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Here is my homework problem. Again, sorry for the formatting:

A continuous random variable $X$ has a symmetric distribution with mean $18$. A brilliant mathematician has estimated that the probability that $X$ is less than $12$ is at most $12.5 %$. Approximate the probability that $X$ lies between $7$ and $29$. Giant hint: what is the standard deviation?

So far I have determined that:


$p(x \lt 12)\le0.125$

I know that

Standard Deviation= $\sigma= \sqrt{\text{Variance of x}}$

and that $Var(x) = E(x^2)-\mu^2 =$ integral from $a$ to $b$ of $x^2 f(x)dx$

Because it has symmetric distribution I feel like I should be able to determine $E(x^2)$ without knowing $f(x)$. But I have nothing in my notes as to how to accomplish this. Can anyone point me in the right direction?

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You will not be able to find the standard deviation exactly from the information given, unless you are told or assume a lot more about the specific distribution of the random variable. But you can use an inequality (Hint: Russian name) that has presumably been discussed in the course to get some information about the standard deviation. – André Nicolas Apr 23 '11 at 18:08
At first I thought Chebyshev's Theorem would require finding σ first. but since P(μ-kσ ≤ x ≤ μ+kσ) ≥ 1-1/(k^2) I can find k first without σ. I got the answer on the first try after this, so thank you! – Ocasta Eshu Apr 23 '11 at 18:44

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