Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
North Seattle Community College BUS210 Business Statistics Chapter 6 Discrete Probability Distributions Learning Objectives In this section, you will learn about: Probability distributions and their properties Important discrete probability distributions: The Binomial Distribution The Poisson Distribution Calculating the expected value and variance Using the Binomial and Poisson distributions to solve business problems BUS210: Business Statistics Discrete Probability Distributions - 2 Definitions Random variable: representation of a possible numerical value from an uncertain event. Discrete random variables: “How many” produce outcomes that come from counting (e.g. number of courses you are taking). Continuous random variables: “How much” produce outcomes that come from a measurement (e.g. your annual salary, or your weight). BUS210: Business Statistics Discrete Probability Distributions - 3 Random Variables Random Variables Discrete Random Variable Ch. 6 BUS210: Business Statistics Continuous Random Variable Ch. 7 Discrete Probability Distributions - 4 Discrete Random Variables Can only assume a countable number of values Examples: Rolling five dice (as in Yahtzee) If X is the number of times that a 6 shows up, then X could be equal to 0, 1, 2, 3, 4, or 5 Flipping a coin 3 times. If X is the number of times that heads comes up, then X could be equal to 0, 1, 2, or 3 BUS210: Business Statistics Discrete Probability Distributions - 5 Discrete Random Variables A probability distribution is… a mutually exclusive listing of all possible numerical outcomes for a variable and the probability of an occurrence associated with each outcome. Example for a discrete random variable: BUS210: Business Statistics Number of Classes Taken Probability 1 0.20 2 0.10 3 0.40 4 0.24 5 or more 0.06 Notice that the total probability adds up to 1.00 Discrete Probability Distributions - 6 Discrete Random Variable Probability Distribution 4 possible outcomes Experiment: Toss 2 Coins. Let X = # heads. T T T H H H BUS210: Business Statistics T H Probability Probability Distribution 0.50 0.25 0 X 1 2 X Freq Probability 0 1 1/4 = 0.25 1 2 2/4 = 0.50 2 1 1/4 = 0.25 Total = 1.00 Discrete Probability Distributions - 7 Discrete Random Variable Expected Value Expected Value is the mean of a discrete random variable E(X) = m = N å X P(X ) i i Also known as the Weighted Average i=1 Example: Experiment of 2 Tossed Coins i X P(X) XiP(Xi) X1 0 1/4 = 0.25 (0)(0.25) = 0.00 X2 1 2/4 = 0.50 (1)(0.50) = 0.50 X3 2 1/4 = 0.25 (2)(0.25) = 0.50 E(X) = 1.00 BUS210: Business Statistics Discrete Probability Distributions - 8 Discrete Random Variable Measuring Dispersion Variance of a discrete random variable N s 2 = å[X i - E(X)]2 P(X i ) i=1 Standard Deviation of a discrete random variable s = s2 = N 2 [X E(X)] P(X i ) å i i=1 Look familiar? This is ( X - m ) BUS210: Business Statistics 2 Discrete Probability Distributions - 9 Discrete Random Variable Measuring Dispersion (continued) Example: standard deviation of 2 tossed coins s= s = 2 N 2 [X E(X)] P(X i ) = 0.50 = 0.707 å i i=1 X E(X) X-E(X) (X-E(X))2 P(X) (X-E(X))2P(X) X1 0 1 -1 1 0.25 0.25 X2 1 1 0 0 0.50 0.00 X3 2 1 +1 1 0.25 0.25 ∑= 0.50 BUS210: Business Statistics Discrete Probability Distributions - 10 Probability Distributions Probability Distributions Discrete Probability Distributions Continuous Probability Distributions Binomial Normal Poisson Exponential Ch. 6 BUS210: Business Statistics Ch. 7 Discrete Probability Distributions - 11 Probability Distributions Binomial A finite number of observations, n 20 tosses of a coin 12 auto batteries purchased from a wholesaler Each observation is categorized as… Success: “event of interest” has occurred, or Failure: “event of interest” has not occurred i.e. - heads (or tails) in each toss of a coin; i.e. - defective (or not defective) battery The complement of a success These two categories, success & failure, are mutually exclusive and collectively exhaustive BUS210: Business Statistics Note: A random experiment with only 2 outcomes is known as a Bernoulli experiment. Discrete Probability Distributions - 12 Probability Distributions Binomial The probability of… success (event occurring) is represented as π failure (event not occurring) is 1 – π Probability of success (π) is constant (continued) Probability of getting tails is the same for each toss of the coin Observations are independent Each event is unaffected by any other event Two sampling methods deliver independence Infinite population without replacement