Question

if x and y are independent random variable with means 2, 3 and variance 1, 2 respectively. find the mean and variance of the random variable z=2x-3y.

Answer 1

SOLUTION

GIVEN

x and y are independent random variable with means 2, 3 and variance 1, 2 respectively

TO DETERMINE

The mean and variance of the random variable z = 2x - 3y

FORMULA TO BE IMPLEMENTED

1. E( ax + by ) = a E(x) + b E(y)

$\sf{2. \: var(ax + by) = {a}^{2} \: var(x) + {b}^{2} \: var(y) }$

EVALUATION

Here it is given that x and y are independent random variable with means 2, 3 and variance 1, 2 respectively

∴ E(x) = 2 & E(y) = 3

∴ var(x) = 1 & var(y) = 2

Now the random variable z is defined by

z = 2x - 3y

CALCULATION OF MEAN

Mean

$\sf{ = E(z)}$

$\sf{ = E(2x - 3y)}$

$\sf{ = 2E(x) - 3E(y)}$

$=( 2 \times 2) - (3 \times 3)$

$= - 5$

CALCULATION OF VARIANCE

Variance

$= \sf{var(z)}$

$= \sf{var(2x - 3y)}$

$= \sf{ {2}^{2} \: var(x) + {( - 3)}^{2} \: var(y) }$

$= \sf{ 4 \: var(x) + 9 \: var(y) }$

$= (4 \times 1) + ( 9 \times 2)$

$= 4 + 18$

$= 22$

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