| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2014 |
| Session | June |
| Marks | 6 |
| Topic | Discrete Probability Distributions |
| Type | Two unknowns from sum and expectation |
| Difficulty | Moderate -0.3 This is a straightforward probability distribution question requiring basic recall of fundamental properties: probabilities sum to 1, definition of expectation, and variance formula. All three parts are direct applications of standard formulas with minimal algebraic manipulation, making it slightly easier than a typical A-level question. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| \(x\) | 1 | 2 | \(n\) | 7 |
| \(\mathrm{P}(X = x)\) | 0.4 | 0.3 | \(p\) | 0.1 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(p = 1 - (0.4 + 0.3 + 0.1) = 0.2\) | B1 | [1] |
| (ii) \((1 \times 0.4) + (2 \times 0.3) + (n \times 0.2) + (7 \times 0.1) = 2.5\) | M1 | Use of formula for \(E(X)\) s.o.i. to set up an equation in \(n\). |
| \(\therefore 0.2n + 1.7 = 2.5\) | A1 | c.a.o. |
| \(\therefore 0.2n = 0.8\) | ||
| \(\therefore n = 4\) | A1 | [2] |
| (iii) \(E(X^2) = (1^2 \times 0.4) + (2^2 \times 0.3) + (4^2 \times 0.2) + (7^2 \times 0.1) = 9.7\) | B1 M1 | Correct expression for \(E(X^2)\) s.o.i. ft c's \(n\). Use of formula for \(\text{Var}(X)\) s.o.i. |
| \(\text{Var}(X) = 9.7 - 2.5^2 = 3.45\) | A1 | c.a.o. [3] |
**(i)** $p = 1 - (0.4 + 0.3 + 0.1) = 0.2$ | B1 | [1]
**(ii)** $(1 \times 0.4) + (2 \times 0.3) + (n \times 0.2) + (7 \times 0.1) = 2.5$ | M1 | Use of formula for $E(X)$ s.o.i. to set up an equation in $n$.
$\therefore 0.2n + 1.7 = 2.5$ | A1 | c.a.o.
$\therefore 0.2n = 0.8$ |
$\therefore n = 4$ | A1 | [2]
**(iii)** $E(X^2) = (1^2 \times 0.4) + (2^2 \times 0.3) + (4^2 \times 0.2) + (7^2 \times 0.1) = 9.7$ | B1 M1 | Correct expression for $E(X^2)$ s.o.i. ft c's $n$. Use of formula for $\text{Var}(X)$ s.o.i.
$\text{Var}(X) = 9.7 - 2.5^2 = 3.45$ | A1 | c.a.o. [3]A discrete random variable $X$ has the following probability distribution.
\begin{tabular}{|c|c|c|c|c|}
\hline
$x$ & 1 & 2 & $n$ & 7 \\
\hline
$\mathrm{P}(X = x)$ & 0.4 & 0.3 & $p$ & 0.1 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\roman*)]
\item Write down the value of $p$. [1]
\item Given that $\mathrm{E}(X) = 2.5$, find $n$. [2]
\item Find $\mathrm{Var}(X)$. [3]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2014 Q3 [6]}}