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Math 3338: Probability Discrete Probability Distributions Pankaj Singh Department of Mathematics, University of Houston [email protected] math.uh.edu/∼pankaj/math3338 Pankaj Singh (University of Houston) Math 33338: Probability Spring, 2016 1 / 12 Probability Space A Probability Space is a triple (S, E, P), where S is the sample space, E is the event space, and P is a real valued function having the following properties: 0 ≤ P(E ) ≤ 1 for all E ∈ E; P(S) = 1; P ∪∞ j=1 Ej = ∞ X P(Ej ) j=1 for any sequence of events E1 , E2 , · · · with the property that Ei ∩ Ej = ∅ whenever i 6= j (sequence of mutually exclusive events). Pankaj Singh (University of Houston) Math 33338: Probability Spring, 2016 2 / 12 Random Variables and Their Probability Distributions A random variable for a probability space (S, E, P) is a real-valued function whose domain is a sample space. Pankaj Singh (University of Houston) Math 33338: Probability Spring, 2016 3 / 12 Example Let X represent the number of heads obtained from tossing a fair coin 3 times. List the possible values of X. Pankaj Singh (University of Houston) Math 33338: Probability Spring, 2016 4 / 12 Discrete Random Variable A random variable X is said to be discrete if it can take on only a finite number - or a countably infinite number -of possible values x. The probability mass function of X , denoted by p(x), assigns probability to each value x of X so that the following conditions are satisfied: P(X = x) = p(x) ≥ 0. X P(X = x) = 1, where the sum is over all possible values of x. x Pankaj Singh (University of Houston) Math 33338: Probability Spring, 2016 5 / 12 Determine the probability mass function for X. Pankaj Singh (University of Houston) Math 33338: Probability Spring, 2016 6 / 12 Cumulative Distribution Function (CDF). The cumulative distribution function F(b) for a random variable X is F (b) = P(X ≤ b) if X is discrete, F (b) = ∞ X p(x), x=−∞ where p(x) is the probability function. The cumulative distribution function (CDF) is sometimes also called the distribution function. The Pankaj Singh (University of Houston) Math 33338: Probability Spring, 2016 7 / 12 Determine the cumulative distribution function for X . Pankaj Singh (University of Houston) Math 33338: Probability Spring, 2016 8 / 12 Sketch the graph of the cumulative distribution function for X . Pankaj Singh (University of Houston) Math 33338: Probability Spring, 2016 9 / 12 Properties of CDF The cumulative distribution function for a random variable X is a function FX : R → [0, 1] with the following properties: FX (x) = P(X ≤ x) lim FX (x + ) = FX (x) for all x →0+ lim FX (x) = 0 x→−∞ lim FX (x) = 1 x→∞ Pankaj Singh (University of Houston) Math 33338: Probability Spring, 2016 10 / 12 0, 0.05, 0.15, Suppose FX (x) = 0.35, 0.65, 0.85, 1, if if if if if if if x <1 1≤x <2 2≤x <3 3≤x <4 4≤x <5 5≤x <6 x ≥6 Graph the cumulative distribution function of X . Find the probability mass function of X . Graph the probability mass function of X . Pankaj Singh (University of Houston) Math 33338: Probability Spring, 2016 11 / 12 Pankaj Singh (University of Houston) Math 33338: Probability Spring, 2016 12 / 12