Question

Discuss how a manager of a retail store can use both the binomial and Poisson distributions to make business decisions. Explain how the distributions differ. Provide examples of the type of data that could be used in calculating the probabilities.

Answer 1

In order to use a binomial distribution, we need to know how many events we're measuring and what the probability of individual success or failure is. Most importantly, each individual result must be independent of the results from any other trial. Given this, a binomial distribution measures the probability of a specified number of positive or negative results.

In our retail store case, we know that on the busiest day we can expect 150 rentals, which forms the number of independent events or trials. We also know that, historically, 60% of our customers rent skis and 40% rent snowboards, which provides our probability. If we decide that we only need to have 65 snowboards in stock, what is the probability that we will run out of snowboard rentals on any specific day?

Formulas and lookup tables can be used for these calculations, but it is common to use a spreadsheet or statistical program to calculate the binomial probability. In this case, we want to know the probability that 66 or more customers out of 150 will want to rent a snowboard.

P(failure>65, trials=150, probability=0.40) = 13.9%.

This number is statistically significant, and indicates that we should increase our stock. For example, simply increasing our stock to 70 results in a much lower chance of failure:

P(failure>70, trials=150, probability=0.40) = 2.8%.

The Poisson Distribution is a tool used in probability theory statistics to predict the amount of variation from a known average rate of occurrence, within a given time frame.

In other words, if the average rate at which a specific event happens within a specified time frame is known or can be determined (e.g., Event “A” happens, on average, “x” times per hour), then the Poisson Distribution can be used as follows:

To determine how much variation there will likely be from that average number of occurrences
To determine the probable maximum and minimum number of times the event will occur within the specified time frame.
store can utilize the Poisson Distribution to examine how they may be able to take steps to improve their operational efficiency. For instance, an analysis done with the Poisson Distribution might reveal how a company can arrange staffing in order to be able to better handle peak periods for customer service calls

difference between binomial and poisson distribution-

1. The difference between the two is that while both measure the number of certain random events (or "successes") within a certain frame, the Binomial is based on discrete events, while the Poisson is based on continuous events.

2. with a binomial distribution you have a certain number, nn, of "attempts," each of which has probability of success pp. With a Poisson distribution, you essentially have infinite attempts, with infinitesimal chance of success.

3.That is, given a Binomial distribution with some n,p, if you let n→∞ and p→0 in such a way that np→λ, then that distribution approaches a Poisson distribution with parameter λ.

examples

-binomial distribution

Q. A coin is tossed 10 times. What is the probability of getting exactly 6 heads?

I’m going to use this formula: b(x; n, P) – nCx Px (1 – P)n – x
The number of trials (n) is 10
The odds of success (“tossing a heads”) is 0.5 (So 1-p = 0.5)
x = 6

P(x=6) = 10C6 0.5^6 0.5^4 = 210 0.015625 0.0625 = 0.205078125.

-poisson distribution-

On a particular river, overflow floods occur once every 100 years on average. Calculate the probability of k = 0, 1, 2, 3, 4, 5, or 6 overflow floods in a 100-year interval, assuming the Poisson model is appropriate.

Because the average event rate is one overflow flood per 100 years, λ = 1

{\displaystyle P(k{\text{ overflow floods in 100 years}})={\frac {\lambda ^{k}e^{-\lambda }}{k!}}={\frac