1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

where x is any real number. If density of distribution p ξ (x) exists, thenxFξ( x) = ∫ pξ( x) dx.−∞I also introduce the formulas for expectation and variance in thiscase:∞∞∫ ξ ∫−∞−∞Eξ = xp ( x) dx, E f (ξ) = f ( x) p ( x) dx,ξ∞2 2var ξ = E(ξ − Eξ) = ∫ ( x − Eξ) pξ( x) dx.−∞3. Bernoulli Trials. The Poisson Jurors3.1. Bernoulli trials. And so, it is incomparably simpler tointroduce probabilities of independent, rather than dependent events.Therefore, stochastic models with independent events have much morechances to be practically applied. The most simple and thus the mostwidely applicable is the model in which we imagine a certain numbern of independent trials, each of them resulting in one of the twopossible outcomes called success and failure. The probability ofsuccess is supposed to be the same throughout and is denoted by p sothat failure will be q = 1 – p. Denote also success and failure by 1 and0, then the result of n trials will be a sequence of these numbers havinglength n.The set of elementary events ω, Ω = {ω}, thus consists of all suchsequences of length n and therefore has 2 n elements. Takingindependence of individual trials into account, we ought to provide adefinition according to which the probability p(ω) of each elementaryevent ω will be calculated by changing each 1 by number p, and eachfailure by changing each 0 by number q and multiply the obtainednumbers. We will then haveP(ω) = p µ(ω) q n−µ(ω)where µ(ω) is the number of unities in the sequence of the ω’s.Experiments described by this stochastic models are calledBernoulli trials, and the random variable µ = µ(ω) is the number ofsuccesses in n such trials. Let us determine the distribution of thatrandom variable. Its possible values are evidently numbers 0, 1, ..., nso that∑ ∑µ(ω) n µ(ω)= = = =P{µ m} P(ω)p q −∑m n−m m n−mp q = p q ⋅ (number of such ω that µ(ω) = m).25