DISCRETE PROBABILITY DISTRIBUTIONS DISCRETE PROBABILITY DISTRIBUTIONS Random Variables
DISCRETE PROBABILITY DISTRIBUTIONS
DISCRETE PROBABILITY DISTRIBUTIONS Random Variables Discrete Probability Distributions Expected Value and Variance The Binomial Probability Distribution The Poisson Probability Distribution
RANDOM VARIABLES A random variable is a numerical description of the outcome of an experiment. A discrete random variable may assume either a finite number of values or an infinite sequence of values.
FINITE AND INFINITE VALUES FOR DISCRETE RANDOM VARIABLES Discrete random variable with a finite number of values: Let x = number of TV sets sold in one day where x can take on 5 values (0, 1, 2, 3, 4) Discrete random variable with an infinite sequence of values: Let x = number of customers arriving in one day where x can take on the values 0, 1, 2, . . . We can count the customers arriving, but there is no finite upper limit on the number that might arrive.
DISCRETE RANDOM VARIABLES A Certified Public Accountant (CPA) examination has 4 parts. We can define a discrete random variable as x = the number of parts of CPA exam passed This discrete random variable may assume finite number of values of 0, 1, 2, 3, 4. We may make a call to sell a product. We can define a discrete random variable x. x = 1 if we sell, x = 0 if we don’t.
CONTINUOUS RANDOM VARIABLES A continuous random variable may assume any numerical value in an interval or collection of intervals. x = the time between arrival of two customers into a supper market. x = the exact weight of an individual
PROBABILITY DISTRIBUTIONS The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. The probability distribution is defined by a probability function, denoted by f(x), which provides the probability for each value of the random variable.
DISCRETE PROBABILITY DISTRIBUTIONS The required conditions for a discrete probability function are: f (x ) > 0 f (x ) = 1 We can describe a discrete probability distribution with a table, graph, or equation.
USING DATA TO COMPUTE PROBABILITIES Using past data on TV sales (below left), a tabular representation of the probability distribution for TV sales (below right) was developed. Number Units Sold of Days x f (x ) 0 80 0. 40 1 50 1. 25 2 40 2. 20 3 10 3. 05 4 20 4. 10 200 1. 00
GRAPHICAL REPRESENTATION OF A DISCRETE PROBABILITY DISTRIBUTION A graphical representation of the probability distribution for TV sales in one day. 50 Probability . 40. 30. 20. 10 0 1 2 3 4 Values of Random Variable x (TV sales)
A DISCRETE UNIFORM PROBABILITY DISTRIBUTION Discrete Uniform Probability Distribution x 1 2 3 4 5 6 f(x) = 1/n (n is the number of values that x takes) Rolling a die f(x) 1/6 1/6 1/6 1 2 3 4 5 6
EXPECTED VALUE AND VARIANCE The expected value, or mean, of a random variable is a measure of its central location. The variance summarizes the variability in the values of a random variable. Expected value of a discrete random variable: E (x ) = = xf (x ) Variance of a discrete random variable: Var(x ) = 2 = (x - )2 f (x ) The standard deviation, , is defined as the positive square root of the variance.
EXAMPLE Expected Value of a Discrete Random Variable x 0 1 2 3 4 f (x ). 40. 25. 20. 05. 10
EXAMPLE Expected Value of a Discrete Random Variable x f (x ) xf (x ) -------------0. 40. 00 1. 25 2. 20. 40 3. 05. 15 4. 10. 40 1. 20 = E (x ) The expected number of TV sets sold in a day is 1. 2
CALCULATE THE STANDARD DEVIATION Variance and Standard Deviation of a Discrete Random Variable x x- (x - )2 f (x ) _________ (x - )2 f (x ) ________
EXAMPLE: JSL APPLIANCES Variance and Standard Deviation of a Discrete Random Variable x x- (x - )2 f (x ) (x - )2 f (x ) _____ 0 1 2 3 4 ___________ -1. 2 -0. 2 0. 8 1. 8 2. 8 1. 44 0. 04 0. 64 3. 24 7. 84 . 40. 25. 20. 05. 10 ________ . 576. 010. 128. 162. 784 1. 660 = The variance of daily sales is 1. 66 TV sets. The standard deviation of sales is 1. 29 TV sets.
James Bernoulli (Jacob I) born in Basel, Switzerland Dec. 27, 1654 -Aug. 16, 1705 He is one 8 mathematicians in the Bernoulli family.
THE BINOMIAL PROBABILITY DISTRIBUTION The probability distribution associated with a random variable is called binomial probability distribution. Properties of a Binomial Experiment The experiment consists of a sequence of n identical trials. Two outcomes, success and failure, are possible on each trial. The probability of a success, denoted by p, does not change from trial to trial. The trials are independent.
For example , Consider the experiment of tossing a coin five times and on each toss observing whether the coin lands with a head or tail on its upward face. Suppose we want to count the number of heads appearing over the five tosses.
TESTING FOR 4 BINOMIAL PROPERTY. The experiment consist of five identical trials; each trial involves the tossing of one coin. Two outcomes are possible for each trial: a head or a tail. We can designate head a success and a tail as failure. The probability of each head and tail are same for each trial, with p =. 5 and 1 -p =. 5. The trials or tosses are independent because the outcome on any one trial is not affected by what happens on other trials or tosses.
