If X is a random variable such that E(X)=3 and E(X 2 )=13, use Chebyshev’s inequality todetermine a lower bound for P(-2<X<8).

Chebyshev's inequality states that for any random variable X, and for any positive k, the probability that X deviates from its mean by more than k standard deviations is at most 1/k^2.

Given that E(X) = 3 and E(X^2) = 13, we can calculate the variance of X, which is E(X^2) - [E(X)]^2 = 13 - 3^2 = 4. Therefore, the standard deviation of X is sqrt(4) = 2.

We want to find a lower bound for P(-2 < X < 8). This is equivalent to finding a lower bound for P(3 - 2.5 < X < 3 + 2.5), where 2.5 is 2.5 standard deviations away from the mean.

By Chebyshev's inequality, we know that the probability that X deviates from its mean by more than 2.5 standard deviations is at most 1/(2.5)^2 = 0.16.

Therefore, the probability that X does not deviate from its mean by more than 2.5 standard deviations is at least 1 - 0.16 = 0.84. This is the lower bound for P(-2 < X < 8).