Poisson Distribution

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Allan Blankenship
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1 Poisson Distribution Activity Name and Table: Poisson Distribution Learning Goals and Objectives 1. Students will understand what it means for biological processes to be Poissonian. 2. Students will appreciate the relationship between the average occurrence and the probability of it occurring. 3. Students will carry out the process of scientific inquiry. 4. Students will define a Poisson distribution. 5. Students will generate a data set on the probability of a T cell being infected with a virus(es). 6. Students will predict the likelihood of one observing the mean value of viruses occurring. 7. Students will evaluate the outcomes of a random process. 8. Students will hypothesize whether a process is Poissonian and design a test for that hypothesis. 9. Students will collect data and create a histogram from their data. Characteristics and Formula The following characteristics help identify a Poisson distribution (PD): 1. The probability that an event occurs in a region is the same for all regions of equal size. A region could be a unit of time, an event, a volume, a particular location, etc. 2. The number of events that occur in a given region is independent of the number of events that occur in a different region. 3. The probability of getting k instances of a given event is: P (k) = λk k! e λ Where the mean of the number of events, < k >, is equal to λ. Also worth noting is that the variance, σk 2, is also equal to the mean, λ.

2 Poisson Distribution Page 2 of 5 Poisson Distribution Preactivity For the following assume that the mean of our PD is equal to one (λ = 1). 1. Calculate the probability that there are zero events i.e. P (0). 2. Calculate the probability that there is one event i.e. P (1). 3. Calculate the probability that there are two or more events i.e. P (2 or more). 4. Plot your results for the probability of each event: Activity: HIV in T cells In the following activity we will be going through a toy example of how HIV infects T cells. T cells will be represented by Easter eggs and HIV will be represented by pieces of candy. Design Question Let s begin with a design question for this experiment: If a million HIV are mixed with a million T cells how likely is a T-cell in the population to be uninfected by HIV? Why? Page 2

3 Poisson Distribution Page 3 of 5 Poisson Distribution Activity Each step will have a discussion question directly associated with it but please go through all the steps sequentially and answer the questions as your group completes the activity. 1. Take an egg out and put a piece of candy in it then put the egg back into the plastic bag. Pass the plastic bag to the next person. Why are eggs drawn one at a time? 2. Before picking out an egg, mix the plastic bag thoroughly then pick out an egg, place a piece of candy in the egg, put the egg back in the plastic bag and then pass the plastic bag to your neighbor. Why is it necessary to mix the plastic bag of Easter eggs before drawing another egg? 3. Continue the above two steps until your group is out of candy. How will the candies be distributed? What will the average number of candies per egg (λ) be? Will there be λ pieces of candy in each egg? 4. Once completed it is now time to record the number of viruses in each cell. In your group, record the number of times a cell had zero viruses, one, or two or more viruses. 5. What you have just created is a histogram meaning a plot that simply shows the number of times something occured. How does one convert from a histogram to a plot of the probability of getting k events? Do your graphs have an interesting shape? Can you explain why or why not? 6. How could this experiment be redesigned so that it is quite unlikely to draw a cell without any viruses in it? Page 3

4 Poisson Distribution Page 4 of 5 Poisson Distribution Virology In Virology the average number of viruses in cells is called the multiplicity of infection, MOI. Now the PD would be: P (n) = mn n! e m Where m is the multiplicity of infection and n is the number of viruses in that specific cell. Discussion Questions For the following hypothetical situations, discuss whether or not each could be predicted via the PD. Give reasons to justify your responses. 1. The arrival of s in your inbox. 2. The number of snowy days during a week in winter. 3. The number of minutes late the shuttle is. 4. The height of people in the room. Page 4

5 Poisson Distribution Page 5 of 5 Poisson Distribution 5. The number of goals in a soccer match. 6. The number of dead limbs on a tree in the forest. 7. The distribution of clouds in the sky. Page 5

Lesson 8: Graphs of Simple Non Linear Functions

Lesson 8: Graphs of Simple Non Linear Functions Student Outcomes Students examine the average rate of change for non linear functions and learn that, unlike linear functions, non linear functions do not have a constant rate of change. Students determine