E[X]= ∑* ( X*∗*P*(*X*) )

= 1 * (0.1) * (0.9)$^0$ + 2 * (0.1) * (0.9)$^1$ + 3 * (0.1) * (0.9)$^2$ + 4 * (0.1) * (0.9)$^3$ .......... upto infinity

E(x) = (0.1) { 1 * (0.9)$^0$ + 2 * (0.9)$^1$ + 3 * (0.9)$^2$ + 4 * (0.9)$^3$ .......... upto infinity }

let S = 1 * (0.9)$^0$ + 2 * (0.9)$^1$ + 3 * (0.9)$^2$ + 4 * (0.9)$^3$ .......... upto infinity

let k = 0.9 ===> **(1-k) = 0.1**, but note that k < 1

S = 1 * (k)$^0$ + 2 * (k)$^1$ + 3 * (k)$^2$ + 4 * (k)$^3$ .......... upto infinity **----------------- (1)**

==> kS = k * {1 * (k)$^0$ + 2 * (k)$^1$ + 3 * (k)$^2$ + 4 * (k)$^3$ .......... upto infinity}

kS = 1 * (k)$^1$ + 2 * (k)$^2$ + 3 * (k)$^3$ + 4 * (k)$^4$ .......... upto infinity **----------------- (2)**

**Perform (1) - (2)**

S - kS = 1 * (k)$^0$ + 1 * (k)$^1$ + 1 * (k)$^2$ + 1 * (k)$^3$ .......... upto infinity

S(1-k) = (k)$^0$ + (k)$^1$ + (k)$^2$ + (k)$^3$ + (k)$^4$ .......... upto infinity

S(1-k) = $\frac{1}{1-k}$ ------------- ( apply infinite sum of GP series where r < 1 )

S = $\frac{1}{(1-k)(1-k)}$

E(X) = 0.1 * S = $\frac{1}{(1-k)(1-k)}$ = 0.1 * $\frac{1}{(0.1)(0.1)}$ = $\frac{1}{(0.1)}$ = 10