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The variance of a random variable is a measure of its spread around the mean . This means that it is a quantity that indicates the average dispersion of the values of said variable on both sides of the mean or the amplitude of its probability distribution. This parameter is an important quantity for any random variable, regardless of its probability distribution.
On the other hand, the Poisson distribution is a discrete probability distribution that serves to model the frequency with which discrete events occur within a time interval , although it can also be referred to in relation to other continuous variables, such as a length of a wire, a surface, etc.
The Poisson distribution is of great importance, since it allows modeling processes as daily as the number of people arriving in a line at the ticket office of an ATM, as well as processes as complex as the number of radioactive decays in a given time interval. from a sample of nuclear waste.
Mathematical definition of the Poisson distribution
A random variable X follows a Poisson distribution if its probability mass function or PMF has the following form:
In the formula, λ is an always positive parameter of the distribution and x represents the different values that the random variable can take. In Poisson processes, the parameter λ generally represents speed or frequency per unit time, per unit area, and so on.
As we will show later, λ is, in turn, the mean of the Poisson distribution, as well as its variance.
Now that we know what this distribution function is and what it is for, let’s look at a more formal definition of variance, the general way to calculate it, and finally, how the variance is calculated for the particular case of the Poisson distribution.
What is the variance?
Mathematically, the variance of a random variable X, denoted in statistics by Var(X) , corresponds to the expected value of the square of the deviation of said variable from its mean, which is expressed with the following formula:
Although the previous definition can be used to calculate the variance of any random variable, it can also be calculated more easily using the first and second ordinary moments, or moments around the origin (m 1 , m 2 ) as follows :
This way of calculating the variance is more convenient than the first, so it will be the one we will use in this article to calculate the variance of the Poisson distribution.
Calculation of the variance of the Poisson distribution
Calculation of the average or first ordinary moment
Let us remember that, for any discrete distribution, the mean or expectation of X can be determined by means of the following expression, which defines the first moment:
We can take this sum from x=1 onwards, since the first term is zero. Also, if we now multiply and divide everything by λ and also replace x!/x with (x-1)! , we obtain:
This expression can be simplified by making the change of variable y = x – 1 , leaving:
The function inside the summation is again the Poisson probability function, which, by definition, is the summation of all probabilities from zero to infinity of any probability function that must equal 1.
We already have the first moment or the mean of the Poisson function. We will now use this result and the expectation of the square of X to find the variance.
Calculation of the second ordinary moment
The second moment is given by:
We can use a little trick to solve this sum that consists of replacing x 2 by x(x-1)+x:
Where we use the previous result in the second term of the summation, we multiply and divide by λ 2 to obtain the exponent λ x-2 and we apply the change of variable y = x – 2 .
Now all that remains is to replace these two moments in the formula for the variance, and we will have the expected result:
References
Devore, J. (2021). Probability and Statistics for Engineering and Science . CENGAGE LEARNING.
Rodó, P. (2020, November 4). Poisson distribution . Economipedia. https://economipedia.com/definiciones/distribucion-de-poisson.html
UNAM [Luis Rincon]. (2013, December 16). 0625 Poisson Distribution [Video]. Youtube. https://www.youtube.com/watch?v=y_dOx8FhHpQ
