Studdy AI Logo
Math

QuestionFind E((X+2Y)2)E\left((X+2 Y)^{2}\right) given E(X)=6E(X)=6, Var⁡(X)=1\operatorname{Var}(X)=1, E(Y)=3E(Y)=3, Var⁡(Y)=9\operatorname{Var}(Y)=9.

Studdy Solution

STEP 1

Assumptions1. The expectation of X,(X), is6. The variance of X, Var(X), is13. The expectation of Y,(Y), is34. The variance of Y, Var(Y), is95. The correlation between X and Y, Corr(X, Y), is not given6. The covariance between X and Y, Cov(X, Y), is defined as[(X-E[X])(Y-E[Y])]
7. The variance of X, Var(X), is also defined as Cov(X, X)

STEP 2

We are asked to find the value of((X+2Y)^2). Let's first expand this expression.
(X+2Y)2=X2+4XY+4Y2(X+2Y)^2 = X^2 +4XY +4Y^2

STEP 3

Now, we apply the expectation operator to each term in the expanded expression.
((X+2Y)2)=(X2)+(XY)+(Y2)((X+2Y)^2) =(X^2) +(XY) +(Y^2)

STEP 4

We can simplify this further by taking the constants out of the expectation operator.
((X+2Y)2)=(X2)+4(XY)+4(Y2)((X+2Y)^2) =(X^2) +4(XY) +4(Y^2)

STEP 5

We know that(X^2) can be calculated as Var(X) + [(X)]^2, and(Y^2) can be calculated as Var(Y) + [(Y)]^2. Let's substitute these into the equation.
((X+2Y)2)=Var(X)+[(X)]2+4(XY)+4[Var(Y)+[(Y)]2]((X+2Y)^2) = Var(X) + [(X)]^2 +4(XY) +4[Var(Y) + [(Y)]^2]

STEP 6

We also know that(XY) can be calculated as Cov(X, Y) +(X)(Y). Let's substitute this into the equation.
((X+2Y)2)=Var(X)+[(X)]2+4[Cov(X,Y)+(X)(Y)]+4[Var(Y)+[(Y)]2]((X+2Y)^2) = Var(X) + [(X)]^2 +4[Cov(X, Y) +(X)(Y)] +4[Var(Y) + [(Y)]^2]

STEP 7

Now we can substitute the given values for(X), Var(X),(Y), and Var(Y) into the equation.
((X+2Y)2)=1+(6)2+4[Cov(X,Y)+6∗3]+4[9+(3)2]((X+2Y)^2) =1 + (6)^2 +4[Cov(X, Y) +6*3] +4[9 + (3)^2]

STEP 8

We don't have the value for Cov(X, Y), but we know that Cov(X, Y) = Corr(X, Y) * sqrt[Var(X) * Var(Y)]. Since Corr(X, Y) is not given, we can't calculate Cov(X, Y). But we know that Cov(X, Y) =0 if X and Y are independent. So, let's assume X and Y are independent and Cov(X, Y) =0.
((X+2Y)2)=1+(6)2+4[0+6∗3]+4[+(3)2]((X+2Y)^2) =1 + (6)^2 +4[0 +6*3] +4[ + (3)^2]

STEP 9

Now, we can calculate the value of((X+2Y)^2).
((X+2Y)2)=+36+4[+18]+4[9+9]=37+72+72=181((X+2Y)^2) = +36 +4[ +18] +4[9 +9] =37 +72 +72 =181So, the value of((X+2Y)^2) is181.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord