{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

CS2800-Probability_part_d_v.3

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

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

1 Discrete Math CS 2800 Prof. Bart Selman [email protected] Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds

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

View Full Document
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 0

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

View Full Document
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 = =

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

View Full Document
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

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

View Full Document
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

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

View Full Document
10 Probability Distribution The probability function, properties ( 29 x x P X each for 0 ( 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 !

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

View Full Document
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 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 = = =

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

View Full Document
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 =
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern