(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Log-ons to a certain computer website occur randomly at a uniform average rate of 2.4 per minute. State the distribution of the number N of log-ons that occur during a period oftminutes.

Obtain the probablity that at least one log on occurs during a period oftminutes.

Hence obtain the probability density function of T, where T minutes is the interval between successive log-ons.

Identify the distribution of T and state its mean and variance.

2. Relevant equations

3. The attempt at a solution

This one's a bit embarrassing really. What I don't get is the "Hence obtain the probability density function of T, where T minutes is the interval between successive log-ons." part. It's only 3 marks for this section of the question so it's probably something simple. I must be missing some key fact or something.

Here's what I did for the first bit.

The distribution of N is Poisson with a mean of 2.4t so X~Po(2.4t)

The probability of at least one log on is P(X≥1) = 1 - P(X=0) = 1 - e^{-2.4t}

And now I am stuck. This is from a really old A-Level Further Mathematics exam so I only had an examiner's report which gave the answers with a short and confusing explanation. It said that "The probability density function is obtained by taking the derivative of their answer 1 - e^{-2.4t}. Which gives a negative exponential distribution 2.4e^{-2.4t}. The mean and variance is easily 1/2.4 and (1/2.4)^{2}respectively."

What I don't understand iswhythe probablity density function is the derivative of that expression.

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# Probability Distribution Problem

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