Question

p(x) = k x (x – 1), x = 1, 2, 3, 4.
Obtain the value of k
Construct the probability distribution table for x
Hence, calculate E(x), E(x2) and Var(x)

Answer 1

$$\sum p(x_i)=1; \\k(1*(1-1)+2*(2-1)+3*(3-1)+ \\+4*(4-1))=1; \\k=0.05;$$

$$\begin{matrix} x & 1 & 2 & 3 & 4 \\ p(x) & 0 & 0.1 & 0.3 & 0.6 \end{matrix}$$

$$E(x)=\sum p(x_i)x_i; \\E(x)=0*1+0.1*2+0.3*3+0.6*4=3.5;$$

$$E(x^2)=\sum p(x_i)x_i^2; \\E(x^2)=0*1+0.1*4+0.3*9+0.6*16=12.7;$$

$$Var(x)=E(x^2)-E(x)^2; \\Var(x)=12.7-12.25=0.45;$$