# SOCR EduMaterials Activities Poisson Distribution

(Difference between revisions)
 Revision as of 17:19, 6 August 2007 (view source)IvoDinov (Talk | contribs)← Older edit Revision as of 17:20, 6 August 2007 (view source)IvoDinov (Talk | contribs) m Newer edit → Line 20: Line 20: * '''Exercise 5:'''  People enter a gambling casino at a rate of 1 for every two minutes. * '''Exercise 5:'''  People enter a gambling casino at a rate of 1 for every two minutes. ** What is the probability that no one enters between 12:00 and 12:05? ** What is the probability that no one enters between 12:00 and 12:05? - * What is the probability that at least 4 people enter the casino during that time? + ** What is the probability that at least 4 people enter the casino during that time? * '''Exercise 6:''' Let $X_1$ denote the number of vehicles passing a particular point on the eastbound lane of a highway in 1 hour. Suppose that the Poisson distribution with mean $\lambda_1=5$ is a reasonable model for $X_1$. Now, let $X_2$ denote the number of vehicles passing a point on the westbound lane of the same highway in 1 hour. Suppose that $X_2$ has a Poisson distribution with mean $\lambda_2= 3$. Of interest is $Y= X_1 + X_2$, the total traffic count in both lanes in one hour.  What is the  $P(Y < 5) ?$ * '''Exercise 6:''' Let $X_1$ denote the number of vehicles passing a particular point on the eastbound lane of a highway in 1 hour. Suppose that the Poisson distribution with mean $\lambda_1=5$ is a reasonable model for $X_1$. Now, let $X_2$ denote the number of vehicles passing a point on the westbound lane of the same highway in 1 hour. Suppose that $X_2$ has a Poisson distribution with mean $\lambda_2= 3$. Of interest is $Y= X_1 + X_2$, the total traffic count in both lanes in one hour.  What is the  $P(Y < 5) ?$

## This is an activity to explore the Poisson Probability Distribution.

• Exercise 1: Use SOCR to graph and print the distribution of a Poisson random variable with λ = 2. What is the shape of this distribution?
• Exercise 2: Use SOCR to graph and print the distribution of a Poisson random variable with λ = 15. What is the shape of this distribution? What happens when you keep increasing λ?
• Exercise 3: Let $X \sim Poisson(5)$. Find $P(3 \le X < 10)$, and $P(X >10 | X \ge 4)$.
• Exercise 4: Poisson approximation to binomial: Graph and print $X \sim b(60, 0.02)$. Approximate this probability distribution using Poisson. Choose three values of X and compute the probability for each one using Poisson and then using binomial. How good is the approximation?

Below you can see the distribution of a Poisson random variable with λ = 5. In this graph you can also see the probability that between 2 and 5 events will occur. • Exercise 5: People enter a gambling casino at a rate of 1 for every two minutes.
• What is the probability that no one enters between 12:00 and 12:05?
• What is the probability that at least 4 people enter the casino during that time?
• Exercise 6: Let X1 denote the number of vehicles passing a particular point on the eastbound lane of a highway in 1 hour. Suppose that the Poisson distribution with mean λ1 = 5 is a reasonable model for X1. Now, let X2 denote the number of vehicles passing a point on the westbound lane of the same highway in 1 hour. Suppose that X2 has a Poisson distribution with mean λ2 = 3. Of interest is Y = X1 + X2, the total traffic count in both lanes in one hour. What is the P(Y < 5)?