Two continuous random variables X and Y are related as

Y = 2X + 3

Let  and  denote the variances of X and Y, respectively. The variances are related as

Concept:

Variance of a random variable ‘y’ is given by:

Var[y] = E[y²] – E²[y]

Properties of mean:

1) E[K] = K, Where K is some constant

2) E[c X] = c. E[X], Where c is some constant

3) E[a X + b] = a E[X] + b, Where a and b are constants

4) E[X + Y] = E[X] + E[Y]

Application:

Variance of y = E[(2x + 3)²] – (E[2x + 3])²

= E[4x² + 12x + 9] – (E[2x + 3])²

= 4E[x²] + 12E[x] + 9 – (E[2x] + E[3])²

= 4E[x2] + 12E[x] + 9 – (2E[x] + 3)2

= 4E[x2] + 12E[x] + 9 – (4E²[x] + 9 + 12E[X])

= 4E[x2] + 12E[x] + 9 – 4E2[x] – 9 – 12E[X]

= 4E[x2] – 4E2[x]

= 4[E[x²] – E²[x]]

This can be written as:

= 4 (variance of x), i.e.

The variance of y = 4 times the variance of x

Properties of Variance:

1) V[K] = 0, Where K is some constant.

2) V[cX] = c2 V[X]

3) V[aX + b] = a2 V[X]

4) V[aX + bY] = a2 V[X] + b2 V[Y] + 2ab Cov(X,Y)

Cov.(X,Y) = E[XY] - E[X].E[Y]