Question

(a) A random variable X has the following probability function

.x : -2 -1 0 1 2 3

p(x): 0.1 k 0·2 2k 0·3 k

(i) Find the value of k. and calculate mean and variance.

(ii) construct the cumulative distribution function c.d.f F(x) and draw its graph.

Answer 1

Solution to i

$\sum p(x)=1$

Thus;

$$0.1+k+0.2+2k+0.3+k=1$$ $$0.6+4k=1 \therefore 4k=0.4$$ $$k=0.1$$

Mean, $\bar{x}$

$\bar{x}=\sum xp(x)$

$$=-2(0.1)-1(0.1)+0(0.2)+1(0.2)+2(0.3)+3(0.1)$$ $$=0.8$$

variance

$var= \sum x^2p(x)- {\bar{x}}^2$

$\sum x^2p(x)$ $=4(0.1)+1(0.1)+0(0.2)+1(0.2)+4(0.3)+9(0.1)$

$=2.8$

$$var=2.8-({0.8^2})=2.16$$

Solution to ii

Cumulative Distribution Function

Cumulative Distribution Graph