Stat 5101 Lecture Notes - School of Statistics

152 Stat 5101 (Geyer) Course NotesDefinition 5.3.1 (Bernoulli Random Vector).A random vector X =(X 1 ,...,X k ) is Bernoulli if the X i are the indicatorsof a partition of the sample space, that is,whereandis the whole sample space.X i = I AiA i ∩ A j = ∅,k⋃A ii=1i ≠ jDefinition 5.3.2 (Bernoulli Random Vector).A random vector X =(X 1 ,...,X k ) is Bernoulli if the X i are zero-or-onevaluedrandom variables andwith probability one.X 1 + ···+X k =1.Definition 5.3.3 (Bernoulli Random Vector).A random vector X =(X 1 ,...,X k ) is Bernoulli if the X i are zero-or-onevaluedrandom variables and with probability one exactly one of X 1 , ..., X k isone and the rest are zero.The equivalence of Definitions 5.3.2 and 5.3.3 is obvious. The only way abunch of zeros and ones can add to one is if there is exactly one one.The equivalence of Definitions 5.3.1 and 5.3.3 is also obvious. If the A i forma partition, then exactly one of theX i (ω) =I Ai (ω)is equal to one for any outcome ω, the one for which ω ∈ A i . There is, of course,exactly one i such that ω ∈ A i just by definition of “partition.”5.3.1 Categorical Random VariablesBernoulli random vectors are closely related to categorical random variablestaking values in an arbitrary finite set. You may have gotten the impressionup to know that probability theorists have a heavy preference for numericalrandom variables. That’s so. Our only “brand name” distribution that is notnecessarily numerical valued is the discrete uniform distribution. In principle,though a random variable can take values in any set. So although we haven’tdone much with such variables so far, we haven’t ruled them out either. Ofcourse, if Y is a random variable taking values in the setS = {strongly agree, agree, neutral, disagree, strongly disagree} (5.36)