III. Poisson Probability Distribution (Chapter 5.3)

Transcription

1 III. Poisson Probability Distribution (Chapter 5.3) The Poisson Probability Distribution is a discrete probability distribution for the counts of events that occur randomly in a given interval. The interval can be a unit of time, area, length, volume, or any physical area in which there can be more than one occurrence of an event. A Poisson random variable X = the number of events in a given interval and has these characteristics: 1) The experiment consists of counting the number of times, k, an event occurs in a given interval. The interval can be and interval of time, space, area or volume. 2) The probability of the event occurring is the same for each interval (of time, space, area, or volume). 3) The number of occurrences of the event in one interval is independent of the number of occurrences in other intervals. 4) The mean number of events (i.e. expected number of events; expected value), denoted λt, is known over the interval (of time, volume, area, length). The variance of a Poisson is also equal to λt, and the standard deviation is equal to. 5) The Poisson random variable can take value from zero to infinity ( ). In contrast, the Binomial distribution always has a finite upper limit equal the number of trials n. Probabilities for a Poisson distribution. 1) The probability of exactly k successes over the same interval of time (volume, area, etc.) is, ( ), k = 0, 1, 2,. Where: t = Number of segments of interest. k = Number of success in t segments. e = Base of natural logarithm (e ), there should a button on your calculator that calculates powers of e. The population mean and standard deviation are given by and σ = 2) The probability of at most k success is P(X k) is given by the Cumulative Poisson Probability Distribution Table Appendix C. Steps for Solving a Poisson Probability Distribution Problem: 1) Define segment unit (minute, hour, page, liter, week, day ) 2) Find λ, the expected number of successes over the given interval of time, area, space, population, etc. 3) Determine t, the number of units to be considered, and then calculate. 4) Determine the event of interest and use the Poisson Formula, Poisson Cumulative Distribution Table or Excel Function POISSON.DIST to find the required probability. Example 1: Suppose that the mean number of customers who arrive at a bank is equal 3 per minute. Use the formula of a Poisson probability distribution to find: (Round off to 4 decimal places) 1) The probability that in a given minute two customers will arrive 2) The probability that in a given minute less than three customers will arrive 3) The probability that in a given minute more than two customers will arrive 4) The probability that in a given two minutes two customers will arrive 5) The probability that in a given two minutes less than three customers will arrive 6) The probability that in a given two minute more than two customers will arrive

2 Solution: ) ) [ ] ) ( ( ) ) ) ) [ ] [ ] Example 2: Solve example 1 above using the Poisson Distribution Tables. 1) 2) 3) 4) ( ( ) ) 5) 6) Example 3: Assume the errors a secretary makes per page are known to follow a Poisson distribution with mean of 2 errors per page. Use the formula of a Poisson probability distribution to find: 1) The probability that no errors will be found in a given page. 2) The probability that in a given page less than two errors will found. 3) The probability that in a given page more than two errors will found. 4) The probability that in a given two pages three errors will be found. 5) The probability that in a given two pages at least three errors will be found. 6) The probability that in a given two pages at most three errors will be found. Solution: ) ( ) ) [ ] [ ]

3 ) [ ( )] [ ] [ ( )] Example 4: Solve example 3 above using the Poisson Cumulative Distribution Tables. ) ) ) ) Example 5: Use the Poisson Cumulative Distribution Table to answer the following questions, for 1) Exactly three 2) Fewer than 3 / less than , direct from the table 3) Three or fewer / less than or equal 3 / no more than 3 ( ), direct from the table 4) More than 3 / greater than 3 5) Three or more / greater than or equal 3 /no less than 3 = ) Between 3 and 6 / more 3 and less than 6/ between 3 and 6 exclusive ( ) 7) Between 3 and 6 inclusive / from 3 but no more than 6 8) From 3 but less than 6 ( ) = ) More than 3 but no more than 6 ( ) =0.4558

4 Example 6: Use Excel Function POISSON.DIST to answer the following questions for answer to 4 decimal places) 1) Exactly three (round the 2) Fewer than 3 / less than ) Three or fewer / less than or equal 3 / no more than 3 4) More than 3 / greater than 3 ( ) 5) Three or more / greater than or equal 3 /no less than 3 6) Between 3 and 6 / more 3 and less than 6/ between 3 and 6 exclusive ( ) 7) Between 3 and 6 inclusive / from 3 but no more than 6 ( ) ( ) 8) From 3 but less than 6 ( ) = ) More than 3 but no more than 6 ( ) Example 7: A boat fisherman on a lake catches on the average 0.6 fish per hour. Suppose you decided to fish the lake on a boat, use the Poisson Cumulative Distribution Table to find: 1) The expected number of fish caught over 5-hour period ( ). 2) The probability that you will catch 0, 1, or 3 fish in 7 hours? ( )

5 3) The probability that you will catch at least 5 fish in 3 hours? ( ( )( ) 4) The probability that you will catch at most 3 fish in 5 hours? ( ( )( ) 5) The probability that you will catch more than 4 fish in 2 hours? ( ( )( ) 6) The probability that you will catch between 3 and 8 fish (inclusive) in 4 hours? ( ( )( ) ) 7) The probability that you will catch between 3 and 8 fish in 4 hours? ( ( )( ) ) 8) The probability that you will catch 3 fish in 30 minutes? ( ( ) ) Example 8: A loom which produces plaid wool fabric is known to produce, on the average, one noticeable flaw per 20 yards of fabric, find: 1) The expected number of flaws in 40 yards of fabric. ( ) 2) The expected number of flaws in 60 yards of fabric. ( ) 3) The probability that there will be no flaws in a fifty-yard piece of the wool fabric. ( ( ) ) 4) The probability that there will be 4 flaws in a fifty-yard piece of the wool fabric. (

6 5) The probability that there will be 4 flaws in a 30-yard piece of the wool fabric. ( 6) The probability that there will be more than 2 flaws in a 10-yard piece of fabric. 7) The probability that there will be no more than 2 flaws in a 30-yard piece of fabric. ( ( ) ) 8) The probability that there will be no less than 2 flaws in a 30-yard piece of fabric. 9) The probability that there will be less than 2 flaws in a 30-yard piece of fabric. 10) The probability that there will be at least 2 flaws in a 30-yard piece of fabric. 11) The probability that there will be at most 2 flaws in a 30-yard piece of fabric. ( ( ) ) 12) The probability that there will be between and 4 and 6 flaws (inclusive) in a 30-yard piece of fabric? ( ( ) ) 13) The probability that there will be between and 4 and 6 flaws (inclusive) in a 30-yard piece of fabric? ( ( ) ) 14) The probability that there will be between and 4 and 6 flaws (exclusive) in a 30-yard piece of fabric? ( ( ) )