Y 3X 5 X has mean 0 and variance 2 What is VarX YSolution

Y = 3X + 5. X has mean 0 and variance 2. What is Var[X – Y]?

Solution

So To Find variance of a Give eq. if the variance, of one of the terms in the equation is given we use the formula

Z=aX+bY

So Variance of Z is

Var[Z]=Var[aX+bY]

Var[Z]=Var[aX]+Var[bY]

Var[Z]=a²Var[X]+b²Var[Y] and Variance of a Constant is Zero.

So for the Given Equation

Y=3X+5

Variance of Y will be

Var[Y]=Var[3X+5]

Var[Y]=Var[3X]+Var[5] So, Var[5]=0, as 5 is a Constant

Var[Y]=Var[3X]=3²Var[X]

Var[Y]=9Var[X] Var[X]=2 (Given in the Question)

So, Var[Y]=9*2=18

So We get, Var[Y]=18.

Now to find Var[X-Y]

Var[X-Y]=Var[X]+Var[-Y]

Var[X-Y]=Var[X]+(-1)²Var[Y] As b=-1 Here

Var[X-Y]=Var[X]+Var[Y]

Var[X-Y]=2+18=20

So Var[X-Y]=20 So, Answer is 20.

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