If we consider the Poisson distribution with λ = 0.5, the plot would show that the probability of observing zero events is the highest, followed by the probability of observing one event, and so on. As λ increases to 1 and 2, the blot shifts to the right, indicating that the probability of observing more events increases. However, this does not imply that larger λ means we have more trials. The Poisson distribution describes a scenario where events occur randomly in time or space, and the value of λ represents the average rate of events per unit of time or space.
For the Poisson distribution it is always the case that the variance is equal to the expectation, namely to λ: E(X) = Var(X) = λ is that because when the Var(X) = np(1-p) and p is small, (1-p) is approaching 1 , correct?
It is actually No, the statement E(X) = Var(X) = λ for the Poisson distribution is actually true regardless of the value of p, as the Poisson distribution is not related to the binomial distribution formula of Var(X) = np(1-p).
The Poisson distribution has a single parameter λ which represents both its mean and variance. Specifically, the mean and variance of a Poisson random variable X are both equal to λ, i.e., E(X) = Var(X) = λ. This is a fundamental property of the Poisson distribution that can be derived mathematically from its probability mass function.
The equality E(X) = Var(X) = λ for the Poisson distribution means that the spread of the distribution around its mean is equal to the mean itself. In other words, the Poisson distribution has a specific "shape" where the probability of observing values around the mean is highest, and this probability decreases as we move further away from the mean. This is often visualized as a bell curve or a symmetric "hump" centered around the mean value.
When E(X) = Var(X) = λ, it also means that the probability of observing very large or very small values is relatively low. For example, if λ = 5, the probability of observing a value of 10 or more is only about 0.02, and the probability of observing a value of 20 or more is less than 0.0001. On the other hand, the probability of observing a value of 4 or less is about 0.26, and the probability of observing a value of 3 or less is about 0.12.
In summary, E(X) = Var(X) = λ for the Poisson distribution tells us about the expected value and the spread of the distribution, as well as the probabilities of observing different values. This property is useful in many applications, such as modeling rare events, counting occurrences of certain phenomena, and analyzing queuing systems.
Reference
Yakir, B. (2011). Introduction to statistical thinking (with R, without Calculus). The Hebrew University of Jerusalem, Department of Statistics.
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