# $\int_{-\infty}^{\infty} x^3 f(x) dx < \infty$ then $Cov(X,X)<\infty$ ? TRUE OR FALSE

$$x \in R$$ is a continuous random variable.

Is the statement : IF $$\int_{-\infty}^{\infty} x^3 f(x) dx < \infty$$ then: $$Cov(X,X)<\infty$$ .TRUE?

My thought was that Var(x)=Cov(x,x) , so $$Var(x)= E(x^2) - E^2(x)$$. Hence

both $$\int_{-\infty}^{\infty} x^2 f(x) dx < \infty$$ and $$\int_{-\infty}^{\infty} x f(x) dx < \infty$$

so I think the Question can be written as:

Is it true that IF $$\int_{-\infty}^{\infty} x^3 f(x) dx < \infty$$ then both $$\int_{-\infty}^{\infty} x^2 f(x) dx < \infty$$ and $$\int_{-\infty}^{\infty} x f(x) dx < \infty$$?.

It looks False to me. But I am not sure how to prove it.

• IF $\int_{-\infty}^{\infty} x^2 f(x) dx = infty$ does that mean that $\int_{-\infty}^{\infty} x f(x) dx = \infty$?.
– Mia
Sep 14 at 11:45
• Hint: where does $|x|³>x²$ occur? Remark: the assumption should be $\int |x|^3 f(x)\,\text dx<\infty$ Sep 14 at 11:56
• stats.stackexchange.com/questions/244202 and stats.stackexchange.com/questions/251431 show several techniques to address this.
– whuber
Sep 14 at 15:17

Is it true that IF $$\int_{-\infty}^{\infty} x^3 f(x) dx < \infty$$ then both $$\int_{-\infty}^{\infty} x^2 f(x) dx < \infty$$ and $$\int_{-\infty}^{\infty} x f(x) dx < \infty$$?.
Instead it sound me true. There is a theorem that warrant us that if moment of order $$k$$ exist then all lower order moments exists too. If third order moment exist, variance must exist.