Discrete Probability Distributions

Discrete Probability Distributions

Discrete Probability Distributions Random variables Discrete probability distributions Expected value and variance Binomial probability distribution Random Variables A random variable is a numerical description of the outcome of an experiment Discrete Random Variables A discrete random variable may assume a finite number of numerical values or an infinite sequence of values such as 0, 1, 2, . . . Example:

1 = passed driving test 2 = failed driving test This is a discrete variable because it is either pass or failyou cant score 1, for example. Notice also it assumes a finite number of values Example: JSL Appliances Discrete random variable with an infinite sequence of values Let Let xx = = number number of of customers customers arriving arriving in in one

one day, day, where where xx can can take take on on the the values values 0, 0, 1, 1, 2, 2, .. .. .. We can count the customers arriving, but there is n finite upper limit on the number that might arrive. Continuous Random Variables Continuous random variables can assume an infinite number of values within a defined interval. The liquid in these bottles (x) must be

between 0 and 32 ounces. But x could be 2 oz., 2.1 oz., 2.01 oz, 2.001 oz., . . . 0 x 32 Random Variables Question Family size Random Variable x Type x = Number of dependents in Discrete family reported on tax return Distance from x = Distance in miles from Continuous home to store home to the store site Own dog

or cat x = 1 if own no pet; Discrete = 2 if own dog(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s) Discrete Probability Distributions The The probability probability distribution distribution is is defined defined by by aa probability probability function function,, denoted denoted by by ff((xx),), which which provides

provides the the probability probability for for each each value value of of the the random random variable variabl The The required required conditions conditions for for aa discrete discrete probability probability function function are: are: f(x) > 0 f(x) = 1

Example: JSL Appliances Using past data on TV sales, a tabular representation of the probability distribution for TV sales was developed. Number Units Sold of Days 0 80 1 50 2 40 3 10 4 20 200

x 0 1 2 3 4 f(x) .40 .25 .20 .05 .10 1.00 80/200 Example: JSL Appliances Graphical Representation of the Probability Distribution Probability

.50 .40 .30 .20 .10 0 1 2 3 4 Values of Random Variable x (TV sales) Discrete Uniform Probability Distribution This is the simplest probability distribution described by a formula. It assumes that possible values of random

variables are equally likely f ( x) 1 / n n = number of values the random variable may assume. Example: Rolling a Die x 1 2 3 4 5 6 f(x) 1/6 1/6 1/6 1/6 1/6 1/6

Note that n = 6 Expected Value E(x) The expected value, or mean, of a random variable, is a measure of central location for the random variable For a discrete random variable x we have E ( x) xf ( x) E ( x) xf ( x) Notice that this is a weighted average of the values a random variable can assume. The weights are the probabilities. Example: JSL Appliances Expected Value of a Discrete Random Variable x

0 1 2 3 4 f(x) xf(x) .40 .00 .25 .25 .20 .40 .05 .15 .10 .40 E(x) = 1.20 expected number of TVs sold in a day

Variance of a Discrete Random Variable The variance of a random variable x is a weighted average of the squared deviations of a random variable from its mean (expected) value. The weights are the probabilities. Var ( x) 2 ( xi ) 2 f ( x) Standard Deviation of a Random Variable () StDev( x) ( xi ) 2 f ( x) Example: JSL Appliances Variance and Standard Deviation of a Discrete Random Variable x x- 0 1 2

3 4 -1.2 -0.2 0.8 1.8 2.8 (x - )2 f(x) (x - )2f(x) 1.44 0.04 0.64 3.24 7.84 .40 .25

.20 .05 .10 .576 .010 .128 .162 .784 TVs square Variance of daily sales = 2 = 1.660 d Standard deviation of daily sales = 1.2884 TVs Using Excel to Compute the Expected Value, Variance, and Standard Deviation 1 2 3

4 5 6 7 8 9 10 Formula Worksheet A Sales 0 1 2 3 4 B Probability 0.40 0.25 0.20 0.05

0.10 Mean =SUMPRODUCT(A2:A6,B2:B6) Variance =SUMPRODUCT(C2:C6,B2:B6) Std.Dev. =SQRT(B9) C Sq.Dev.from Mean =(A2-$B$8)^2 =(A3-$B$8)^2 =(A4-$B$8)^2 =(A5-$B$8)^2 =(A6-$B$8)^2 Using Excel to Compute the Expected Value, Variance, and Standard Deviation 1 2 3 4 5

6 7 8 9 10 Value Worksheet A Sales 0 1 2 3 4 Mean 1.2 Variance 1.66 Std.Dev. 1.2884 B Probability 0.40 0.25 0.20

0.05 0.10 C Sq.Dev.from Mean 1.44 0.04 0.64 3.24 7.84 The Binomial Distribution This is a very useful tool for multi-step experiments where each step has 2 outcomes hence the term binomial. Properties of Binomial Experiment 1. The experiment consists of a sequence of n identical trials. 2. Two outcomes are possible on each trial. We refer to one outcome as a success and the other outcome as a failure.

