# PDF Questions - expected value? 2nd moment? variance?

I am trying to figure out how to solve the questions above for the given PDF but im not sure how to do the steps.

I'm not looking for answers here as I want to know how to do it, just looking for the equations and steps for each part.

I know the standard deviation is the square root of the variance correct?

• Can you show what you've done so far? Can you write out the formula for the $n^{th}$ moment $\mathbb{E}[Y^n]$ in terms of an integral? – Rookatu Feb 19 '14 at 19:26

(a) graph the continuous function from -1 to 1

(b) The definition of E(x) for continuous functions is $\int_a^b xf(x)~dx$. In your case this is y instead.

(c) same as b, instead use $\int_a^b x^2f(x)~dx$

(d) use answer in c, subtract the square of answer in b (definiton of variance is $E(x^2) - [E(x)]^2$)

In general, for a pdf $f_Y(y)$, the expected value of $Y$ is $E[Y] = \int_{-\infty}^\infty yf_Y(y)\ dy$.
More generally, the expected value $E[g(Y)]$ is $\int_{-\infty}^\infty g(y)f_Y(y)\ dy$. And so forth...