| Exam Board | SPS |
|---|---|
| Module | SPS FM Statistics (SPS FM Statistics) |
| Year | 2022 |
| Session | February |
| Marks | 4 |
| Topic | Discrete Probability Distributions |
| Type | Calculate E(X) from given distribution |
| Difficulty | Easy -1.3 This is a straightforward application of standard formulas for E(X) and Var(X) from a discrete probability distribution, followed by simple linear transformations. All values are given in a table requiring only arithmetic calculations (∑xP(x) and ∑x²P(x)), with no problem-solving, conceptual insight, or algebraic manipulation needed. This is below average difficulty even for A-level, being purely computational. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| \(r\) | 2 | 3 | 4 | 5 | 6 | 7 |
| \(\mathrm { P } ( X = r )\) | 0.03 | 0.07 | 0.27 | 0.49 | 0.13 | 0.01 |
\begin{enumerate}
\item The random variable $X$ represents the clutch size (the number of eggs laid) by female birds of a particular species. The probability distribution of $X$ is given in the table.
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
$r$ & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
$\mathrm { P } ( X = r )$ & 0.03 & 0.07 & 0.27 & 0.49 & 0.13 & 0.01 \\
\hline
\end{tabular}
\end{center}
(a) Find each of the following.
\begin{itemize}
\item $\mathrm { E } ( X )$
\item $\operatorname { Var } ( X )$
\end{itemize}
On average $65 \%$ of eggs laid result in a young bird successfully leaving the nest.\\
(b) (i) Find the mean number of young birds that successfully leave the nest.\\
(ii) Find the standard deviation of the number of young birds that successfully leave the nest.\\[0pt]
\\
\hfill \mbox{\textit{SPS SPS FM Statistics 2022 Q1 [4]}}