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Probability distributions are mathematical functions that describe the likelihood of different outcomes in a random event or experiment. Two common probability distributions are the Binomial Distribution and Poisson Distribution. These distributions are frequently used in statistics to model the behavior of discrete data.

This article will explore the definitions, formulas, applications, mean and variance, probability density functions, relationship between the two distributions, limitations, and assumptions of Binomial and Poisson distributions.

The Binomial Distribution is a probability distribution that describes the number of successes in a fixed number of independent trials with two possible outcomes, usually labeled as success (S) or failure (F). The trials must be independent and have the same probability of success. The probability of success is denoted as p, and the probability of failure is denoted as q = 1 - p.

The formula for the Binomial Distribution is:

```
P(X = k) = (n choose k) * p^k * q^(n-k)
```

where:

- P(X = k) is the probability of getting exactly k successes in n trials
- n is the total number of trials
- k is the number of successes
- (n choose k) is the binomial coefficient, calculated as n!/(k!(n-k)!)
- p is the probability of success
- q is the probability of failure (q = 1 - p)

For example, suppose a company is launching a new product and conducts a survey of 100 potential customers. The probability of a customer buying the product is 0.2. The company wants to know the probability that exactly 25 customers will buy the product. Using the Binomial Distribution formula, we have:

```
P(X = 25) = (100 choose 25) * 0.2^25 * 0.8^75
= 0.086
```

Therefore, the probability of exactly 25 customers buying the product is 0.086 or 8.6%.

The Poisson Distribution is a probability distribution that describes the number of events that occur in a fixed interval of time or space, assuming that the events occur independently and at a constant rate. The rate of occurrence is denoted as λ.

The formula for the Poisson Distribution is:

```
P(X = k) = (e^-λ * λ^k) / k!
```

where:

- P(X = k) is the probability of getting exactly k events in a fixed interval
- e is the mathematical constant 2.71828...
- λ is the rate of occurrence of events
- k is the number of events

For example, suppose a store receives an average of 10 customers per hour. The store wants to know the probability of having exactly 12 customers in the next hour. Using the Poisson Distribution formula, we have:

```
P(X = 12) = (e^-10 * 10^12) / 12!
= 0.094
```

Therefore, the probability of having exactly 12 customers in the next hour is 0.094 or 9.4%.

Binomial Distribution and Poisson Distribution have different applications in real-world scenarios. Here are a few cases:

- Binomial Distribution can be utilized to calculate the likelihood of victory or disappointment in a fixed number of trials. For case, it can be utilized to demonstrate the likelihood of winning or losing a game, the likelihood of passing or failing a test, or the likelihood of getting a certain number of heads or tails in a series of coin flips.
- Poisson Distribution can be utilized to calculate the likelihood of a certain number of occasions happening in a fixed interim of time or space. For illustration, it can be utilized to demonstrate the likelihood of mishaps or incidents happening in a certain range, the likelihood of phone calls received by a call centre in a certain period, or the likelihood of clients entering a store in a certain time outline.

The mean and variance of a probability distribution are measures of its central tendency and variability, respectively. The mean of the Binomial Distribution is given by:

```
μ = np
```

where:

- μ is the mean or expected value of the distribution
- n is the total number of trials
- p is the probability of success

The variance of the Binomial Distribution is given by:

```
σ^2 = npq
```

where:

- σ^2 is the variance of the distribution
- n is the total number of trials
- p is the probability of success
- q is the probability of failure

The mean and variance of the Poisson Distribution are both equal to the rate parameter λ, i.e.,

```
μ = σ^2 = λ
```

For example, using the same scenario as before, the mean and variance of the Binomial Distribution for the company launching a new product are:

```
μ = 100 * 0.2 = 20
σ^2 = 100 * 0.2 * 0.8 = 16
```

Therefore, the expected number of customers buying the product is 20, and the variance is 16.

Using the same scenario for the store receiving an average of 10 customers per hour, the mean and variance of the Poisson Distribution are:

```
μ = σ^2 = λ = 10
```

Therefore, the expected number of customers in the next hour is 10, and the variance is also 10.

