Exponential distribution problem

[JoskeJos]

Hello forum,

I recently lost my notes on this matter, so I hope someone can fill in the gaps in my memory.

The problem I am working on is the following:

For one ticket window, the waiting time for one people satisfies an exponential distribution λe-λt and expected waiting time is 4 min.
a. What’s the probability of the waiting time less than 7 min for 2 persons?
b. What’s the probability of the waiting time longer than 165 min for 36 persons?

As far as my memory goes:
a) P(X>7) = 1 - (1-e^(-0,25*7)) = ... -> for one person

However I thought the amount of people influences the result, but I can't remember how.

 

[chiro]

Hey JoskeJos and welcome to the forums.

I'm assuming that you're process refers to one that models one passenger and I am also assuming that every passenger has the same process characteristics.

So what you need to do is calculate P(X + Y < 7) where X and Y are your exponential. This distribution gives a Gamma distribution with n = 2 and the parameter should be your rate (look this up if you haven't come across this).

Using this information, can you now find the probability?

The other one should be similar. Also you should tell me if there are extra assumptions because if there are, then you will need to use another model.

 

[JoskeJos]

Hey chiro, sorry for my late reply (I've been really busy).
First and foremost thanks for your reply.

For question a:
λ = 1/4
X, Y -> λe-λt
X+Y = ((λt)^(2-1) / (2-1)!)* λe-λt
P(X+Y <7) = integral of 0 to infinity of (X+Y)

Is this what you meant?

For question b, can't I solve this by normal distribution?
Because 36! is a pretty big number?

 

[chiro]

You will be able to use the Central Limit Theorem for b) since you are summing a lot of random variables which allows you to use a normal approximation. It's only because you have 36 and not 3 samples.

For the first one, you need a distribution for X + Y and my suggestion was to look into the PDF of what a Gamma distribution represents (in particular for the sum of exponential random variables).

P(Z < 7) = Integral from (-infinity to 7) f(z)dz from the definition of a CDF of a continuous random variable.

Also I don't know whether you mean 36 factorial or just 36. The CLT concerns results of adding I.I.D random variables of the same distribution.

 

[JoskeJos]

I checked the PDF and everything cleared up.
Thank you for your time and help.

Btw I did mean factorial :-)