The diagram shows the graph of the probability density function, f, of a random variable X. The graph is a quarter circle entirely in the first quadrant with centre (0,0) and radius a, where a is a positive constant. Elsewhere f(x)=0.

(a)

Show that $a=\frac{2}{\sqrt{\pi}}$ .

[ 2 ]

(b)

Show that $\mathrm{f}(x)=\sqrt{\frac{4}{\pi}-x^{2}}$ .

[ 2 ]

(c)

Show that $\mathrm{E}(X)=\frac{8}{3 \sqrt{\pi^{3}}}$ .

[ 4 ]

[Maximum number: 10]

The graph of the probability density function f of a random variable X is symmetrical about the line x=2. It is given that $\mathrm{P}(2<X<5)=\frac{117}{256}$ .

(a)

Using only this information show that $\mathrm{P}(X>-1)=\frac{245}{256}$ .
It is now given that, for x in a suitable domain,

\mathrm{f}(x)=k\left(12+4 x-x^{2}\right), \text { where } k \text { is a constant. }

[ 2 ]

(b)

Find the value of k.

[ 3 ]

(c)

A different random variable X has probability density function $\mathrm{g}(x)=\frac{2}{9}\left(2+x-x^{2}\right)$ . The domain of X is all values of x for which $\mathrm{g}(x) \geqslant 0$ .

Find $\operatorname{Var}(X)$ .

[ 5 ]

[Maximum number: 10]

The probability density function, f, of a random variable X is given by

f(x)= \begin{cases}k(1+\cos x) & 0 \leqslant x \leqslant \pi \\ 0 & \text { otherwise }\end{cases}

where k is a constant.

(a)

Show that $k=\frac{1}{\pi}$ .

[ 3 ]

(b)

Verify that the median of X lies between 0.83 and 0.84 .

[ 3 ]

(c)

Find the exact value of E(X).

[ 4 ]

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