I have a question:
I need to prove variance can be infinite, then that variance can be greater or equal to zero.
For the second question I proceeded like this: $Y=X-\mu$
$\phi(X)=X^2$
Then, using Jensen's inequality I get
$\phi(E[Y]) \le E [\phi(Y)]$
$(E[Y])^2 \le E[(X-\mu)^2]$
Since the left hand side is equal to $0$ then I know that variance has to be greater or equal to zero
For the first question I don't know ho to proceed. I mean, I know how to do the reverse, if I have a r.v $X \in L^p$ with $p=3$
then I know that this rv has third moment finite so it will also have second moment (variance) finite. But I don't know how to prove that the variance of a general r.v can be infinite.
Can you please help me?
You don't need Jensen's inequality for the second question. $(X-\mu)^{2} $ is a non-negative random variable and that implies that $E(X-\mu)^{2} \geq 0$.
Let $X$ take the values $1,2,...$ and $P(X=n)=\frac c {n^{2.5}}$ where $c$ is chosen such that $c \sum\limits_{n=1}^{\infty} \frac 1 {n^{2.5}}=1$. Then $X$ has finite mean but infinite variance.
This is because $\sum \frac n {n^{2.5}}<\infty$ but $\sum \frac {n^{2}} {n^{2.5}}=\infty$