| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2017 |
| Session | June |
| Marks | 14 |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Piecewise PDF with k |
| Difficulty | Standard +0.3 This is a straightforward piecewise PDF question requiring standard techniques: integrating to find the constant a (using ∫f(x)dx = 1), constructing the CDF by integrating each piece, and comparing F(0.25) to 0.75 to determine the upper quartile position. All steps are routine applications of core probability theory with no novel insight required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
Total: 10 marks
**Part (i):**
Use total area $= 1$ with integration **M1**
$a + a\left[1 - \frac{1}{3}\right] = 1$ **A1** Correct integration
$a = \frac{3}{5}$ **A1** 0.6 or $\frac{3}{5}$ only
**Part (ii):**
Integrate for one non-zero region ($a$ can remain, constants can be omitted) **M1**
$$F(x) = \begin{cases} \frac{3}{5}(x+1) & -1 \leqslant x < 0 \\ \frac{3}{5}\!\left(x - \dfrac{x^3}{3}+1\right) & 0 \leqslant x \leqslant 1 \\ 0 & x < -1 \\ 1 & x > 1 \end{cases}$$
**A1** One formula correct; **A1** Other formula correct; **A1** Ranges $-1 \leqslant x < 0$ and $0 \leqslant x \leqslant 1$ correct, allow $\leqslant$/$<$; **B1** 0 and 1. (Total **4** marks for this part)
**Part (iii):**
$F(0.25) = 0.746875 < 0.75$ **M1** Evaluate $F(0.25)$ and compare with 0.75
$UQ > 0.25$ **A1** Correct conclusion from correct values. Alt method by finding UQ: $4u^3 - 12u + 3 = 0$ (oe) M1; Correct solution (e.g. by GC $u = 0.25556...$) (or sign change to demonstrate root between 0.25 and 1) and correct conclusion A1. Alt method: $u = 0.25 + u^3/3 > 0.25$ and conclusion B2.
Total: **10 marks**4 A continuous random variable $X$ has probability density function
$$\mathrm { f } ( x ) = \begin{cases} a & - 1 \leqslant x < 0 \\ a \left( 1 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$
where $a$ is a constant.\\
(i) Find the value of $a$.\\
(ii) Find the cumulative distribution function of $X$.\\
(iii) Determine whether the upper quartile is greater than or less than 0.25 .
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2017 Q4 [14]}}