CAIE A-Level Further Math 4.1.2 Expected values of functions

Practise finding expected values for functions of continuous random variables by integrating g(x)f(x) over the correct intervals.

[Maximum number: 5]

The continuous random variable X has probability density function f given by

f(x)= \begin{cases}0 & x<0 \\ \frac{6}{5} x & 0 \leqslant x \leqslant 1 \\ \frac{6}{5} x^{-4} & x>1\end{cases}

(a)

Given that E(X)=1, find the variance of X.

[ 3 ]

$\mathrm{E}\left(X^{2}\right)=\int_{0}^{1} \frac{6}{5} x^{3} \mathrm{~d} x+\int_{1}^{\infty} \frac{6}{5} x^{-2} \mathrm{~d} x$

1

M1

Attempt evaluation of the sum of these two
integrals

$\mathrm{E}\left(X^{2}\right)=1.5$

1

A1

Correct value seen or implied

$\operatorname{Var}(X)=1.5-1^{2}=0.5$

1

A1FT

Follow through their 1.5 providing variance is
positive

3

(b)

Find $\mathrm{E}(\sqrt{X})$ .

[ 2 ]

$\mathrm{E}(\sqrt{X})=\int_{0}^{1} \frac{6}{5} x^{\frac{3}{2}} \mathrm{~d} x+\int_{1}^{\infty} \frac{6}{5} x^{-\frac{7}{2}} \mathrm{~d} x$

1

M1

Attempt evaluation of the sum of these two
integrals

$\frac{24}{25}$ or 0.96

1

A1

2

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4.1.2—Expected values of functions question 1

Question (a)

Question (b)