Hypergeometric. SetP(X=i) =miN-mn-iNn.This comes up in sampling without replacement: if there areNballs, of whichmare one color and the otherN-mare another, and we choosenballs at random without replacement, thenXrepresents the probabilityof havingiballs of the first color.6. Continuous distributions.A r.v.Xis said to have a continuous distribution if there exists a nonnegative functionfsuch thatP(a≤X≤b) =baf(x)dxfor everyaandb. (More precisely, such anXis said to have an absolutely continuous distribution.)fiscalled the density function forX. Note∞-∞f(x)dx=P(-∞< X <∞) = 1. In particular,P(X=a) =aaf(x)dx= 0 for everya.Example.Suppose we are givenf(x) =c/x3forx≥1. Since∞-∞f(x)dx= 1 andc∞-∞f(x)dx=c∞11x3dx=c2,we havec= 2.DefineF(y) =P(-∞< X≤y) =y-∞f(x)dx.Fis called the distribution function ofX.Wecan defineFfor any random variable, not just continuous ones, by settingF(y) =P(X≤y). In the caseof discrete random variables, this is not particularly useful, although it does serve to unify discrete andcontinuous random variables. In the continuous case, the fundamental theorem of calculus tells us, provided
This is the end of the preview.
access the rest of the document.
Probability distribution, Probability theory, probability density function, dx, absolutely continuous distribution.