Poisson distribution

This entry explores Poisson distribution. By the end of this entry you’ll be able to know when a Poisson distribution is a suitable model to use, understand how to use probabilities using a Poisson distribution, be able to use tables of cumulative frequency distribution of a Poisson, know how and when to use the Poisson distribution as an approximation to the binomial distribution. Here are the key points that you’ll understand by the end of this entry.
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Knowledge of binomial distribution and factorial notation will be beneficial. First we will see how the exponential series relate to the Poisson distribution.
You may use your calculator to evaluate the exponential function e^x for various values of x. For instance e^-2> = 0.1353.. and e^0.5 = 1.6487… But we can also define the exponential function e^x as a series.
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…we know that λ⁰ = 1, if we let x = λ we get;
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…we can divide both sides by e to get;
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Above the sum of the infinite series on the right-hand side is equals to 1 and therefore we could use the values as probabilities to define a probability distribution. The table below illustrates this. Let X be a random variable such that X takes the values 0, 1, 2, 3, … then the probability distribution for X is;

The probability function is there;
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…and we say that X has a Poisson distribution with parameter λ and we write this as;
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…where…

• ~ means ‘is distributed’
• Po is for the Poisson distribution
• λ is the parameter

Below we shall look at some examples;

Example

The random variable X~Po(1.2) Find;
a) P(X = 3)
Here we simply use the following formula with λ = 1.2 and x=3.
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…we get;
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b) P(X ≥ 1)
For this we know that;
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…this is because;
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…so using the formula we can simplify the expression to;
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c)P(2Poisson distribution mean and variance

Next we shall explore how to find the mean and variance of Poisson distribution. Here are some key points to note;

• If the random variable X ~ Po(λ) then it can be shown that;
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In the following example we shall use the tables of the Poisson cumulative distribution.