If X is a non-negative random variable for which E(X) exists,show that for every t>0.

$\displaystyle P[X\geq t] \leq\frac {E(X)}{t}$

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- Jul 15th 2009, 06:56 AMroshanheroChebyshev's inequality
If X is a non-negative random variable for which E(X) exists,show that for every t>0.

$\displaystyle P[X\geq t] \leq\frac {E(X)}{t}$ - Jul 15th 2009, 09:45 AMCaptainBlack
$\displaystyle \frac{E(X)}{t}=\int_0^{\infty} \frac{x}{t}p(x)\; dx = \int_0^t\frac{x}{t}p(x)\; dx + \int_t^{\infty}\frac{x}{t}p(x)\; dx$

.........$\displaystyle \ge \int_t^{\infty}\frac{x}{t}p(x)\; dx$

and as $\displaystyle \frac{x}{t}\ge 1$ for $\displaystyle x\ge t$ hence:

$\displaystyle \frac{E(X)}{t}\ge \int_t^{\infty}\frac{x}{t}p(x)\; dx\ge \int_t^{\infty}p(x)\; dx$

CB - Jul 15th 2009, 08:27 PMroshanhero
How can we prove chebyshev inequality from this relation?

- Jul 15th 2009, 08:28 PMmatheagle
And you didn't notice that under Mr Fantasy's link...

http://people.csail.mit.edu/ronitt/COURSE/S07/lec25.pdf

under your last post...

http://www.mathhelpforum.com/math-he...nequality.html

it's Theorem ONE, on page ONE. - Jul 15th 2009, 08:38 PMCaptainBlack
- Jul 15th 2009, 08:45 PMroshanhero
Thanks,but,what should i use in t.

- Jul 15th 2009, 09:22 PMCaptainBlack