Probability Distributions

Transcription

1 Probability Distributions

2 Series of events Previously we have been discussing the probabilities associated with a single event: Observing a 1 on a single roll of a die Observing a K with a single card selection Now, we ll extend this concept to more complex situations.

3 Some Definitions Sample Space: : the set of all possible values the events may assume: HH, HT, TH, TT Random Variable: : A rv is a real-valued function of events in the ss: The sample space S = {S, F} for success or failure. A random variable X is X(S) = 1 and X(F) = 0 Define a rv X by letting X = the student GPA. The possible values of X are between and such that {x:{ : x 4.333}.

4 Discrete Random Variables A rv whose possible values either constitute a finite set or can be listed in an infinite sequence in which there is a first element, a second element, and so on is a discrete random variable

5 Probability Distns for Discrete RVs It is often of interest to see how the entire probability of events is distributed. That is, how is the total amount of 0 1 probability assigned? The probability distribution (pdf or pmf) of X shows this. The pdf of a discrete rv is defined for every number x by p(x) ) = P(X = x).

6 Properties 1) 0 P(X = x) ) 1 2) P( X = x) = 1 2)

7 Produce Example A produce company supplies lots of produce to grocers. Each lot refers to a different farm. The number of bad pieces of produce on average in each shipment is recorded for quality control: lot # bad pieces

8 If a grocer selects a random lot for purchase, let X be the number of bad items: The possible values of X are 0, 1, 2, and 4. Let p(x) ) be the probability that X = x. Where x can be 0, 1, 2, or 4: p(0) = P(X = 0) = P(lot 1, 4, 6 selected) = 3/6 = 0.50 p(1) = P(X = 1) = P(lot 2 is selected) = 1/6 = p(2) = P(X = 2) = P(lot 3 is selected) = 1/6 = p(4) = P(X = 4) = P(lot 5 is selected) = 1/6 = 0.167

9 Graphically Probability

10 Video Store Example A video store has kept track of the number of videos that customers rent in week to help determine potential promotional programs to increase the number of rentals. Over the course of a year, 50% of the customers rented 1 video, 30% 2 videos, 10% 3 videos, 4% 4 videos, and 3% 5 or more videos (called 5 videos).

11 The pdf can now be defined as: p(1) = P(X = 1) = P(1 video) = 0.50 p(2) = P(X = 2) = P(2 videos) = 0.30 p(3) = P(X = 3) = P(3 videos) = 0.10 p(4) = P(X = 4) = P(4 videos) = 0.07 p(5) = P(X = 5) = P(5 videos) = 0.03

12 Or as: 0.50 x = x = 2 Px ( ) = 0.10 x = 3, 0 if x 1, 2, 3, 4, x = x = 5

13 Graphically # videos

14 Bernoulli RV If a rv only has two options, it is called a binary rv. If the options are 0 or 1, it is called a Bernoulli rv. For a Bernoulli random variable, the pdf can be generally defined as: 1 α if x = 0 Px ( = x) = α if x = 1 0 otherwise

15 For comedy vs non-comedy Px ( = x) = if x = 0 (non-comedy) = 0.35 if x = 1 (comedy) 0 otherwise 0.65 if x = 0 (non-comedy) = 0.35 if x = 1 (comedy) 0 otherwise

16 Calculating Probability what is the probability that a customer will rent 1 or 2 videos in a week? ( 1 2) = ( 1) + ( 2) P P P = = 0.80

17 Cumulative Distributions Cumulative probability is adding successive probabilities to obtain a probability up to and including a certain event. For example, the probability that a customer rents 3 or fewer videos is = 0.90.

18 Statistically, this is defined as: F( x) = P( X x ) = p( x ) i i i

19 For the video example, the CDF is: # videos CDF P(X? x)

20 Graphically 1.0 Cum Prob # videos

21 What is the probability that a customer will rent 4 or fewer videos in a week? # videos CDF P(X? x)

22 Or: 1.0 Cum Prob # videos

23 Obtaining the probability that X falls in between two numbers can also be obtained from the CDF. In general, ( ) = ( ) ( )-1),, for for a a b P a X b F b F a

24 Expected Value Often we re interested in the most likely outcome, or the value we expect to see: ( ) ( ), = = E X µ x p x i X i i i

25 Properties Some properties of expected value: for any constant a, E(aX) ) = ae(x) for any constant b, E(X + b) = E(X) + b

26 Variance ( ) ( 2 ) ( ) 2 Var X = E X E X

27 Properties Some properties of variance: for any constant a, Var(aX) ) = a2var(x) for any constant b, Var(X + b) = Var(X) Notice that the addition of a constant will change the location of the expected value (mean) of X but NOT the variance (spread of values) of X.

28 Summary Therefore, the video store data can be summarized as follows: A video store recorded the number of videos rented per customer in a week. The average number of videos rented was 1.83 (sd =1.06), with a range of 1 5 or more videos in a week and 80% of the customers rented 1 or 2 videos in a week.