1. ## chebyshev's inequality problem

How can I use Chebyshev's Inequality to show that P[0 < X < 3] >= 0.75?
note: X ~ Exp (lamba = 1)

and does the Cheby. inequality improve as k --> infinity?

it's a little confusing. can someone help?

2. Originally Posted by silentrain
How can I use Chebyshev's Inequality to show that P[0 < X < 3] >= 0.75?
note: X ~ Exp (lamba = 1)

and does the Cheby. inequality improve as k --> infinity?

it's a little confusing. can someone help?
There are things you need to make it your business to know:

1. The statement of Chebyshev's Inequality.

2. The mean of an exponential distribution.

3. The standard deviation of an exponential distribution.

Once you know these things you need to see that:

$\displaystyle \Pr(0 < X < 3) = \Pr(0 < X < \mu + 2 \sigma)$

$\displaystyle = \Pr(\mu - 2 \sigma < X < \mu + 2 \sigma)$

since $\displaystyle \mu - 2 \sigma < 0$ and $\displaystyle \Pr(X < 0) = 0$

$\displaystyle = \Pr( - 2 \sigma < X - \mu < 2 \sigma) = \Pr(|X - \mu| < 2 \sigma)$

where $\displaystyle \mu$ and $\displaystyle \sigma$ are respectively the mean and standard deviation of X.