Last modified: July 04, 2018
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Poisson Distribution (Discrete)
A discrete random variable X follows a Poisson distribution if the events occur independently and at a constant average rate. The Poisson distribution is denoted as X∼Poisson(λ), where λ is the average rate (or mean) of events occurring in a given interval.
Probability Mass Function (PMF)
The PMF of a Poisson distribution is given by:
P(X=k)=e−λλkk!
where k∈0,1,2,….
Cumulative Distribution Function (CDF)
The CDF of a Poisson distribution is given by:
F(k)=P(X≤k)=k∑i=0e−λλii!
Expected Value and Variance
The expected value (mean) and variance of a Poisson distribution are both equal to the average rate parameter λ:
E[X]=Var(X)=λ
Moment Generating Functions and Moments
The moment generating function (MGF) of a Poisson distribution is:
MX(t)=E[etX]=eλ(et−1)
To find the n-th moment, we take the n-th derivative of the MGF with respect to t and then evaluate it at t=0:
E[Xn]=dnMX(t)dtn|t=0
- First Moment (Mean):
E[X]=λ
- Second Moment (Variance + Mean^2):
E[X2]=λ2+λ
Example: Hospital Emergency Room Visits
A hospital's emergency room experiences an average of 5 visits per hour due to accidents. We can model this using a Poisson distribution.
Given:
- Average rate of success (visits per hour) λ=5
I. What is the probability exactly 3 accidents occur in an hour?
For exactly 3 accidents:
P(X=3)=e−5⋅533!=0.1404
II. What is the probability of more than 7 accidents in an hour?
For more than 7 accidents:
P(X>7)=1−7∑k=0e−5⋅5kk!=0.1339
III. What is the probability of at most 2 accidents in an hour?
For at most 2 accidents:
P(X≤2)=2∑k=0e−5⋅5kk!=0.1247
Poisson Approximation to the Binomial Distribution
If the number of trials in a binomial distribution is large (n) and the probability of success (p) is small, then the binomial distribution can be approximated by a Poisson distribution with parameter λ=np:
Binomial(n,p)≈Poisson(np)