Continuous random variables — WST02 IAL Mathematics practice questions · Study-IAL

Continuous random variables

WST02Topic 2.23 questions

Probability density and cumulative distribution functions, mean, variance, median and mode.

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Question 17 marks
The continuous random variable $X$ has probability density function $f(x) = \begin{cases} \dfrac{x}{8} & 0 \le x \le 4 \\ 0 & \text{otherwise.} \end{cases}$

(a) Find $E(X)$ and $\text{Var}(X)$ .

(b) Find the cumulative distribution function $F(x)$ for $0 \le x \le 4$ , and hence find the median of $X$ .

(c) Write down the mode of $X$ .
Mark scheme
- (a) M1 $E(X) = \displaystyle\int_0^4 \dfrac{x^2}{8}\,dx$ (1)
Question 24 marks
The continuous random variable $X$ has cumulative distribution function $F(x) = \dfrac{x^2 + 2x}{15}$ for , with for and for .

E(X) = \tfrac{8}{3}

(1)

E(X^2) = \displaystyle\int_0^4 \dfrac{x^3}{8}\,dx = 8

(1)

\text{Var}(X) = 8 - \left(\tfrac{8}{3}\right)^2 = \tfrac{8}{9}

(1)

(b)

F(x) = \dfrac{x^2}{16}

(1)

\dfrac{m^2}{16} = 0.5

, so median

m = 2\sqrt{2} = 2.83

(1)

= 4

(1)

0 \le x \le 3

F(x) = 0

x < 0

F(x) = 1

x > 3

(a) Find the probability density function of $X$ .

(b) Find $P(1 < X < 2)$ .

Mark scheme

(a) $f(x) = F'(x) = \dfrac{2x + 2}{15}$ for $0 \le x \le 3$ (and 0 otherwise) (1)
(b) M1 $P(1 < X < 2) = F(2) - F(1) = \tfrac{8}{15} - \tfrac{3}{15}$ (1)
A1 $= \tfrac{1}{3}$ (1)
(c) $m^2 + 2m = 7.5$ gives $m = -1 + \sqrt{8.5} = 1.92$ (1)

158

−

153

8.5

1.92