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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:

mean=μ=18

p(x<12)≤0.125

I know that Standard Deviation= σ= Square Root of Variance of x

and that Var(x) = E(x^2)-μ^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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