If $X$ has mean $3$ and variance $8$, find $E\left[(3 + 2X)^2\right]$ and $E(4 + 7X)$.

Question

this question is throwing me off and my textbook doesn't include any solutions, or any examples relating to these kinds of question so any help on a solutuion would be great

If $X$ has mean $3$ and variance $8$, find $E\left[(3 + 2X)^2\right]$ and $E(4 + 7X)$.

Answer 1

By linearity of $E$, $$\begin{align} E[(3+2X)^2]&=E[9+12X+4X^2]\\& = E[9]+E[12X]+E[4X^2]\\&=9+12E[X]+4E[X^2].\end{align}$$ Now recall that $$V[X] = E[X^2]-E[X]^2. $$

Answer 2

hint: expand everything, use the linearity of $E$ and the relations $$ E[1] = 1\\ \text{var }X = E[X^2] - E[X]^2 $$

Answer 3

You know that:

$$E[X]=3$$

and

$$Var[X]=E[X^2]-E[X]^2=8$$

So

$$E[X^2]=17$$

Now using linearity of $E$ you have:

$$E[(3+2X)^2]=E[9+12X+4X^2]=9E[1]+12E[X]+9E[X^2]=9+12 \cdot 3 + 9 \cdot 17$$

$$E[4+7X]=4E[1]+7E[X]=4+7 \cdot 3.$$