3. The probability of success, denoted by , does not change from trial to trial. Consequently, the probability of failure, denoted by 1 , does not change from trial to trial. 4. The trials are independent. Stationarity assumption We are interested in computing the number of successes (x) for n number of trials Binomial Probability Distribution The binomial distribution is given by: n! x (n x) f ( x) (1 ) x!(n x)!

Where: f(x) = probability of success in n trials n = number of trials p = probability of success in any one trial. Example: Evans Electronics Binomial Probability Distribution Evans is concerned about a low retention rate for employees. In recent years, management has seen a turnover of 10% of the hourly employees annually. Thus, for any hourly employee chosen at random, management estimates a probability of 0.1 that the person will not be with the company next year. Evans electronics If we selected three (3) employees at random, what is the probability that one(1) will leave the company within

the year? Notice that: The experiment has three identical trialsthat is, n = 3. There are two outcomes for each trialthe employee leaves (S) or the employee stays (F). The probability that an employee will leave is .1that is, = .1 The decision of each employee to leave is independent of the decisions made by the other employees. Counting the Number of Outcomes First Employee Second Employee Third Experimental Employee Outcome (S,S,S) S

S F F S S F F S S F S F 3

(S,S,F) 2 (S,F,S) 2 1 (S,F,F) (F,S,S) F Value of x (F,S,F) (F,F,S) (F,F,F) 2 1 1

0 Number of Experimental Outcomes Providing Exactly x Successes in n trials n n! x x!(n x)! In our Evans Electronics example, n = 3 and x = 1. Thus: 3 (3)(2)(1) 6 3 1 (1)(2)(1) 2 Refer to the tree diagram to verify this is right What is the probability the first employee selected will leave and second and third will stay? Note this is outcome (S, F, F)

Because these events are independent, we can multiply probabilities. . Thus the probability of (S, S, F) is given by: (1 )(1 ) (1 ) 2 Thus we have: 2 .1(1 .1) .1(.81) .081 Trial Outcomes Experimental Outcome Probability of Experimental Outcome (S,F,F) (1 )(1 ) (1 ) 2 .1(1 .9) 2 .081

(F,S,F) (1 )(1 ) (1 ) 2 .1(1 .9) 2 .081 (F,F,S) (1 )(1 ) (1 ) 2 .1(1 .9) 2 .081 Binomial Probability Distribution Binomial Probability Function n! f (x) px (1 p)(n x) x!(n x)! n! x!(n x)!

px (1 p)(n x) Probability of aa particular Probability of particular Number of experimental Number of experimental sequence of trial outcomes sequence of trial outcomes outcomes providing exactly outcomes providing exactly with x successes in n trials with x successes in n trials xx successes in n trials successes in n trials

Probability Distribution for the Number of Employees Leaving Within the Year x f(x) 0 3! (.1) 0 (.9) 3 .729 0!3! 1 3! (.1)1 (.9) 2 .243 1!2! 2 3! (.1) 2 (.9)1 .027

2!1! 3! 3 (.1) (.9) 0 .001 3! 3 Using Excel to Compute Binomial Probabilities Formula Worksheet A 1 2 3 4 5 6 7 8 9

x 0 1 2 3 B 3 = Number of Trials (n ) 0.1 = Probability of Success (p ) f (x ) =BINOMDIST(A5,$A$1,$A$2,FALSE) =BINOMDIST(A6,$A$1,$A$2,FALSE) =BINOMDIST(A7,$A$1,$A$2,FALSE) =BINOMDIST(A8,$A$1,$A$2,FALSE) Using Excel to Compute Binomial Probabilities Value Worksheet A 1 2

3 4 5 6 7 8 9 x 0 1 2 3 B 3 = Number of Trials (n ) 0.1 = Probability of Success (p ) f (x ) 0.729 0.243 0.027 0.001

Using Excel to Compute Cumulative Binomial Probabilities Formula Worksheet A 1 2 3 4 5 6 7 8 9 x 0 1 2 3 B

3 = Number of Trials (n ) 0.1 = Probability of Success (p ) Cumulative Probability =BINOMDIST(A5,$A$1,$A$2,TRUE) =BINOMDIST(A6,$A$1,$A$2,TRUE) =BINOMDIST(A7,$A$1,$A$2,TRUE) =BINOMDIST(A8,$A$1,$A$2,TRUE) Using Excel to Compute Cumulative Binomial Probabilities Value Worksheet A 1 2 3 4 5 6 7 8 9

x 0 1 2 3 B 3 = Number of Trials (n ) 0.1 = Probability of Success (p ) Cumulative Probability 0.729 0.972 0.999 1.000 Expected Value and Variance for a Binominal Distribution The expected value is computed by: E ( x) n The variance is computed by:

Var ( x) 2 n (1 ) Evans Electronics Example Remember that n = 3 and = .1. Thus: E ( x) 3(.1) .3 Var ( x) 3(.1)(.9) .27 Note also that: .27 .52

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