If E[X]=1 and Var(X)=5, find (a) E[(2+X)^2] (b) Var(4+3X)

If E[X]=1 and Var(X)=5, find (a) E[(2+X)^2] (b) Var(4+3X)


H
Asked by 2 years ago
240 points

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

(a) $E[(2+X)^2]$
(b) $Var(4+3X)$

If E
hansel

1 Answer

Answered by 2 years ago
8.7k points

part (a)

use the following identity (where a and b are constants),

$$E[aX+b]=aE[X]+b$$

to get,

$$E[(2+X)^2]=E[(2+X)(2+X)]=E[4+4X+X^2]=4+4E[X]+E[X^2]$$

and since

$$Var(X)=E[X^2]-(E[X])^2 \longrightarrow E[X^2]=Var(X)+E[X]^2$$

$$E[X]^2=5+(1)^2=5$$

$$E[(2+X)^2]=4+4(1)+5=13$$

part (b)

use the identity,

$$\text{Var}(aX+b)=a^2\text{Var}(X)$$

to get,

$$Var(4+3X)=3^2Var(X)=9Var(X)=9 \cdot 5=45$$

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Asked: 2 years ago

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