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

Practise integrating continuous densities to find means and variances, and use stated moments with verified Edexcel IAL questions.

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