| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Find constant k in PDF |
| Difficulty | Easy -1.3 This is a straightforward application of the basic property that a PDF must integrate to 1, requiring only simple integration of a constant over an interval. Part (a) involves elementary algebra (k × 10 = 1), and part (b) is direct substitution into the uniform CDF. This is below average difficulty as it tests only definitional knowledge with minimal calculation. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03e Find cdf: by integration |
| Answer | Marks |
|---|---|
| (a) \(k = \frac{1}{10}\), so \(F(x) = 0\) \((x < 5)\), \(F(x) = \frac{x-5}{10}\) \((5 \le x \le 15)\), \(F(x) = 1\) \((x > 15)\) | B1 B1 M1 A1 |
| (b) \(P(X < 8) = F(8) = \frac{3}{10}\) | B1 |
(a) $k = \frac{1}{10}$, so $F(x) = 0$ $(x < 5)$, $F(x) = \frac{x-5}{10}$ $(5 \le x \le 15)$, $F(x) = 1$ $(x > 15)$ | B1 B1 M1 A1 |
(b) $P(X < 8) = F(8) = \frac{3}{10}$ | B1 |
**Total: 6 marks**2. A continuous random variable $X$ has the probability density function
$$\begin{array} { l l }
\mathrm { f } ( x ) = k & 5 \leq x \leq 15 , \\
\mathrm { f } ( x ) = 0 & \text { otherwise. }
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item Find $k$ and specify the cumulative density function $\mathrm { F } ( x )$.
\item Write down the value of $\mathrm { P } ( X < 8 )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q2 [6]}}