6 The random variable \(T\) is the time, in suitable units, between two successive arrivals in a hospital casualty department. The probability density function of \(T\) is f , where $$\mathrm { f } ( t ) = \begin{cases} 0.2 \mathrm { e } ^ { - 0.2 t } & t \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$ State the expected value of \(T\). Write down the distribution function of \(T\) and find \(\mathrm { P } ( T > 10 )\).

## Question 6:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $E(T) = \frac{1}{0.2} = 5$ | B1 | State or find $E(T)$ |
| $F(t) = 1 - \exp(-0.2t)$ for $t \geq 0$ | B1 | State or find distribution function |
| $= 0$ (otherwise, $t < 0$) | B1 | |
| $P(T > 10) = 1 - F(10) = 1 - (1 - e^{-2})$ | M1 | Find $P(T > 10)$ |
| $= e^{-2}$ or $0.135$ | A1 | |

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6 The random variable $T$ is the time, in suitable units, between two successive arrivals in a hospital casualty department. The probability density function of $T$ is f , where

$$\mathrm { f } ( t ) = \begin{cases} 0.2 \mathrm { e } ^ { - 0.2 t } & t \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$

State the expected value of $T$.

Write down the distribution function of $T$ and find $\mathrm { P } ( T > 10 )$.

\hfill \mbox{\textit{CAIE FP2 2013 Q6 [5]}}