If X is a random variable such that E(X)= 3 and E(X2)=13, use Chebychev's inequality to determine a lower bound for P(-2<X<8).

Chebyshev's inequality is a tool in probability theory that provides a lower bound for the probability that a random variable falls within a certain range around its mean.

The inequality is stated as follows: For any random variable X with finite expected value μ and variance σ^2, and for any positive real number k, the probability that X falls within k standard deviations of its mean is at least 1 - 1/k^2. Mathematically, this is expressed as:

P(|X - μ| < kσ) ≥ 1 - 1/k^2

In this case, we are given that the expected value E(X) = μ = 3 and E(X^2) = 13. The variance of X, σ^2, is given by E(X^2) - [E(X)]^2 = 13 - 3^2 = 13 - 9 = 4. So, the standard deviation σ is the square root of the variance, which is 2.

We want to find a lower bound for P(-2 < X < 8), which is the same as P(|X - 3| < 5). This is within 5/2 = 2.5 standard deviations of the mean.

Applying Chebyshev's inequality with k = 2.5, we get:

P(|X - 3| < 5) ≥ 1 - 1/(2.5)^2 = 1 - 1/6.25 = 1 - 0.16 = 0.84

Therefore, a lower bound for the probability that X falls between -2 and 8 is 0.84.