Poisson and Geometric Distribution in Statistics.pptx

Statistics

1. Poisson and Geometric Distribution By Jeremiah Blessed Mudhangu M226102

2. Poison Distribution The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given that these events occur with a known constant mean rate and independently of the time since the last event

3. Key Characteristics Parameter (λ) : The distribution is characterized by a single parameter  λ λ  (lambda), which is the average number of events in the interval. Probability Mass Function (PMF) : The probability of observing  k k  events in an interval is given by the formula: P(X=k)= e−λλkk! P ( X = k )= k ! e − λλk ​ where: P(X=k) P ( X = k ) is the probability of  k k  events, e e  is Euler's number (approximately 2.71828), k! k ! is the factorial of  k k . Mean and Variance : Both the mean and variance of a Poisson distribution are equal to  λ λ .

4. Applications The Poisson distribution is commonly used in various fields, such as: Traffic flow : Modeling the number of cars passing through a toll booth in an hour. Call centers : Analyzing the number of incoming calls in a given time period. Biology : Counting the number of mutations in a given stretch of DNA.

5. Example If the average number of emails received per hour is 5 (i.e., λ=5 ), What is the probability of receiving exactly 3 emails in the next hour

6. Geometric Distribution The geometric distribution is a discrete probability distribution that models the number of trials needed to achieve the first success in a series of independent Bernoulli trials (experiments with two possible outcomes: success or failure).

7. Key Characteristics

8. Applications The geometric distribution is applicable in various scenarios, such as: Quality Control: Determining the number of items inspected until the first defective item is found. Sales: Calculating how many sales calls are needed until the first successful sale occurs. Games: Modeling the number of rolls of a die until the first "six" is rolled.