binomial_notes - Notes on Bernoulli and Binomial random...

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Notes on Bernoulli and Binomial random variables October 1, 2010 1 Expectation and Variance 1.1 Definitions I suppose it is a good time to talk about expectation and variance, since they will be needed in our discussion on Bernoulli and Binomial random variables, as well as for later disucssion (in a forthcoming lecture) of Poisson processes and Poisson random variables. Basically, given a random variable X : S → R , having a pdf f ( x ), define the expectation to be E ( X ) := integraldisplay −∞ xf ( x ) dx. In other words, it is kind of like the “average value of X weighted by f ( x )”. If X is discrete, where say X can take on any of the values a 1 , a 2 , ... , we would have E ( X ) := summationdisplay i a i p ( a i ) , where p here denotes the mass function. More generally, if ν : B ( R ) [0 , 1] is a probability measure associated to a random variable X (continuous, discrete, or otherwise), so that for A ∈ B ( R ) we have P ( X A ) = integraldisplay R ( x ) = ν ( A ) , 1
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we define E ( X ) := integraldisplay R xdν ( x ) . Of course, we really haven’t worked much with the sort of r.v.’s for which such a ν cannot be relized in terms of pdf’s, but I thought I would point it out anyways. We likewise define the variance operator V ( X ) to be V ( X ) := integraldisplay R ( x μ ) 2 f ( x ) dx, where μ = E ( X ) . If X is discrete, then V ( X ) := summationdisplay i ( a i μ ) 2 p ( a i ); and, of course, there is the even more general version V ( X ) := integraldisplay R ( x μ ) 2 ( x ) . We know that the expectation is a kind of average, and now I want to give you a feel for what the variance is: Basically, it is a measure of how “flat” the pdf is – the flatter it is, the more the values of X away from the mean μ get weighted; and therefore the larger the variance V ( X ) is. For example, consider the two random variables X and Y having pdf’s f ( x ) and g ( x ), respectively, given by f ( x ) = braceleftbigg 1 / 2 , if x [ 1 , 1]; 0 , if x < 1 or x > 1 . and g ( x ) = braceleftbigg 1 / 4 , if x [ 2 , 2]; 0 , if x < 2 or x > 2 .
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