| Exam Board | SPS |
|---|---|
| Module | SPS FM Statistics (SPS FM Statistics) |
| Year | 2024 |
| Session | September |
| Marks | 10 |
| Topic | Geometric Distribution |
| Type | Variance of geometric distribution |
| Difficulty | Moderate -0.3 This question tests standard properties of variance and expectation with linear transformations, plus basic geometric distribution formulas. Part (a) uses Var(aX+b)=a²Var(X), part (b) requires finding p from the given variance then computing E(3D+5), and part (c) applies the geometric distribution probability formula. All steps are routine applications of memorized formulas with straightforward algebra, making it slightly easier than average. |
| Spec | 5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)5.02h Geometric: mean 1/p and variance (1-p)/p^25.04a Linear combinations: E(aX+bY), Var(aX+bY) |
3.
The random variable $D$ has the distribution $\operatorname { Geo } ( p )$. It is given that $\operatorname { Var } ( D ) = \frac { 40 } { 9 }$.\\
Determine
\begin{enumerate}[label=(\alph*)]
\item $\operatorname { Var } ( 3 D + 5 )$,
\item $\mathrm { E } ( 3 D + 5 )$,
\item $\mathrm { P } ( D > \mathrm { E } ( D ) )$.\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Statistics 2024 Q3 [10]}}