STATS 426 Lecture Notes - Lecture 5: Probability Distribution, Universal Windows Platform Apps, Standard Deviation
Document Summary
Some inequalities and the weak law of large numbers. We rst introduce some very useful probability inequalities. Markov"s inequality: let x be a non-negative random variable and let g be a increasing non-negative function de ned on [0, ). We will prove the inequality assuming that x is a continuous random variable with density functionf ; an analogous proof holds in the discrete case. The theorem of course holds more generally, but a completely rigorous proof is outside the scope of this course. Now, x g(x) g( ) ( ) Z[ , ) g(x) f (x) dx g(x) f (x) dx +z[ , ) g(x) f (x) dx g(x) f (x) dx g( ) f (x) dx by . This is equivalent to the assertion of markov"s inequality. Note that markov"s inequality gives us an upper bound on the tail-probailities of. X and is more useful for smaller values of (and consequently smaller values of g( )).