Edexcel A-Level Mathematics A2 S2.2.4 Mean and Variance of Continuous Random Variables Questions

Practise continuous-variable moments by integrating density functions to find E(X), E(X²), variance and transformed expectations.

Exam points

Integrate xf(x) and x²f(x) over each support interval before applying Var(X).
Use E(aX+bY+c) or a stated variance to form equations for unknown constants.

[Maximum number: 4]

A random variable X has probability density function given by

f(x)=\left\{\begin{array}{cc} \frac{1}{4} & -\frac{1}{2} \leqslant x<\frac{1}{2} \\ 2 x-\frac{3}{4} & \frac{1}{2} \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{array}\right.

where k is a positive constant.

Use calculus to find E(X)

$\left[\int_{-0.5}^{0.5} \frac{1}{4} x \mathrm{~d} x\right]+\int_{0.5}^{1.25} 2 x^{2}-\frac{3}{4} x \mathrm{~d} x=[0]+\left[\frac{2 x^{3}}{3}-\frac{3}{8} x^{2}\right]_{0.5}^{1.25}$ $=\left(\frac{2 \times 1.25^{3}}{3}-\frac{3}{8} \times 1.25^{2}\right)-\left(\frac{2 \times 0.5^{3}}{3}-\frac{3}{8} \times 0.5^{2}\right)$ $=\frac{93}{128}$
(4)

All question bank results loaded

Edexcel A-Level Mathematics A2 S2.2.4 Mean and Variance of Continuous Random Variables Questions question 1