Question

let x be a random variable with E(x)=1 and e[x(x-1)]=4. find var(x)

Answer 1

We are given that,

$E(x)=1$ and $E(x(x-1))=4$

We can write $E(x(x-1))$ as,

$E(x(x-1))=E(x^2-x)=E(x^2)-E(x)=4$, but $E(x)=1$. So,$E(x^2)-E(x)=E(x^2)-1=4\implies E(x^2)=5$  

Now,

$var(x)=E(x^2)-(E(x))^2=5-(1)^2=5-1=4$

Therefore, var(x)=4.