CAIE A-Level Mathematics 6.3.1 Continuous random variables

Practise recognising valid continuous distributions, using total area 1 and interpreting constants as limits or contextual bounds.

Exam points

use a diagram or integral limits to link constants defining a single interval

[Maximum number: 3]

A random variable X has probability density function f given by

f(x)= \begin{cases}a x-x^{3} & 0 \leqslant x \leqslant \sqrt{2} \\ 0 & \text { otherwise }\end{cases}

where a is a constant.

Show that a=2.

$\int_{0}^{\sqrt{2}}\left(a x-x^{3}\right) \mathrm{d} x=1$

M1

Attempted integration of f(x) and equated to 1.

$\left[a \frac{x^{2}}{2}-\frac{x^{4}}{4}\right]_{0}^{\sqrt{2}}=1$ $a-\frac{4}{4}=1$

A1

Correct integration and substitute correct limits.

a=2

A1

AG Convincingly obtained and no errors seen.

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6.3.1—Continuous random variables question 1