Question on exponential distribution?

[theBEAST]

Homework Statement

Homework Equations

f(x) = e^-λλ^x/x!

The Attempt at a Solution

Initially I thought I could solve this problem using the Law of Memoryless. That, the solution would just be P(X <= 2). However, I was wrong. Turns out the solution is P(X <= 4.5) - P(X<= 2.5). Does anyone know why?

 

[kduna]

First of all the equation you listed is actually a poisson distribution. As the Poisson distribution is discrete, it really wouldn't make sense in this situation.

http://en.wikipedia.org/wiki/Exponential_distribution

The an exponential distribution has pdf of the form:

f(x) = λe^-λx for x ≥ 0.

Anyway, what you said isn't quite right.

Turns out the solution is P(X = 4.5) - P(X=2.5)

Since this is a continuous distribution, P(X = 4.5) = 0. In fact, P(X = c) = 0 for any c.

The solution would actually be P(X ≤ 4.5) - P(X ≤ 2.5) = F(4.5) - F(2.5).

Where F(x) is the cumulative distribution function.

The reason for this is that you want X to be less that 4.5, but all the stuff below 2.5 is not wanted , so you subtract off the probability that X is less that 2.5.