Poisson Distribution
The Poisson distribution for count data, where independent events occur at a uniform average rate and X∼Po(λ) with λ=E(X)=Var(X), including probability calculations from technology and the fact that sums of independent Poisson variables are Poisson.
Definition of a Poisson random variable
The Poisson distribution is a discrete probability distribution used to calculate the number of occurrences of an event in a given interval of time or space. In order to use a Poisson distribution, the following conditions must be satisfied:
Events are independent
Events occur at some average rate which is uniform during the period of interest
If a random variable X follows a Poisson distribution, we write X∼Po(λ), where λ is the average number of occurrences of an event in a given interval. X takes on non-negative integer values, x∈{0,1,2,…}.
To model count data in a different interval than the one given, we can construct a different random variable Y, whose parameter depends on how much larger or smaller its interval is than X's. In particular, if the interval of X is considered one "unit" and Y models count data for m units, then Y∼Po(m⋅λ).
Calculating Poisson probabilities using technology
To calculate P(X=x),x∈{0,1,2,…} for X∼Po(λ) using a calculator, press 2nd →distr (on top of vars) to open the probability distribution menu. Select poissonpdf( with your cursor (or press alpha →C ). Type the value of λ after "λ" and the value of x after "x value." Then click paste and enter and the calculator will return the value of P(X=x).
To calculate P(X≤x),x∈{0,1,2,…} for X∼Po(λ) using a calculator, press 2nd →distr (on top of vars) to open the probability distribution menu. Select poissoncdf( with your cursor (or press alpha →D ). Type the value of λ after "λ" and the value of x after "x value." Then click paste and enter and the calculator will return the value of P(X≤x). Notice that using the cdf function on a calculator will always assume an inclusive less than or equal to "≤", so if you want to find a different inequality, you must adjust your calculation accordingly.
Mean and Variance of a Poisson are both λ
For a random variable X with X∼Po(λ), the parameter λ is equivalent to both the expectation and variance of the random variable X:
Sum of Poisson distributions is Poisson
The sum of independent, Poisson distributed random variables follows a Poisson distribution.
If X and Y are independent random variables with X∼Po(λX) and Y∼Po(λY), the random variable Z=X+Y follows the distribution
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