How to Prove Markov’s Inequality

Statement

Let X be a non-negative random variable, and E(X) exists, c > 0, then,  P(X>=c)<=E(X)/c  

or  P(X<=c)>=1-E(X)/c” width=”179″ height=”39″>  .</span><span id=

 

Proof

Let X be a non-negative random variable,

E(X)=∫∞0xf(x)dx

=∫c0xf(x)dx+∫∞cxf(x)dx
>=∫∞cxf(x)dx” width=”117″ height=”43″></span></figure><figure class=>=c∫∞cf(x)dx” width=”118″ height=”43″></span></figure><figure class==cP(X>=c)” width=”102″ height=”19″></span></figure><p><span style=then  P(X>=c)<=E(X)/c  , 

and because  P(X>=c)=1-P(X<=c)  , 

P(X<=c)>=1-E(X)/c” width=”179″ height=”39″>  Q.E.D.</span></p><p></p><p><span style=Corollaries

Chebyshev’s Inequality

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