used to model continuous data.

3. What is the probability mass function of a Poisson random variable?

The probability mass function of a Poisson random variable is given by P(X=x) = (λ^x * e^-λ) / x!, where λ is the mean number of events in the given time interval or space.

4. How do you calculate the mean and variance of a Poisson random variable?

The mean of a Poisson random variable is equal to λ, and the variance is also equal to λ. This means that the standard deviation is equal to the square root of λ.

5. In what situations is a Poisson random variable commonly used?

A Poisson random variable is commonly used in situations where the number of events occurring in a given time interval or space is of interest, such as in the fields of biology, finance, and telecommunications.

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