Here’s another one. Let’s suppose that and are independent binomial random X Y variables. Say that ~ Bin(m, and ~ Bin(n, Observe that the are the X p) Y p). p’s same. Let random variable be the total; = + What is the probability law of T T X Y. X, given that = ? T t This asks us to find P[ = | = ]. Certainly ~ Bin(m+n, Then X x T t T p). P[ = | = ] = = X x T t = = = = It is interesting that the symbol completely disappeared! Later, we’ll identify this as p a case in which is a sufficient statistic for T p. Let’s suppose that random variables and have the joint probability law X Y fXY(x, y). This could be a joint probability function for two discrete random variables, it could be a joint density function for two continuous random variables, it could be a mixed law. We know all about the following (Yes…..is this review?) = marginal density of = fX(x) X The integral is written from to +∞, but the actual work will depend on ∞ what set of values takes. Y = marginal density of = fY(y) Y = mean of = expected value of = X X X = Let’s look at the last form. It really means . The order of integration can usually be reversed (assuming it’s a convergent integral) as = . The last form has in { } the density so this all makes sense. fX(x), = = = and are defined in an analogous way. Y
Here’s another one. Let’s suppose that X and Y are independent binomial random variables. Say that X ~ Bin(m, p) and Y ~ Bin(n, p). Observe that the p’s are the same. Let random variable T be the total; T = X + Y. What is the probability law of X, given that T = t ? This asks us to find P[ X = x | T = t ]. Certainly T ~ Bin(m+n, p). Then P[ X = x | T = t ] = = = = = = It is interesting that the symbol p completely disappeared! Later, we’ll identify this as a case in which T is a sufficient statistic for p. Let’s suppose that random variables X and Y have the joint probability law fXY(x, y). This could be a joint probability function for two discrete random variables, it could be a joint density function for two continuous random variables, it could be a mixed law. We know all about the following (Yes…..is this review?) fX(x) = marginal density of X = The integral is written from ∞ to +∞, but the actual work will depend on what set of values Y takes. fY(y) = marginal density of Y = X = mean of X = expected value of X = = Let’s look at the last form. It really means . The order of integration can usually be reversed (assuming it’s a convergent integral) as = . The last form has in { } the density fX(x), so this all makes sense. = = = Y and are defined in an analogous way.
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