Finite population with replacement BUS210: Business Statistics Discrete Probability Distributions - 13 Probability Distributions Binomial Applications: (continued) Situations where are there only two outcomes Apple marks newly manufactured iPads as either defective or acceptable Visitors to Amazon’s website will either buy an item or not buy an item. Asking if voters will approve a referendum, a pollster receives responses of “yes” or “no” Federal Express marks a delivery as either damaged or not damaged BUS210: Business Statistics Discrete Probability Distributions - 14 Probability Distributions Binomial (continued) Counting Revisited: You toss a coin three times. In how many ways can you get two heads? Possible ways: HHT, HTH, THH, so there are three ways you can getting two heads. This situation is fairly simple….but, what about 10 times?…a 100?...or even a thousand? For more complicated situations we need a method to be able to count the number of ways (combinations). BUS210: Business Statistics Discrete Probability Distributions - 15 Counting Techniques Rule of Combinations The number of combinations of selecting x objects out of n objects is n! n Cx = x!(n - x)! where: n! means “n factorial” 2! = (1)(2) 4! = (1)(2)(3)(4) Note: 0! = 1 (by definition) BUS210: Business Statistics Discrete Probability Distributions - 16 Counting Techniques Rule of Combinations You visit Baskin & Robbins. How many different possible 3 scoop combinations could you decide on for your ice cream cone if you select from their 31 flavors? The total choices is n = 31, and we select x = 3. 31! 31! 31·30 ·29 ·28! = = 31 C3 = 3!(31- 3)! 3!28! 3·2 ·1·28! = BUS210: Business Statistics 31·30 ·29 = 31·5·29 = 4495 combinations 3·2 ·1 Discrete Probability Distributions - 17 Probability Distributions Binomial n! P(x) = π x (1-π) n-x n!(n-x)! π = probability of “event of interest” n = sample size (number of trials or observations) x = number of “events of interest” in sample for the number of heads in 3 coin flips, we would use… π = 0.5 n =4 X = can be 0, or 1, or 2, or 3 and (1 – π) = 0.5 Each value of X will have a different probability æ nö Note: other sources may show this equation as f (x) = ç ÷ p x (1- p)(n-x) or similar. è xø BUS210: Business Statistics Discrete Probability Distributions - 18 Probability Distributions Binomial Example What is the probability of 2 successes in 7 observations if the probability of the event of interest is 0.4? x = 1, n = 7, and π = 0.4 n! P(x = 2) = p x (1- p )n-x x!(n - x)! 7! = (0.4)2 (1- 0.4)7-2 2!(7 - 2)! = (21)(0.4)2 (0.6)5 = 0.2613 BUS210: Business Statistics Discrete Probability Distributions - 19 Probability Distributions Binomial Example The probability of a single customer purchasing an extended warranty is 0.35. What is the probability of having 3 of the next 10 customers purchase an extended warranty? x=3 ; n = 10 ; p = 0.35 n! so, P(X=3) = p X (1- p )n-X X!(n - X)! 10! = (.35)3 (1-.35)10-3 3!(10 - 3)! = (120)(.042875)(0.04902) = 0.2522 BUS210: Business Statistics Discrete Probability Distributions - 20 Probability Distributions Using the Binomial Table n x … π=.20 π=.25 π=.30 π=.35 π=.40 π=.45 π=.50 10 0 1 2 3 4 5 6 7 8 9 10 … … … … … … … … … … … 0.1074 0.2684 0.3020 0.2013 0.0881 0.0264 0.0055 0.0008 0.0001 0.0000 0.0000 0.0563 0.1877 0.2816 0.2503 0.1460 0.0584 0.0162 0.0031 0.0004 0.0000 0.0000 0.0282 0.1211 0.2335 0.2668 0.2001 0.1029 0.0368 0.0090 0.0014 0.0001 0.0000 0.0135 0.0725 0.1757 0.2522 0.2377 0.1536 0.0689 0.0212 0.0043 0.0005 0.0000 0.0060 0.0403 0.1209 0.2150 0.2508 0.2007 0.1115 0.0425 0.0106 0.0016 0.0001 0.0025 0.0207 0.0763 0.1665 0.2384 0.2340 0.1596 0.0746 0.0229 0.0042 0.0003 0.0010 0.0098 0.0439 0.1172 0.2051 0.2461 0.2051 0.1172 0.0439 0.0098 0.0010 10 9 8 7 6 5 4 3 2 1 0 … π=.80 π=.75 π=.70 π=.65 π=.60 π=.55 π=.50 x Examples: BUS210: Business Statistics n = 10, π = .35, x = 3 P(x = 3) = .2522 n = 10, π = .75, x = 2 P(x = 2) = .0004 Discrete Probability Distributions - 21 Probability Distributions Shape of the Binomial The shape of the binomial distribution… depends on the values of π and n n = 5 π = 0.5 n = 5 π = 0.1 P(X) P(X) .6 .4 .6 .4 .2 .2 X 0 0 1 BUS210: Business Statistics 2 3 4 5 0 X 0 1 2 3 4 5 Discrete Probability Distributions - 22 Binomial Distribution Characteristics m = E(x) = np Mean Variance and Standard Deviation s = np (1- p ) 2 s = np (1- p ) Where n = sample size π = probability of the event of interest for any trial (1 – π) = probability of no event of interest for any trial BUS210: Business Statistics Discrete Probability Distributions - 23 Binomial