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If E[X] = 1 and Var(X) = 5, find

(a) E[(2 + X)2];

(b) Var(4 + 3X).

Short Answer

Expert verified

In the given information the answers are (a) E2+X2=14

(b)Var4+3X=45

Step by step solution

01

Given Information(Part-a)

Linearity of expectation is the property that the expected value of the total of random variable is same to the total of their individual expected values, regardless of whether they are independent

02

Step 2:Calculation (Part-a)

E(2+X)2=E4+4X+X2=4+4E(X)+EX2

=4+4E(X)+Var(X)+E(X)2=14

Where we have used the identityEX2=Var(X)+E(X)2

03

Final answer (Part-a)

The final Answer is E2+X2=14.

04

Step 4:Given Information (Part-b)

Linearity of expectation is the property that the expected value of the total of random variable is same to the total of their individual expected values, regardless of whether they are independent

.VarCX=C2.VaraX+b=a2

05

Step 5:Calculation(Part-b)

Var4+3X=Var3X

= 9VarX

=45

06

Step 6:Final Answer (Part-b)

The final answer isVar4+3X=45

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