Edexcel IAL Mathematics S2.2.4 mean and variance of continuous random variables

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)

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

M1A1

$=\left(\frac{2 \times 1.25^{3}}{3}-\frac{3}{8} \times 1.25^{2}\right)-\left(\frac{2 \times 0.5^{3}}{3}-\frac{3}{8} \times 0.5^{2}\right)$

dM1

$=\frac{93}{128}$

A1

(4)

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S2.2.4 - Mean and variance of continuous random variables question 1