Poisson Distribution Characteristics

• Probability Mass Function (PMF): PMF describes the likelihood of observing a specific number of events in a fixed interval. It is given by:
  P(X = k) = (e^-λ × λ^k) / k!

• Cumulative Distribution Function (CDF): CDF gives the probability that the random variable is less than or equal to a certain value. It is expressed as:
  F(x) = ∑(from k=0 to ⌊x⌋) (e^-λ × λ^k) / k!

• Moment Generating Function (MGF): MGF provides a way to derive moments of the distribution. It is represented by:
  M(t) = e^{(λ(e^t – 1))}

• Characteristic Function (CF): CF is an alternative way to describe the distribution and is given by:
  ϕ(t) = e^{(λ(e^(it) – 1))}

• Probability Generating Function (PGF): PGF generates the probabilities of the distribution and is expressed as:
  G(z) = e^{(λ(z – 1))}

• Median: Median, which represents the central value, is approximately λ+ (1/3)​−0.02/λ​.

• Mode: Mode, or the most probable value, is simply the integer part of λ, denoted as ⌊λ⌋.

• Mean and Variance: The mean (λ) and variance (λ) of a Poisson distribution are equal. This means that both the average number of events and the spread or variability around this average are characterized by the same parameter.

• Non-negative and Discrete: The Poisson distribution describes the probability of non-negative integer values only, as it models counts of events. It is a discrete probability distribution.

• Memorylessness: Events in a Poisson process are memoryless, meaning the probability of an event occurring in the future is independent of the past, given the current state. For example, if you’re waiting for a bus, the probability of the bus arriving in the next minute doesn’t depend on how long you’ve already been waiting.

• Independent Increments: The number of events occurring in non-overlapping intervals is independent. For instance, if you’re counting the number of cars passing through an intersection in one minute, the number of cars in the next minute is independent of the number in the previous minute.

• Rare Events Approximation: When the average rate of occurrence (λ) is large and the probability of a single event is small, the Poisson distribution can approximate the binomial distribution. This is known as the “rare events” approximation, where the binomial distribution with a large number of trials and a small probability of success converges to a Poisson distribution.

• Skewness and Kurtosis: Poisson distribution is positively skewed (skewness > 0) and leptokurtic (kurtosis > 0), meaning it has a longer tail on the right side and heavier tails than the normal distribution. However, for large values of λ, it becomes increasingly symmetric and bell-shaped, resembling a normal distribution.

Some other properties are:

• Poisson distribution has only one parameter “λ” where λ = np.

• Poisson distribution is positively skewed and leptokurtic.

Note: Here leptokurtic means values greater kurtosis than the normal distribution, and kurtosis is the nothing but the sharpness of the peak of the frequency distribution curve.

Poisson Distribution is one of the types of discrete probability distributions like binomial distribution in probability. It expresses the probability of a given number of events occurring in a fixed interval of time.

Poisson distribution is a type of discrete probability distribution that determines the likelihood of an event occurring a specific number of times (k) within a designated time or space interval. This distribution is characterized by a single parameter, λ (lambda), representing the average number of occurrences of the event.

In this article, we will discuss the Poisson Distribution including its definition, Poisson Distribution formula, Poisson Distribution examples, and properties of Poisson Distribution in detail.