Stat 5101 Lecture Notes - School of Statistics

5.3. BERNOULLI RANDOM VECTORS 153you can’t talk about expectations or moments, E(Y ) is defined only for numerical(or numerical vector) random variables, not for categorical random variables.However, if we number the categoriesS = {s 1 ,s 2 ,...,s 5 }with s 1 = strongly agree, and so forth, then we can identify the categoricalrandom variable Y with a Bernoulli random vector XX i = I {si}(Y )that isX i = 1 if and only if Y = s i .Thus Bernoulli random variables are an artifice. They are introduced toinject some numbers into categorical problems. We can’t talk about E(Y ),but we can talk about E(X). A thorough analysis of the properties of thedistribution of the random vector X will also tell us everything we want toknow about the categorical random variable Y , and it will do so allowing us touse the tools (moments, etc.) that we already know.5.3.2 MomentsEach of the X i is, of course, univariate Bernoulli, writeX i ∼ Ber(p i )and collect these parameters into a vectorp =(p 1 ,...,p k )Then we abbreviate the distribution of X asX ∼ Ber k (p)if we want to indicate the dimension k or just as X ∼ Ber(p) if the dimension isclear from the context (the boldface type indicating a vector parameter makesit clear this is not the univariate Bernoulli).Since each X i is univariate Bernoulli,E(X i )=p ivar(X i )=p i (1 − p i )That tells usE(X) =p.To find the variance matrix we need to calculate covariances. For i ≠ j,cov(X i ,X j )=E(X i X j )−E(X i )E(X j )=−p i p j ,