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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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1 Answer 1

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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