Probability Distributions: Poisson Distribution Flashcards
Poisson Distribution
- Poisson distribution is a discrete probability distribution
- It describes the mean number of events occurring in a fixed time interval
- Often groups with the Binomial Distribution
- The mean number of occurrences in Poisson distribution is denoted by λ.
- Lambda (λ) = Mean = Variance
- Unlike Binomial Distribution, Poisson distribution continues forever, and it is bounded by 0 and ∞
- The Poisson Distribution is quite useful when you desire to estimate the probabilities of events that occur randomly in some unit of measure
- e.g. the number of traffic accidents at a particular intersection per month
- Basis for the C and U control charts
- E.g. Number of patients arriving in emergency room between 11 and 12 pm.
When to use the Poisson Distribution
- Discrete function
- It measures only occurring or not occurring
- In other words, it measures only whole numbers and no fractions.
- Example: number of telephone calls in a month
- In other words, it measures only whole numbers and no fractions.
- Poisson distribution is also a very convenient distribution as it takes only one parameter
Assumptions of Poisson Distribution
- The probability of an occurrence is constant over time.
- In other words, the rate does not change based on time.
- Occurrences are independent
- In other words, the occurrence of one event does not affect the occurrence of a subsequent event
- It is also a discrete probability distribution
- No upper limit with the number of occurrences of an event during the specified time interval
- The probability of a single occurrence of an event within a specified time eriod is proportional to the length of the time period
- The probability of each occurrence is less than 0.1
- It is a positively skewed distribution
The Poisson Equation
The mean and variance are equal for the Poisson distribution
Example of Poisson Distribution
- XYZ is a call center and the operational process has 3.5% error rate. So, what is the probability of k (0, 1, 2, 3, 4, 5) errors?
- λ = 3.5
- x = 0, 1, 2, 3, 4, 5
Poisson and Time Intervals
- The number of occurrences of some vent follows a Poisson Distribution, the time between successive occurrences will follow an Exponential Distribution
Poisson Distribution Question Tips
- When we hear “What is the probability of occurrences?” in a question, we know it’s time to use Poisson. Also, there are examples of seeing or hearing the word ‘PER.’
- What is the probability of zero occurrences?
- Bugs PER megabyte of code
- Numbers of ‘sick days’ PER school year
Standard Deviation Formula
Square root of the mean
Given the following information:
Probability of 1 or more defects = 0.69
Probability of 2 or more defects = 0.34
Probability of 3 or more defects = 0.12
Probability of 4 or more defects = 0.03
What is the probability of 2 or fewer defects?
The probability of 3 or more defects is 0.12, the probability of 2 or fewer defects must be 1 - 0.12 = 0.88
Question 7.13
The average number of flaws in large plate glass is 0.25 per pane. The standard deviation of this Poisson distribution is:
C bar = Poisson average
Poisson sigma = square root of c bar
Poisson sigma = square roof of 0.25 = 0.500
The standard deviation is the square root of the mean/average
Example 1 (nobody, Or fewer, standard deviation)
Example 2 (or more)
Example 3 (Less than)
Example 4 (More than)
Example 5 (Exactly)
Poisson Distribution Formula and Example
