Q.4.38 - A First Course in Probability

Question

If E[X] = 1 and Var(X) = 5, find 

(a) E[(2 + X)2]; 

(b) Var(4 + 3X). 

Step-by-Step Solution

Answer

In the given information the answers are (a) $E\left({\left(2+X\right)}^2\right)=14$

(b) $Var\left(4+3X\right)=45$

Step 1: 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

Step 2:Calculation (Part-a)

                  $E\left((2+X{)}^2\right)=E\left(4+4X+X^2\right)=4+4E\left(X\right)+E\left(X^2\right)$

              $=4+4E\left(X\right)+Var\left(X\right)+E(X{)}^2=14$

Where we have used the identity$E\left(X^2\right)=Var\left(X\right)+E(X{)}^2$

Step 3: Final answer (Part-a)

The final Answer is $E\left({\left(2+X\right)}^2\right)=$$14$.

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

 $$\begin{gathered} .Var\left(CX\right)=C^2 \\ .Var\left(aX+b\right)=a^2 \end{gathered}$$

Step 5:Calculation(Part-b)

$Var\left(4+3X\right)=Var\left(3X\right)$

                   = $9Var\left(X\right)$

                    $=45$

Step 6:Final Answer (Part-b)

The final answer is $Var\left(4+3X\right)=$$45$