Expectation of a product of random variables

Question

Let $X$ be a positive and real random variable such that $E[X] <\infty$. My question is if $E[X^2]<\infty$.

Thank you in advance.

Answer 1

In general, it is not true. Take for example a random variable $X$ with a density $f(x) = \cfrac{\alpha}{x^{\alpha +1}} \mathbf{1}_{\mathbb{R_+}}(x)$ and $ 1 < \alpha < 2$. Such variable admits a first moment since $\mathbb{E}[X] = \cfrac{\alpha}{\alpha -1} $ however $\mathbb{E}[X^2] = + \infty $.

Answer 2

No. If $X$ takes the values $2,3,...$ with probabilities $\frac c {n^{2}(ln \, n)^{2}}$ then we can see that $EX<\infty$ but $EX^{2}=\infty$. [$c$ is chosen so that the probabilities add up to $1$].

Answer 3

No. You only have $L^2\subset L^1$. A counter example is $X(\omega )=\frac{1}{\sqrt \omega }$ on $((0,1), \mathcal B, m)$ where $m$ is the Lebesgue measure and $\mathcal B$ the borel set.