Poisson Distribution in Statistics and Mathematics

Contents hide

1 Poisson Distribution-

2 Poisson Distribution is Limiting Case of Binomial Distribution

3 Moment Generating Function of Poisson Random Variable

4 Expectation Poisson Distribution Random Variable

5 Variance of Poisson Random Variable

6 Skewness

7 Kurtosis

8 Skewness and Kurtosis of Poisson Random Variable

9 References

10 See Also:

Poisson Distribution-

The Poisson distribution is defined by a single parameter, usually denoted by λ (lambda), which represents the average rate of occurrence of the events. The probability mass function (PMF) of the Poisson distribution is given in the image above.

Poisson Distribution is Limiting Case of Binomial Distribution

It’s worth noting that the Poisson distribution is a limiting case of the binomial distribution for rare events as the number of trials ( $n$ ) approaches infinity and the probability of success ( $p$ ) approaches zero, while keeping the product $n p$ constant.

Moment Generating Function of Poisson Random Variable

Expectation Poisson Distribution Random Variable

The expectation, or mean, of a Poisson random variable is equal to its parameter λ. In mathematical terms, if X is a Poisson random variable, then its expectation E(X) is given by:

$E (X) = λ$

Where λ is the average rate at which events occur in a fixed interval of time or space. See how expectation is calculated using moment generating function.

Variance of Poisson Random Variable

The variance of a Poisson random variable is also equal to its parameter λ. In mathematical terms, if X is a Poisson random variable, then its variance Var(X) is given by:

$Va r (X) = λ$

So, the variance of a Poisson distribution is also λ, which is the average rate at which events occur in a fixed interval of time or space.

Skewness

Skewness is another statistical measure used to describe the shape of a distribution. It indicates the asymmetry of the probability distribution of a real-valued random variable about its mean.

A distribution can be positively skewed, negatively skewed, or approximately symmetric (zero skewness).

Positive skewness: The tail on the right side of the distribution is longer or fatter than the left side, and the mass of the distribution is concentrated on the left side. The mean is typically greater than the median in a positively skewed distribution.
Negative skewness: The tail on the left side of the distribution is longer or fatter than the right side, and the mass of the distribution is concentrated on the right side. The mean is typically less than the median in a negatively skewed distribution.
Zero skewness: The distribution is symmetric, meaning the two sides of the distribution are mirror images of each other. In this case, the mean and the median are equal.

Skewness provides insight into the direction and degree of asymmetry present in a distribution, which can be important for understanding and analyzing data.

Kurtosis

Kurtosis is a statistical measure that describes the shape, or peakedness, of the probability distribution of a real-valued random variable. It measures the tails of a distribution compared to the normal distribution.

A distribution with positive kurtosis (leptokurtic) has heavier tails and a higher peak than the normal distribution, indicating more extreme values. Conversely, a distribution with negative kurtosis (platykurtic) has lighter tails and a lower peak, indicating fewer extreme values. A distribution with zero kurtosis (mesokurtic) has a shape similar to that of the normal distribution.

In essence, kurtosis tells us how much of the variance is the result of infrequent extreme deviations, compared to frequent modest deviations from the mean. High kurtosis indicates that most of the variance comes from infrequent extreme deviations, while low kurtosis indicates most of the variance comes from frequent modest deviations.

Skewness and Kurtosis of Poisson Random Variable

The skewness of a Poisson distribution can vary depending on the parameter λ, which represents the mean and variance of the distribution. The formula for the skewness of a Poisson distribution is:

This formula indicates that the skewness of a Poisson distribution decreases as the parameter λ increases. When λ is small, the distribution is more positively skewed, meaning it has a longer tail on the right side. As λ increases, the distribution becomes less skewed and approaches symmetry.

The skewness of a Poisson distribution is always positive, indicating that it is always positively skewed. This is because the Poisson distribution is bounded on the left side but extends indefinitely to the right, leading to more variability and probability mass on the right side of the distribution.

The kurtosis of a Poisson distribution can be calculated using the following formula:

This formula indicates that the kurtosis of a Poisson distribution is inversely proportional to the parameter λ, which represents the mean and variance of the distribution. As λ increases, the kurtosis decreases, and as λ decreases, the kurtosis increases.

The kurtosis of a Poisson distribution is always positive, meaning it is always leptokurtic (having heavier tails than a normal distribution). This indicates that the tails of the distribution are more extreme compared to a normal distribution, which is consistent with the characteristics of a Poisson distribution.

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References

https://en.wikipedia.org/wiki/Poisson_distribution

https://proofwiki.org/Expectation_of_Poisson_Distributio

https://www.math.net/poisson-distribution