Distribution Characteristics EXAMPLES m = np = (5)(.1) = 0.5 s = np (1- p ) = (5)(.1)(1- .1) = 0.6708 m = np = (5)(.5) = 2.5 s = np (1- p ) = (5)(.5)(1- .5) = 1.118 BUS210: Business Statistics n = 5 π = 0.1 P(X) .6 .4 .2 X 0 0 1 2 3 4 5 n = 5 π = 0.5 P(X) .6 .4 .2 0 X 0 1 2 3 4 5 Discrete Probability Distributions - 24 Binomial Distribution Characteristics Compound Events Individual probabilities can be added to obtain any desired combined event probability. Examples: •The probability that less than 3 of the next 10 customers will purchase an extended warranty is P(x<3) = P(x=0) + P(x=1) + P(x=2) = 0.0135 + 0.0725 + 0.1757 = 0.2617 • The probability that 3 or fewer of the next 10 customers will purchase an extended warranty is P(x<3) = P(x=0) + P(x=1) + P(x=2) + P(x=3) = 0.0135 + 0.0725 + 0.1757 + 0.2522 = 0.5139 BUS210: Business Statistics Discrete Probability Distributions - 25 Binomial Distribution Using Excel Cell Formulas nxπ n x π x (1-π) x BUS210: Business Statistics n π cum Discrete Probability Distributions - 26 Probability Distributions Probability Distributions Discrete Probability Distributions Continuous Probability Distributions Binomial Normal Poisson Exponential Ch. 6 BUS210: Business Statistics Ch. 7 Discrete Probability Distributions - 27 Probability Distributions The Poisson Distribution Used when you are interested in… the number of times an event occurs within a continuous unit or interval. Was actually used in a study of malaria Such as time, distance, volume, or area. Examples: The number of… potholes in mile of road (distance) computer crashes in a day (time) chocolate chips in a cookie (volume) mosquito bites on a person (area) BUS210: Business Statistics Discrete Probability Distributions - 28 Probability Distributions The Poisson Distribution Defining features: Independent: The occurrence of any single event does not impact the occurrence of any other event. Constant: The average probability of an event occurring in one area of opportunity remains the same for all other similar areas. The average number of events per unit is represented by (lambda) The average probability of an event decreases or increases proportionally to any decrease or increase in the size of the area of opportunity. BUS210: Business Statistics Discrete Probability Distributions - 29 Probability Distributions The Poisson Distribution -λ λx e P(x) = x! where: x = number of events in an area of opportunity = expected value (mean) of number of events e = base of the natural logarithm system (2.71828...) Note: other sources may show this equation as f (x) = BUS210: Business Statistics m x e- m x! or similar. Discrete Probability Distributions - 30 Poisson Distribution Characteristics Mean E(x) = m = l Variance and Standard Deviation Isn’t that interesting? s =l 2 s= l where = expected number of events BUS210: Business Statistics Discrete Probability Distributions - 31 Probability Distributions Using the Poisson Table X 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0 1 2 3 4 5 6 7 0.9048 0.0905 0.0045 0.0002 0.0000 0.0000 0.0000 0.0000 0.8187 0.1637 0.0164 0.0011 0.0001 0.0000 0.0000 0.0000 0.7408 0.2222 0.0333 0.0033 0.0003 0.0000 0.0000 0.0000 0.6703 0.2681 0.0536 0.0072 0.0007 0.0001 0.0000 0.0000 0.6065 0.3033 0.0758 0.0126 0.0016 0.0002 0.0000 0.0000 0.5488 0.3293 0.0988 0.0198 0.0030 0.0004 0.0000 0.0000 0.4966 0.3476 0.1217 0.0284 0.0050 0.0007 0.0001 0.0000 0.4493 0.3595 0.1438 0.0383 0.0077 0.0012 0.0002 0.0000 0.4066 0.3659 0.1647 0.0494 0.0111 0.0020 0.0003 0.0000 Example: Find P(X = 2) if = 0.50 e- l lX e-0.50 (0.50) 2 P(X = 2) = = = 0.0758 X! 2! BUS210: Business Statistics Discrete Probability Distributions - 32 Poisson Distribution Using Excel Cell Formulas x BUS210: Business Statistics λ cum Discrete Probability Distributions - 33 Probability Distributions Graphing the Poisson 0.70 0.60 0.50 X P(X) 0 1 2 3 4 5 6 7 0.6065 0.3033 0.0758 0.0126 0.0016 0.0002 0.0000 0.0000 BUS210: Business Statistics P(x) = 0.50 0.40 0.30 0.20 0.10 0.00 0 1 2 3 4 5 6 7 x P(X=2) = 0.0758 Discrete Probability Distributions - 34 Probability Distributions Shape of the Poisson The shape of the Poisson Distribution depends on the parameter : 0.70 0.25 = 0.50 0.60 = 3.00 0.20 0.40 P(x) P(x) 0.50 0.30 0.15 0.10 0.20 0.05 0.10 0.00 0.00 0 1 2 3 4 x BUS210: Business Statistics 5 6 7 1 2 3 4 5 6 7 8 9 10 11 12 x Discrete Probability Distributions - 35 Chapter Summary Probability distributions Discrete random variables The Binomial distribution The Poisson distribution BUS210: Business Statistics Discrete Probability Distributions - 36