Using tables of the Poisson cumulative distribution function

This article is a continuation from the previous poisson cumulative article, so it may be a good idea to read through it before continuing. For this article you’ll need a copy of the Poisson cumulative distribution table. It has been attached to this article.
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This article explores using tables of Poisson cumulative distribution function. Below are some examples and conversations;

Example

In this example the random variable X ~ Po(3). Suppose we wanted to find the following;

• P(X ≤ 5)
• P(X = 2)
• P(X > 3)
• P(4 ≤ X < 8)

First we will start with P(X ≤ 5). Here it is best to note down what values we have; We know that;
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However using the Poisson cumulative distribution table is a lot quicker. Here we can simply use the table of Poisson cumulative distribution function with λ = 3 and x = 5. So we get;
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Next we shall find P(X = 2). To find this we use the idea that;
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We find P(X ≤ 2) the find P(X ≤ 1) and find the difference of the two.
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P(X > 3) is the same as P(X ≥ 4) and P(X ≥ 4) is equal to 1 – P(X ≤ 3) so we have the expression as;
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Next we shall find P(4 ≤ X ≤ 7), this is the same as;
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We shall write the probability as a difference of two cumulative probabilities;
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Example

Suppose we wanted to find x given the probability. For example suppose the random variable X ~ Po(7.5). And suppose we wanted to find the values of a, b, c such that;

• P(X ≤ a) = 0.2414
• P(X < b) = 0.5246
• P(X ≥ c) = 0.338

For P(X ≤ a) = 0.2414 we look in the tables of Poisson cumulative distribution to find where P(X ≤ x) = 0.2414 and λ = 7.5 and use the value of x. You will find that;
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For P(X < b) = 0.5246 we use; [IMAGE] We use the tables of Poisson cumulative distribution to find where P(X ≤ x) = 0.5246 and λ = 7.5, we find that x = 7, therefore we have; [IMAGE] Next we shall work with P(X ≥ c) = 0.338. So we can rearrange the expression above to make the term with the ≤ symbol the subject. Remember we can only use the tables of Poisson cumulative distribution when P(X ≤ x), so we have; [IMAGE] But we know P(X ≥ c) = 0.338, so we have; [IMAGE] ...therefore we use the tables of Poisson of cumulative distribution to find x where P(X ≤ x) = 0.6620 and λ = 7.5, we get; [IMAGE]

Applications

This is an application of Poisson distribution; In a survey the number of accidents on a stretch of motorway was conducted over a period of one year. Statistics shows that the mean number of accidents per month was 1.5 and the standard deviation was 1.2, it was also predicted that the number of accidents per month could be modelled by a Poisson distribution.
A Poisson distribution is used to model the number of accidents on the motorway in a randomly chosen month.
In this example we can see that;
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Since the variance is similar to the mean a Poisson distribution may be appropriate. This is a very important property to remember about Poisson distribution. A situation can be modelled by Poisson distribution if the mean = the variance.
Suppose in this example we wanted to find the probability of no more than 2 accidents; Here we let;
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So that we have;
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In this example no more than means “less or equal to” i.e X ≤ 2. So we’re trying to find;
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We use the Poisson cumulative tables with λ = 1.5 and x = 2. We find that;
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So the probability of no more than 2 accidents is 0.81.