Edexcel A-Level Mathematics A2 S2.2.2 Probability Density and Cumulative Distribution Functions Questions

Practise using probability density functions and cumulative distribution functions to find probabilities, constants and values of k.

Exam points

integrate f(x) over the support to find constants or prove a density function is valid
calculate interval probabilities using ∫f(x) dx or differences

[Maximum number: 6]

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.

(a)

Sketch the graph of f(x)

[ 2 ]

(b)

By forming and solving an equation in k, show that k=1.25

[ 4 ]

Area $=\frac{1}{4}+\frac{1}{2}\left(\frac{1}{4}+2 k-\frac{3}{4}\right)\left(k-\frac{1}{2}\right)=1$ or $\frac{1}{4}\left(k+\frac{1}{2}\right)+\frac{1}{2}(2 k-1)\left(k-\frac{1}{2}\right)=1$ or
$\frac{1}{4}+\left[\left(k^{2}-\frac{3}{4} k\right)-\left(\frac{1}{2}^{2}-\frac{3}{8}\right)\right]=1$

A1

$k=1.25^{*}$

A1 *

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Edexcel A-Level Mathematics A2 S2.2.2 Probability Density and Cumulative Distribution Functions Questions question 1

Question (a)

Question (b)