# Discrete uniform distribution vs. binomial distribution; what's the difference?

I read online that a uniform distribution gives to all its values the same probability to occur. In the discrete case, an example of this would be a coin flip. (as they have the same probability to occur) Doesn't this also fall under the binomial distribution, as they are independent trials, and the probability of success stays constant? (0.5)

Sorry if this is a ridiculous question, I am just learning the distributions and having trouble making them intuitive yet.

Thank you!

## 1 Answer

A uniform distribution on $\{0,1\}$ and a Bernoulli distribution with $p=0.5$ (or alternatively a binomial distribution with $n=1$ and $p=0.5$) are the same distribution.

• Yes, and to further clarify: A discrete uniform distribution could be something like {0, 1, 2}. That is, it can contain more than two discrete outcomes, while a Bernoulli is always {0, 1}. Of course, we could generalize to a multinomial distribution, where a multinomial distribution {0, 1, 2} with probabilities of .3333... each would be the same as a discrete uniform along {0, 1, 2} – Mark White Jan 6 '18 at 0:54
• Thank you very much! Would rolling a die be an example of this (uniform, but not Bernoulli)? – Stats_anon Jan 6 '18 at 20:03
• Yes it would (if it was a fair die). This would correspond to the discrete uniform distribution on $\{1,2,3,4,5,6\}$, which is the same distribution as the (6 category) categorical distribution (en.wikipedia.org/wiki/Categorical_distribution, a special case of the multinomial distribution) with $p_1=\cdots=p_6=1/6$. – aleshing Jan 6 '18 at 22:33