| Exam Board | SPS |
|---|---|
| Module | SPS FM Statistics (SPS FM Statistics) |
| Year | 2021 |
| Session | May |
| Marks | 14 |
| Topic | Exponential Distribution |
| Type | Expectation of function of X |
| Difficulty | Standard +0.8 This is a multi-part Further Maths Statistics question requiring: (a) transformation of CDFs using Y=2T, (b) deriving the moment generating function with convergence conditions, and (c) connecting exponential and Poisson distributions conceptually. Part (b) requires integration by parts and understanding of MGF convergence, while part (c) requires recognizing the memoryless property link between distributions. More demanding than standard A-level but routine for FM Statistics students who know MGFs. |
| Spec | 2.04b Binomial distribution: as model B(n,p)5.03d E(g(X)): general expectation formula5.03e Find cdf: by integration |
5. The continuous random variable $T$ has cumulative distribution function
$$\mathrm { F } ( t ) = \begin{cases} 0 & t < 0 , \\ 1 - \mathrm { e } ^ { - 0.25 t } & t \geqslant 0 . \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Find the cumulative distribution function of $2 T$.
\item Show that, for constant $k , \quad \mathrm { E } \left( \mathrm { e } ^ { k T } \right) = \frac { 1 } { 1 - 4 k }$.
You should state with a reason the range of values of $k$ for which this result is valid.
\item $T$ is the time before a certain event occurs.
Show that the probability that no event occurs between time $T = 0$ and time $T = \theta$ is the same as the probability that the value of a random variable with the distribution $\operatorname { Po } ( \lambda )$ is 0 , for a certain value of $\lambda$. You should state this value of $\lambda$ in terms of $\theta$.\\[0pt]
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\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Statistics 2021 Q5 [14]}}