Question

From the content section: 5.13 Weekly Activities:

Norman called today to say he had found a way for us to model the number of customer phone calls per hour. He had been thinking about other statistical factors that might play into our planning, so he got out some of his old stats texts and started browsing through them. Apparently by using something called a Poisson distribution, we could find the likelihood of large volumes of phone calls during any hour of the day. This could really help with staffing decisions.

Norman had already done a little research into other new bakeries and determined that we could expect an average of 4.4 phone calls per hour. Using a Poisson table, I think I can figure this out.

1. What is the probability of more than 3 phone calls per hour?
2. With current staffing plans, the bakery could handle up to 6 phone calls per hour during its busiest periods. What is the probability of more than 6 phone calls during one of the busy hours of the day? Is this a concern?
3. Where do we encounter Binomial or Poisson processes in everyday life?
4. Are there any tools in Excel to help us work with Binomial or Poisson distributions?
5. In what areas of your daily lives are normal curves used to describe things?

HINT: Remember - to find the probability of more than 3 calls you will need to use the concept of complimentary events, i.e. probability of more than 3 would be P(X>3) = 1 - P(X<=3)

Answer 1

x	poisson P(X<x)
0	0.012277
1	0.066298
2	0.185142
3	0.359448
4	0.551184
5	0.719912
6	0.843645
7	0.921421
8	0.964197
9	0.98511
10	0.994312
11	0.997992
12	0.999342
13	0.999799
14	0.999942
15	0.999984
16	0.999996
17	0.999999

These cumulative probability values for Poisson distribution are found using Excel with the command

=Poisson.dist(corresponding x value, parameter, 1)

The probability mass function for Poisson distribution is given by

$P\left ( X=x \right ) = \frac{e^{^{-\alpha }}\alpha ^{x}}{x!} , x=0,1,2,3,.... and \alpha > 0$

Here, since Poisson mean is the parameter itself.

a = 4.4

We need P(more than 3 calls),ie.

$P(X> 3)=1-P(X\leq 3)$ = 1- 0.359448 = 0.6405

Similarly , $P(X> 6)=1-P(X\leq 6)$ =1- 0.843645 = 0.15635

Binomial distribution is used in daily life in any situation that needs a probability of the certain no. of successes or failures. Eg:the no. of male/female workers in acompany.

For Poisson, the most commonly said example is that which is given in the question. Besides, the no. of errors in a book also follow Poisoon distribution

Applying central limit Theorem, as n becomes large almost all distributions tends to normality. Deciding the probability of error in any clinical experiment is done using normal distribution