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CS2800-Probability_part_d_v.3

CS2800-Probability_part_d_v.3 - Discrete Math CS 2800 Prof...

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1 Discrete Math CS 2800 Prof. Bart Selman [email protected] Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds
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2 Discrete Random variable Discrete random variable Takes on one of a finite (or at least countable) number of different values. X = 1 if heads, 0 if tails Y = 1 if male, 0 if female (phone survey) Z = # of spots on face of thrown die
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3 Continuous Random variable Continuous random variable (r.v.) Takes on one in an infinite range of different values W = % GDP grows (shrinks?) this year V = hours until light bulb fails For a discrete r.v., we have Prob(X=x), i.e., the probability that r.v. X takes on a given value x. What is the probability that a continuous r.v. takes on a specific value? E.g. Prob(X_light_bulb_fails = 3.14159265 hrs) = ?? However, ranges of values can have non-zero probability. E.g. Prob(3 hrs <= X_light_bulb_fails <= 4 hrs) = 0.1 0
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4 Probability Distribution The probability distribution is a complete probabilistic description of a random variable. All other statistical concepts (expectation, variance, etc) are derived from it. Once we know the probability distribution of a random variable, we know everything we can learn about it from statistics.
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5 Probability Distribution Probability function One form the probability distribution of a discrete random variable may be expressed in. Expresses the probability that X takes the value x as a function of x (as we saw before): ( 29 ) ( x X P x P X = =
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6 Probability Distribution The probability function May be tabular: = 6 / 1 . . 3 3 / 1 . . 2 2 / 1 . . 1 p w p w p w X
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7 Probability Distribution The probability function May be graphical: 1 2 3 .50 .33 .17
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8 Probability Distribution The probability function May be formulaic: ( 29 1,2,3 for x 6 4 = - = = x x X P
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9 Probability Distribution: Fair die = 6 / 1 . . 6 6 / 1 . . 5 6 / 1 . . 4 6 / 1 . . 3 6 / 1 . . 2 6 / 1 . . 1 p w p w p w p w p w p w X 1 2 3 .50 .33 .17 4 5 6
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10 Probability Distribution The probability function, properties ( 29 x x P X each for 0 ( 29 = x X x P 1
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11 Cumulative Probability Distribution Cumulative probability distribution The cdf is a function which describes the probability that a random variable does not exceed a value. ( 29 ( 29 x X P x F X = Does this make sense for a continuous r.v.? Yes !
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12 Cumulative Probability Distribution Cumulative probability distribution The relationship between the cdf and the probability function: ( 29 ( 29 ( 29 = = = x y X X y X P x X P x F
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13 Cumulative Probability Distribution Die-throwing ( 29 < < < < < < = 6 6 / 6 6 5 6 / 5 5 4 6 / 4 4 3 6 / 3 3 2 6 / 2 2 1 6 / 1 1 0 x x x x x x x x F X 1 2 3 4 5 6 1 graphical tabular ( 29 ( 29 ( 29 = = = x y X X y X P x X P x F ( 29 ( ) 1/ 6 X P x P X x = = =
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14 Cumulative Probability Distribution The cumulative distribution function May be formulaic (die-throwing): ( 29 ( 29 ( 29 ( 29 min max x,0 ,6 6 floor P X x =
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