Question

A discrete random variable has the following distribution
X 1 2 3 4
f(x) k 2k 2/3 k 1/3 k

i).Determine the value of k (2mks)
ii).Find the expectation of X (2mks)
iii).Find the variance of X

Answer 1

i) $\Sigma p_i=1$

$k+2k+\frac{2}{3}k+\frac{1}{3}k=1$

$4k=1, k=\frac{1}{4}$

ii) $E(X)=\Sigma x_ip_i=1*\frac{1}{4}+2*\frac{2}{4}+3*\frac{2}{12}+4*\frac{1}{12}=\frac{25}{12}$

iii)

$Var(X)=\Sigma x_i^2 p_i-(E(X))^2=1^2*\frac{1}{4}+2^2*\frac{2}{4}+3^2*\frac{2}{12}+\\+4^2*\frac{1}{12}-(\frac{25}{12})^2=\frac{107}{144}$