Poisson Probability Distributions PDF

Section 5.5 Poisson Probability Distributions  Learning Targets The Poisson distribution is another discrete probability distribution which is important because it is often used for describing the behavior of rare events (with small probabilities). Some events are rather rare - they don'...

Section 5.5 Poisson Probability Distributions  Learning Targets The Poisson distribution is another discrete probability distribution which is important because it is often used for describing the behavior of rare events (with small probabilities). Some events are rather rare - they don't happen that often. For instance, car accidents are the exception rather than the rule. Still, over a period of time, we can say something about the nature of rare events. An example is the improvement of traffic safety, where the government wants to know whether seat belts reduce the number of death in car accidents. Here, the Poisson distribution can be a useful tool to answer questions about benefits of seat belt use. Other phenomena that often follow a Poisson distribution are death of infants, the number of misprints in a book, the number of customers arriving, and the number of activations of a Geiger counter. The distribution was derived by the French mathematician Siméon Poisson in 1837, and the first application was the description of the number of deaths by horse kicking in the Prussian army. Examples of Poisson Distributions Patients arriving at an emergency room. Radioactive decay. Crashes on a highway. Internet users logging onto a website.  Poisson Probability Distribution What is it? Requirements: A Poisson probability 1. The random variable x is distribution is a discrete the number of occurrences probability distribution that of an event over some applies to occurrences of interval. some event over a specified 2. The interval can be time, interval. The Poisson distance, area, volume, or distribution is often used for same similar unit. describing rare events (small 3. The occurrences must be probabilities). random. Parameters 4. The occurrences must be independent of each other. 5. The occurrences must be uniformly distributed over the interval being used. A formula for Calculating Probability The Formula The Components P(x): The probability of an event occurring x times over an interval. e: The number “e” where e = 2.71828. x: The number of occurrences of the event in an interval. μ: The mean which indicates the average number of events in the given time interval.  Poisson or not? Which of the following are likely to be well modelled by a Poisson distribution? (can check more than one) 1. Number of duds found when I test four components 2. The number of heart attacks in Oswego each year 3. The number of planes landing at O’Hare Airport between 8 and 9am 4. The number of cars getting punctures on the Route 59 each year 5. Number of people in the UK flooded out of their home in July Are they Poisson? Answers: Number of duds found when I test four components - NO: this is Binomial (it is not the number of independent random events in a continuous interval) The number of heart attacks in Oswego each year - YES: large population, no obvious correlations between heart attacks in different people The number of planes landing at O’Hare Airport between 8 and 9am - NO: 8-9am is rush hour, planes land regularly to land as many as possible (1-2 a minute) – they do not land at random times or they would hit each other! The number of cars getting punctures on the Route 59 each year - YES (roughly): If punctures are due to tires randomly wearing thin, then expect punctures to happen independently at random But: may not all be independent, e.g. if there is broken glass in one lane Number of people in the UK flooded out of their home in July - NO: floodings of different homes not at all independent; usually a flood will make many homes flooded at once.  MERCY 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? μ = 6/hour = 3/half-hour, X = 4  Example: Telecommunications Messages arrive at a switching centre at random and at an average rate of 1.2 per second. Find the probability of 5 messages arriving in a 2-sec interval.  Example Radioactive atoms are unstable because they have too much energy. When they release their extra energy, they are said to decay. When studying cesium-137, a nuclear engineer found that over 365 days, 1,000,000 radioactive atoms decayed to 977,287 radioactive atoms. Find the mean number of radioactive atoms that decayed in a day. μ = (1000000 – 977287)/365 =22713/365 = 62.2274 Find the probability that on a given day, 50 radioactive atoms decayed. P(50) = poissonpdf(62.2274, 50) = 0.0155 Example Suppose that trucks arrive at a receiving dock with an average arrival rate of 3 per hour. What is the probability exactly 5 trucks will arrive in a two-hour period? Example Suppose that trucks arrive at a receiving dock with an average arrival rate of 3 per