Question

X and Y are two independent continuous random variables with variances 10 and 5 respectively. Another random variable Z is defined as Z=(X+Y-5). If the mean values of X and Y are 1 and 2 respectively, then the variance of Z will be_____.

• A. 15

Answer 1

The correct option is A 15
$\overline{Z}=E\left[Z\right]=E[X+Y-5]=\overline{X}+\overline{Y}-5$

${\sigma }_Z^2=E\left[\right(Z-\overline{Z}{)}^2]=E[\left(\right(X+Y-5)-(\overline{X}+\overline{Y}-5\left){)}^2\right]$

$=E\left[\right(X+Y-\overline{X}-\overline{Y}{)}^2]=E[\left(\right(X-\overline{X})+(Y-\overline{Y}\left){)}^2\right]$

$=E\left[\right(X-\overline{X}{)}^2+(Y-\overline{X}{)}^2+2(X-\overline{X}\left)\right(Y-\overline{Y}\left)\right]$

$=E\left[\right(X-\overline{X}{)}^2]+E[(Y-\overline{Y}{)}^2]\because E[X-\overline{X}]=E[Y-\overline{Y}]=0$

$\text{So,}{\sigma }_Z^2={\sigma }_X^2+{\sigma }_Y^2=10+5=15$