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

(a) $E[(2+X)^2]$

(b) $Var(4+3X)$

If E

hansel

Answered by Aaron 2 years ago

**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,

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

Surround your text in `*italics*`

or `**bold**`

, to write a math equation use, for example, `$x^2+2x+1=0$`

or `$$\beta^2-1=0$$`

Stats

Views: 60

Asked: 2 years ago