BINOMIAL PROBABILITY FUNCTION Where = The probability of x success in n trials. N P 1 -p = The number of trials. = n! x!(n-x)! = The probability of a success on any one trial = The probability of a failure on any one trial
EXAMPLE: MARTIN CLOTHING STORE PROBLEM Consider the purchase decision of next three customers who enter the martin clothing store. On the basis of past experience , the store manager estimates the probability that any one customer will make a purchase is. 30. What is the probability that two of the next three customer will make a purchase?
Using a three diagram, we can see that the experiment of observing the three customers each making a purchase decision has eight possible outcomes. Using S to denote success(a purchase) and F to denote failure(no purchase), we are interested in experimental outcomes involving two success in the three trials(purchase decisions). Next, we verify that the experiment involving the sequence of three purchase decision can be viewed as binomial experiment. Checking the four requirement for a binomial experiment.
The experiment can be described as a sequence of three identical trials , one trial for each of the three customers who will enter the store. Two outcomes- the customer makes a purchase (success) or the customer does not make a purchase(failure) – are possible for each trial. The probability that the customer will make a purchase (. 30) or will not make a purchase (. 70) is assumed to be the same for all customer. The purchase decision of each customer is independent of the decision of the other customers.
EXAMPLE: SALES MAN Using a Tree Diagram First customer Second customer Third customer S S F F F S F = No purchase X = number of customer making a purchase 1 1 F S= purchase 1 2 S S 2 2 S F Value of x 3 0
PROBABILITY DISTRIBUTION FOR THE NUMBER OF CUSTOMERS MAKING PURCHASE X f(x) 0 . 343 1 . 441 2 . 189 3 . 027 =1. 000
THE BINOMIAL PROBABILITY DISTRIBUTION Expected Value E (x ) = = np Variance Var(x ) = 2 = np (1 - p ) Standard Deviation Example: Evans Electronics E (x ) = = 3(. 1) =. 3 employees out of 3 Var(x ) = 2 = 3(. 1)(. 9) =. 27
Siméon Denis Poisson June 21, 1781 -April 25, 1840
The Poisson Probability Distribution The Poisson experiment typically fits cases of rare events that occur over a fixed amount of time or within a specified region. Properties of a Poisson Experiment • The probability of an occurrence is the same for any two intervals of equal length. • The occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval.
Typical cases o. The number of errors a typist makes per page o. The number of customers entering a service station per hour o. The number of telephone calls received by a witch board per hour. o. Arrival of customers to a service station generally has Poisson distribution. o. Arrival of cars to a service station. o. Arrival of people to a restaurant. o. Arrival airplanes to an airport.
THE POISSON VARIABLE AND DISTRIBUTION The Poisson Random Variable The Poisson variable indicates the number of successes that occur during a given time interval or in a specific region in a Poisson experiment Probability Distribution of the Poisson Random Variable.
POISSON DISTRIBUTION The number of Typographical errors in new editions of textbooks is Poisson distributed with a mean of 1. 5 per 100 pages of a new book are randomly selected. What is the probability that there are no errors? Solution P(X=0)= e- x e-1. 50 = =. 2231 x! 0!
POISSON DISTRIBUTION Example For There are no type error. a 400 page book calculate the following probabilities There are five or fewer type error. Solution P(X=0)= P(X 5)=<use the formula to find p(0), p(1), …, p(5), then calculate p(0)+p(1)+…+p(5) =. 4457 e- x e-660 = =. 002479 x! 0! 33 Important! A mean of 1. 5 typos per 100 pages, is equivalent to 6
EXAMPLE: MERCY HOSPITAL Patients arrive at the emergency room of Mercy Hospital at the average rate of 6 per hour on weekend evenings. What is the probability of 4 arrivals in 30 minutes on a weekend evening? Using the Poisson Probability Function = 6/hour = 3/half-hour, x = 4 e- x x! e-3 3! 0. 168031
EXAMPLE: MERCY HOSPITAL Using the Tables of Poisson Probabilities
Hypergeometric Probability Distribution The hypergeometric distribution is closely related to the binomial distribution. The key differences are: the trials are not independent probability of success changes from trial to trial
Hyper geometric Probability Distribution for 0 < x < r where: f(x) = probability of x successes in n trials n = number of trials N = number of elements in the population r = number of elements in the population labeled success
Hypergeometric Probability Function is the number of ways a sample of size n can be selected from a population of size N. is the number of ways x successes can be selected from a total of r successes in the population. is the number of ways n – x failures can be selected from a total of N – r failures in the population.
Example: Neveready Hypergeometric Probability Distribution Bob Neveready has removed two dead batteries from a flashlight and accidentally place them with the two good batteries he intended as replacements. The four batteries look identical. Bob now randomly selects two of the four batteries. What is the probability he selects the two good batteries?
x = 2 = number of good batteries selected n = 2 = number of batteries selected N = 4 = number of batteries in total r = 2 = number of good batteries in total
Example: Neveready Hypergeometric Probability Distribution
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