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Unformatted text preview: Stat 5101 Lecture Slides Deck 3 Charles J. Geyer School of Statistics University of Minnesota 1 Deja Vu Now we go back to the beginning and do everything again. 2 Probability Mass Functions A probability mass function (PMF) is a function S f→ R whose domain S , which can be any nonempty set, is called the sample space , whose codomain is the real numbers, and which satisfies the following conditions: its values are nonnegative f ( x ) ≥ , x ∈ S and sum to one X x ∈ S f ( x ) = 1 . (Exactly the same as slide 20, deck 1.) 3 Infinite Sample Spaces This time we allow infinite sample spaces. That means the sum X x ∈ S f ( x ) = 1 is an infinite series. So we are now using calculus. 4 Bernoulli Process A Bernoulli process is an infinite sequence of random variables X 1 , X 2 , ... (a stochastic process), that are IID Ber( p ). 5 Geometric Distribution The number of zeros (failures) before the first one (success) in a Bernoulli process is a random variable Y that has the geometric distribution with success probability p , denoted Geo( p ) for short. Clearly, Y takes values in N = { , 1 , 2 ,... } . Its PMF is given by f p ( y ) = Pr( Y = y ) because that is the formula for any PMF. 6 Geometric Distribution (cont.) If Y = y , then we know that the first y variables in the Bernoulli process have the value zero and that X y +1 = 1, and we don’t know anything else about the rest of the infinite sequence X 1 , X 2 , ... . The probability of observing y failures and one success in that order is (1 p ) y p . There is no binomial coefficient, because there is only one order considered. Hence the PMF of the Geo( p ) distribution is f p ( y ) = p (1 p ) y , y = 0 , 1 , 2 ,.... 7 Geometric Distribution (cont.) With every brand name distribution comes a theorem that says the probabilities sum to one. For the geometric distribution, this theorem is ∞ X y =0 p (1 p ) y = 1 . This is a special case of the geometric series ∞ X n =0 s n = 1 1 s whenever 1 < s < 1. Here s = 1 p . 8 Geometric Distribution (cont.) The geometric series only converges when 1 < s < 1, which is 1 < 1 p < 1, which is 0 < p < 2. Of course, we know p ≤ 1 because p is a probability. Thus the parameter space of the geometric family of distributions is { p ∈ R : 0 < p ≤ 1 } unlike the Bernoulli and binomial distributions p = 0 is not al lowed. What goes wrong is that when we try to sum the infinite series ∞ X y =0 (1 p ) y = 1 + 1 + 1 + ··· it does not converge. 9 Geometric Distribution (cont.) So we had to be careful. The phrase “number of failures before the first success in a Bernoulli process” does not define a random variable when the success probability is p = 0 because the first success never happens!...
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 Fall '02
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 Statistics, Normal Distribution, Probability, Probability theory

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