# Probability density function

## Probability density function

The length of time, in minutes, that a customer queues in a Post Office is a random variable, T, with probability density function

$f(t)=\left\{\begin{matrix}&space;c(81-t^2)&space;&&space;0\leqslant&space;t\leqslant&space;9\\&space;0&&space;\text{otherwise}&space;\end{matrix}\right.$

where c is a constant

(a) Show that the value of c is $\inline&space;\frac{1}{486}$

(b) Show that the cumulative distribution function F(t) is given by

$F(t)=\left\{\begin{matrix}&space;0&space;&&space;t<&space;0\\&space;\frac{t}{6}&space;-\frac{t^3}{1458}&&space;0\leqslant&space;t\leqslant&space;9\\&space;1&&space;t>&space;9&space;\end{matrix}\right.$

(c) Find the probability that a customer will queue for longer than 3 minutes.

A customer has been queueing for 3 minutes.

(d) Find the probability that this customer will be queueing for at least 7 minutes.

Three customers are selected at random.

(e) Find the probability that exactly 2 of them had to queue for longer than 3 minutes.

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