The probability density function (PDF) of a probability distribution is a function that describes the probability of a random variable taking on a certain value. The PDF of the Binomial Distribution is given by:

```
f(k) = (n choose k) * p^k * q^(n-k)
```

where:

- f(k) is the probability density function
- k is the number of successes
- n is the total number of trials
- p is the probability of success
- q is the probability of failure (q = 1 - p)

The PDF of the Poisson Distribution is given by:

```
f(k) = (e^-λ * λ^k) / k!
```

where:

- f(k) is the probability density function
- k is the number of events
- e is the mathematical constant 2.71828...
- λ is the rate of occurrence of events

Poisson Distribution PDF

The Binomial Distribution can be approximated by the Poisson Distribution when the number of trials is large and the probability of success is small. Specifically, if np < 10, the Binomial Distribution can be approximated by the Poisson Distribution with a rate parameter of λ = np. In this case, the formula for the Poisson Distribution can be used to calculate the probabilities.

For example, suppose a company sells 10,000 products and the probability of a defective product is 0.001. Using the Binomial Distribution formula, the probability of having exactly 5 defective products is:

```
P(X = 5) = (10,000 choose 5) * 0.001^5 * 0.999^9,995
= 0.036
```

However, since np = 10, which is less than 10, the Binomial Distribution cannot be approximated by the Poisson Distribution. Therefore, we need to use the Binomial Distribution formula to calculate the probability.

Both the Binomial and Poisson Distributions have certain limitations and assumptions that need to be considered when using them to model real-world phenomena. Some of the main limitations and assumptions are:

- Binomial Distribution:
- The trials must be independent and identically distributed.
- The number of trials n must be fixed.
- The probability of success p must be constant for all trials.
- Each trial must have only two possible outcomes (success or failure).

- Poisson Distribution:
- The events must occur independently of each other.
- The rate of occurrence of events λ must be constant over time.
- The probability of an event occurring in a small time interval is proportional to the length of the interval.
- The events must be rare, i.e., the rate of occurrence λ must be small.

Violation of these assumptions can lead to inaccurate modelling and predictions. For example, if the probability of success in the Binomial Distribution varies for different trials or if the events in the Poisson Distribution are not independent, the resulting distribution may not accurately reflect the real-world phenomenon being modelled.

The Binomial and Poisson Distributions are powerful tools for modeling and predicting the probabilities of discrete events. While they have different applications and characteristics, they are both important tools in probability theory and statistics. Understanding their formulas, properties, and assumptions can help researchers, analysts, and decision-makers make informed decisions and predictions in a wide range of fields, from business and finance to healthcare and engineering.

- The Binomial Distribution models the probability of a fixed number of successes in a fixed number of independent trials with a constant probability of success.
- The Poisson Distribution models the probability of a certain number of events occurring in a fixed interval of time or space, assuming a constant rate of occurrence.
- The mean and variance of the Binomial Distribution are determined by the number of trials and the probability of success, while the mean and variance of the Poisson Distribution are both equal to the rate parameter λ.
- The probability density function of the Binomial Distribution is given by a formula that takes into account the number of trials and the probability of success, while the probability density function of the Poisson Distribution is based on the rate of occurrence of events.
- The Binomial Distribution can be approximated by the Poisson Distribution when the number of trials is large and the probability of success is small, with a rate parameter of λ = np.
- Both the Binomial and Poisson Distributions have assumptions and limitations that need to be considered when using them to model real-world phenomena.
- Understanding the formulas, properties, and assumptions of these distributions can help researchers, analysts, and decision-makers make informed decisions and predictions in various fields.

**1. What type of distribution models the probability of a certain number of events occurring in a fixed interval of time or space, assuming a constant rate of occurrence?**

A. Binomial Distribution

B. Poisson Distribution

C. Normal Distribution

D. Exponential Distribution

**Answer**: B. Poisson Distribution

**2. What is the mean of the Binomial Distribution?**

A. np

B. p

C. q

D. n

**Answer**: A. np

**3. Which distribution can be approximated by the Poisson Distribution when the number of trials is large and the probability of success is small?**

A. Poisson Distribution

B. Binomial Distribution

C. Normal Distribution

D. Exponential Distribution

**Answer**: B. Binomial Distribution

**4. What is one of the assumptions of the Poisson Distribution?**

A. The trials must be independent and identically distributed.

B. The number of trials must be fixed.

C. Each trial must have only two possible outcomes.

D. The rate of occurrence of events must be constant over time.

**Answer**: D. The rate of occurrence of events must be constant over time.

Module 5: Probability Distributions

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