CS2800-Probability_part_d_v.3

CS2800-Probability_part_d_v.3 - 1 Discrete Math CS 2800...

Info iconThis preview shows pages 1–15. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds 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 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 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. 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 = = 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 7 Probability Distribution The probability function May be graphical: 1 2 3 .50 .33 .17 8 Probability Distribution The probability function May be formulaic: ( 29 1,2,3 for x 6 4 =- = = x x X P 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 10 Probability Distribution The probability function, properties ( 29 x x P X each for ( 29 = x X x P 1 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 ! 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 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 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 = = = 14 Cumulative Probability Distribution...
View Full Document

This note was uploaded on 10/25/2009 for the course CS 2800 at Cornell University (Engineering School).

Page1 / 51

CS2800-Probability_part_d_v.3 - 1 Discrete Math CS 2800...

This preview shows document pages 1 - 15. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online