Question

2. Let us assume that the population X has the mean μ and the variance σ2 and the population Y h 2σ. If X and Y are independent, express the following quantities by and ơ as the mean u and the variance (2.2) V[X-Y] (2.3) V[2X+3Y] (2.4) VIX-3Y-5]

Answer 1

Question 2

Here as we know that

E[aX + bY] = a E[X] + b E[Y]

V[aX + bY] = a²V[x] + b²V[Y]

V[aX - bY] = a²V[x] + b²V[Y]

(a) E[X -2] = E[X] - 2 = $\mu$ -2

(b) V[X - Y] = V[X] + V[Y] = $\sigma^2 + 2 * \sigma^2$ = 3$\sigma^2$

(c) V[2X + 3Y] = 2²V[X] + 3² V[Y] = 4$\sigma^2$ + 9 * 2$\sigma^2$ = 22 $\sigma^2$

(d) V[X - 3Y - 5] = V[X] + 3² V[Y] = $\sigma^2$ + 9 * 2 $\sigma^2$ = 19 $\sigma^2$