What is the difference between the two? They seem to mean the same thing to me. The probability mass function can be used to find the probability of getting a tail from a coin flip: X{1 tail}=P(X)=P(H),P(T)=1/2
The Bernoulli distribution equation goes like: (1/2)^1(1-.5)^(1-1)=1/2 because p(success) is 1/2, and p(failure) is 1-(1/2).
Are they the same thing? If not what types of problems would one shine over the other? I know that we can use the Bernoulli distribution as the basis of the Binomial distribution to find the number of tails in n trials but is that it? Is the difference that the bernouli distribution equation can be used to find n tails until a head comes up? Are the bernoulli and the probability mass function just used for the probability of a fixed random variable X, which is a subset of the bionomial distribution?
Best Answer
I think your question is a little confusing.
A probability mass function is a function for a discrete random variable which returns probabilities. We denote this by $Pr(X=x)$ which means the probability of the random variable $X$ being $x$. Some examples:
$$ \begin{align} P(X=x)&=\frac{e^{-\lambda}\lambda^x}{x!}, \quad x=1, 2, \dots \tag{Poisson}\\ P(X=x)&={n \choose x}\theta^x(1-\theta)^{n-x}, \quad x=0, 1, \dots, n \tag{Binomial}\\ P(X=x)&=\frac{1}{10}, \quad x=1, 2, \dots, 10 \tag{Uniform}\\ P(X=x)&=\theta^x(1-\theta)^{1-x}, \quad x=0, 1 \tag{Bernoulli} \end{align} $$
The last one is the probability mass function of the Bernoulli distribution, but all of these are probability mass functions. So your question "Are [the Bernoulli distribution and the probability mass function] the same thing?" does not really make sense.
As you point out, if you let $n=1$ in the binomial then you have a Bernoulli distribution. Furthermore, with $\theta=1/2$ you get your example of coin flips. But $\theta$ can be anything between 0 and 1.