21M.2.AHL.TZ2.6

pestleMathematicsAAHLPaper 221M· ahl-4-14-properties-of-discrete-and-continuous-random-variablessource ↗

A continuous random variable $X$ has the probability density function $f_{n}$ given by

$f_{n} (x) = \{\begin{array}{l} (n + 1) x^{n}, \\ 0, \end{array} \begin{matrix} 0 \leq x \leq 1 \\ otherwise \end{matrix}$

where $n \in ℝ, n \geq 0$ .

Show that $E (X) = \frac{n + 1}{n + 2}$ .

[2]

Show that $Var (X) = \frac{n + 1}{{(n + 2)}^{2} (n + 3)}$ .

[4]

Markscheme / solution

$E (X) = (n + 1) \int_{0}^{1} x^{n + 1} d x$ M1

$= (n + 1) {[\frac{x^{n + 2}}{n + 2}]}_{0}^{1}$ A1

leading to $E (X) = \frac{n + 1}{n + 2}$ AG

[2 marks]

METHOD 1

use of $Var (X) = E (X^{2}) - {[E (X)]}^{2}$ M1

$Var (X) = (n + 1) \int_{0}^{1} x^{n + 2} d x - {(\frac{n + 1}{n + 2})}^{2}$

$= (n + 1) {[\frac{1}{n + 3} x^{n + 3}]}_{0}^{1} - {(\frac{n + 1}{n + 2})}^{2}$

$= \frac{n + 1}{n + 3} - {(\frac{n + 1}{n + 2})}^{2}$ A1

$= \frac{(n + 1) {(n + 2)}^{2} - {(n + 1)}^{2} (n + 3)}{{(n + 2)}^{2} (n + 3)}$ M1

EITHER

$= \frac{(n + 1) (n^{2} + 4 n + 4 - (n^{2} + 4 n + 3))}{{(n + 2)}^{2} (n + 3)}$ A1

$= \frac{(n^{3} + 5 n^{2} + 8 n + 4) - (n^{3} + 5 n^{2} + 7 n + 3)}{{(n + 2)}^{2} (n + 3)}$ A1

THEN

so $Var (X) = \frac{n + 1}{{(n + 2)}^{2} (n + 3)}$ AG

METHOD 2

use of $Var (X) = E {(X - E (X))}^{2}$ M1

$Var (X) = (n + 1) \int_{0}^{1} {(x - \frac{n + 1}{n + 2})}^{2} x^{n} d x$

$= (n + 1) {[\frac{1}{n + 3} x^{n + 3} - \frac{2 (n + 1)}{{(n + 2)}^{2}} x^{n + 2} + \frac{n + 1}{{(n + 2)}^{2}} x^{n + 1}]}_{0}^{1}$

$= \frac{n + 1}{n + 3} - {(\frac{n + 1}{n + 2})}^{2}$ A1

$= \frac{(n + 1) ({(n + 2)}^{2} - (n + 1) (n + 3))}{{(n + 2)}^{2} (n + 3)}$ M1

EITHER

$= \frac{(n + 1) (n^{2} + 4 n + 4 - (n^{2} + 4 n + 3))}{{(n + 2)}^{2} (n + 3)}$ A1

$= \frac{(n^{3} + 5 n^{2} + 8 n + 4) - (n^{3} + 5 n^{2} + 7 n + 3)}{{(n + 2)}^{2} (n + 3)}$ A1

THEN

so $Var (X) = \frac{n + 1}{{(n + 2)}^{2} (n + 3)}$ AG

[6 marks]

Examiners’ report

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