# Why do different probability distributions have different restrictions on their parameters?

Is it correct that the parameters of the following distributions must be taken from the intervals given below?

1. Bernoulli. $p$ from $[0, 1]$

2. Binomial. $n$ positive integer, $p$ from $[0, 1]$

3. Geometric. $p$ from $(0, 1)$

4. Poisson. $\lambda \geq 0$

I am not sure about parameters of geometric distributions.

Could you please explain why the ranges of parameters are different? If what I wrote for geometric is true, why can't $p$ be $0$ or $1$?

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You are considering probability distributions for $\Bbb N_0$.

For any $n\in \Bbb N_0$ such a distribution gives the probability for a random variable to be $n$. In order to be a probability distribution the sum of all these probabilities, that is the probability that the random variable is any natural number, must be $1$.

This works fine for your 1) and 2) (the parameter ranges are indeed right).

For 3) you could even include $1$, meaning that the geometrically distributed random variable has always value $0$. However, $p=0$ cannot be included. The reason is, that this would cause the probability of all natural numbers to be $0$. So for $p=0$ we don't get a probability distribution.

Same problem for 4) with $\lambda = 0$, however in this case we could define $P(X=0):=1$ so this might work depending on the definition.

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