Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Having trouble with this question.

Suppose $X$ is a random variable with probability function: $f_X(x)=k/x^2$

I need to use a "Basel Problem" to find k and prove that the expected value $E(X)$ does not exist.

share|improve this question
    
Is $f$ density function and is there some bound on $x$, like $x\ge 1$? –  Berci Nov 1 '12 at 23:23
    
I think you cannot set simply a $k$ such that you will get a density. There is a problem as Berci indicated around $x$ close to zero. –  Seyhmus Güngören Nov 1 '12 at 23:27
    
Presumably $x \in \mathbb{N}$ and $f_X$ is a pmf. –  Niels Diepeveen Nov 1 '12 at 23:54

2 Answers 2

up vote 1 down vote accepted

Use that $\displaystyle\int_{\Bbb R}f_X(x)dx =1$ and that for any (measurable) function $g$, we have $$E(g(X))=\int_{\Bbb R} g(x)\cdot f(x)dx$$ In particular, $E(X)=\displaystyle\int^\infty_{1}x\cdot \frac k{x^2} dx = \infty$ (assuming, $X\ge 1$ always).

share|improve this answer

If we are really supposed to use the "Basel Problem," then our probability is defined only on the positive integers, and $\Pr(X=n)=\dfrac{k}{n^2}$,

The sum of the probabilities over th sample space must be $1$. By Euler's solution to the Basel Problem, we have $$\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots=\sum_1^\infty \frac{1}{n^2} =\frac{\pi^2}{6}.$$ It follows that we must have $k=\dfrac{6}{\pi^2}$.

The expectation of $X$ is then $$\sum_1^\infty n\frac{k}{n^2}=\frac{6}{\pi^2}\sum_1^\infty \frac{1}{n}.$$ But by the divergence of the harmonic series, the expectation does not exist, or, if one prefers, is infinite.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.