Question

1. The Binomial and Poisson Distributions

Both the Binomial and Poisson Distributions deal with discrete data where we are counting the number of occurrences of an event. However, they are very different distributions.  This problem will help you be able to recognize a random variable that belongs to the Binomial Distribution, the Poisson Distribution or neither.

Characteristics of a Binomial Distribution

Characteristics of a Poisson Distribution

The Binomial random variable is the count of the number of success in n trials:   number of successes in n trials of an experiment. There are only two possible outcomes in each trial.  One outcome is denoted as a success (S) and the other is a failure (F). There are a fixed number of  identical trials. The probability of S remains the same from trial to trial. The probability of success is denoted by . The  trials are independent. (The result for one trial does not depend on the results of other trials.) The Binomial distribution can be described by two parameters:  – number of trials, and         – probability of success.

The Poisson random variable is the count of the number of success that occur in a segment:              number of successes in one segment. The segment can be a unit of time, space or any well-defined unit of concern.   We know the average number of success in one segment.  The average is denoted by . The probability of a success in one segment is the same for all segments of the same size. We do not know the number of non-successes. Successes occur one at a time. The occurrence of a success in any segment is independent of the occurrence of a success in any other segment. The Poisson distribution can be described with one parameter: – the mean.

1. In the following situations, indicate whether the random variable is a Binomial Random Variable, Poisson Random Variable, or Neither and explain your choice using the above characteristics. If the random variable is Binomial what are the values of the two parameters  and ?  If the random variable is Poisson what is the value of the parameter  and what is the segment of time or space?

1. According to flightstats.com, American Airlines flights from Dallas to Chicago are on time 80% of the time.  Suppose 15 flights are randomly selected, and let  be the number of on-time flights.  Assume that the arrivals of the planes are independent of one another.  Is a Binomial random variable, a Poisson random variable, or neither?

2. A basketball player makes any given free throw with probability 0.6. During practice the player attempts 10 free throws.  Let  be the number of free throws the player makes.  Assume that the free throw attempts are independent of one another. Is  a Binomial random variable, a Poisson random variable, or neither?

3. You are taking a 1-day African safari. The safari company claims that on average, its clients see 5 lions.  Let  be the number of lions seen during the safari.  Assume that the number of lions seen during each safari is independent of another.  Is  a Binomial random variable, a Poisson random variable, or neither?

4. Allen Thompson is a life insurance salesman for the XYZ Life Insurance Company. On average, he sells 3 policies per week.  Let  be the number of policies sold during the week.  Assume that the number of policies sold during each week is independent of any other week.  Is  a Binomial random variable, a Poisson random variable, or neither?

5. Suppose a baseball team needs to win 4 out of 7 games in a championship game series. Once a team reaches the 4^thwin it is declared the winner of the series and the series ends.  Suppose Clemson is playing USC.  Assume the probability that Clemson wins a game remains the same for each game in the series and that winning a game is independent of winning any other game during the series.  Let  be the number of games played in the series.  Is  a Binomial random variable, a Poisson random variable, or neither?

2. According to the Statistical Abstract of the United States, traffic fatalities occur at the rate of 1.5 deaths per 100 million miles. Assume that traffic fatalities follow a Poisson distribution and that  = 1.5 deaths per 100 million miles. We have discussed the fact that for a Poisson random variable, if we double the length of the segment, the new value for  is doubled.  Follow the steps below to perform a simulation to demonstrate that the number of traffic fatalities in 200 million miles follow a Poisson distribution with an average of 2 = 3 deaths per 200 million miles.

1. Open Minitab and select Calc >> Random Data >> Poisson.  In Number of rows of data to generateenter 1000.  Enter C1 into Store in column(s).  Enter 1.5 for the Mean.  Give C1 a title.  You now have 1000 observations of the number of deaths that occur in 100 million miles.

2. Generate another 1000 observations for a second road segment of length 100 million miles by repeating step i and saving the data in C2. Give C2 a title to distinguish it from C1.

3. Add the values in columns 1 and 2 and put the results into column 3.  To add columns 1 and 2 select Calc >> Calculator.  Enter C3 into Store result in variable.  Enter C1+C2 into Expressionand click OK.  

4. Create a histogram from the data in column 1 and a separate histogram for the data in column 3. Be sure to create the histograms separately.  Using Minitab calculate the Mean, Standard Deviation, Median and IQR for columns 1 and 3. Paste the histograms and requested statistics in the space below.
   
   

5. Write a few sentences to compare the two distributions.  Remember to compare the shapes, centers, spread, and any unusual values.
   
   

6. Create a 4^thcolumn of data that comes from a Poisson distribution with λ=3.   Use the same procedure as step i, except use a Meanof 3 and store the data in column 4. Give the column a title.  Compare the mean and standard deviation of column 3 where we added the two Poisson random variables together to the mean and standard deviation of column 4 whose data came from a Poisson distribution with .  Is this what you expected to see?  Explain.   

Answer 1

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