hour. What is the probability exactly 5 trucks will arrive in a two-hour period? = 0.1606 Example Arrivals at a bus-stop follow a Poisson distribution with an average of 4.5 every quarter of an hour. Calculate the probability of fewer than 3 arrivals in a quarter of an hour. The probabilities of 0 up to 2 arrivals can be calculated directly from the formula with μ = 4.5 So P(0) = 0.01111, P(1)=0.04999, and P(2)=0.11248 So the probability of fewer than 3 arrivals is P(0) + P(1) + P(2) = 0.01111+ 0.04999 + 0.11248 = 0.17358 P(x < 3) = poissoncdf(4.5, 2) = 0.173578 Calculator Shortcut TI-83+ and TI-84 Direction Your Toolbox Press 2nd VARS (to get 1. Use a TI-83/84 Plus. DISTR), then select the 2. Use the formulas option identified as 3. BOOM DONE! poissonpdf / poissoncdf Complete the entry of poissonpdf( µ(λ ), x) or poissoncdf( µ(λ ), x) then press ENTER. Calculator Shortcut “2 nd ” and “VARS” Select “poisson pdf” to calculate individual probability value P (0) = poisson pdf(4.5, 0), P (1) = poisson pdf(4.5, 1), P (2) = poisson pdf(4.5, 2). Select “poisson cdf” to calculate the cumulative probability values P (0) + P (1) + P (2) = poisson cdf(4.5, 2).  Example Dutchess County, New York, has been experiencing a mean of 35.4 motor vehicle deaths each year. Find the mean number of deaths per day. µ = 35.4/365 = 0.0970 Find the probability that on a given day, there are more than 2 motor vehicle deaths. Is it unusual to have more than 2? It is very unusual to have more than 2 vehicle deaths on a given day.  More Practice Consider a Poisson probability distribution with an average number of occurrences of 2 per period. a. Write the appropriate Poisson distribution b. What is the average number of occurrences in three time periods? c. Write the appropriate Poisson function to determine the probability of k occurrences in three time periods. d. Compute the probability of two occurrences in one time period. e. Compute the probability of six occurrences in three time periods. f. Compute the probability of five occurrences in two time periods.  Answe r (a) (b) (c) (d) (e) (f)  Difference from a Binomial Distribution The Poisson distribution differs from the binomial distribution in these fundamental ways: ❖ The binomial distribution is affected by the sample size n and the probability p, whereas the Poisson distribution is affected only by the mean μ. ❖ In a binomial distribution the possible values of the random variable x are 0, 1,... n, but a Poisson distribution has possible x values of 0, 1, 2,... , with no upper limit. Condition of Poisson Distribution as an Approximation of Binomial Distribution: In Binomial Distribution, if n is large and p is small, then the Binomial distribution with parameters n and p, is well approximated by the Poisson distribution with parameter μ = np, i.e. by the Poisson distribution with the same mean.  Poisson Rule of Thumb The Poisson distribution Requirements for can also be used to Using the Poisson approximate the binomial Distribution as an distribution when using Approximation to the the binomial distribution Binomial if both would result in a tedious 1. n ≥ 100 calculation. However, 2. np ≤ 10 certain requirements must In this situation, we need a be met in order to do this. value for mu. That value can Not all problems can use a be calculated by using the Poisson distribution. formula present in section 5.4  Example In the Illinois Pick 3 game, you pay $0.50 to select a sequence of three digits, such as 729. If you play this game once every day, find the probability of winning exactly once in 365 days. n = 365 > 100, p = 1/1000 = 0.001, and μ = np = 0.365 < 10 P(1) = poissonpdf(0.365, 1) = 0.2534 (Poisson) P(1) = binompdf(365, 0.001, 1) = 0.2536 (Binomial)  Example The probability of a certain part failing within ten years is 10-6. Five million of the parts have been sold so far. What is the probability that three or more will fail within ten years?  A Lucky Die An experiment involves rolling a die 6 times and counting the number of 2’s that occur. If we calculate the probability of x = 0 occurrences of number 2 using the Poisson distribution we get 0.368, but we get 0.335 if we use the binomial distribution. Which is the correct probability of getting no 2’s when a die is rolled 6 times? Why is the other probability wrong? Mean and Standard Deviation of the Poisson Probability Distribution Formulas Remember: The mean is Mean is the same as The standard deviation is expected value This is what you would expect the average to be after an infinite number of trials Standard deviation represents the spread of the data that you would expect to see  Recap In this section we have discussed: ❖ Definition of the Poisson distribution. ❖ Requirements for the Poisson distribution. ❖ Difference between a Poisson distribution and a binomial distribution. ❖ Poisson approximation to the binomial.  Homework Pg. 238-239 #10, 12, 15 (use Binomial and Poisson distributions for #15)