Bernoulli experiment - A coin toss - How to mathematically notate the experiment?

Question

The Bernoulli random variable has a probability function: $$f_X(x) = p^x\,(1-p)^{1-x}~\mathbf 1_{x\in\{0,1\}}\\ f_X(0)=1-p\\ f_X(1)=p\qquad$$

Provide an example of a Bernoulli experiment and a Bernoulli random variable.

$$f_X(0)=1-p $$ $$f_X(1) = p $$ If I take a coin toss for an example how should I mathematically notate the experiment?

Thanks

Answer 1

You have the correct expressions for the pmf at those arguments.

$X\sim\mathcal{Ber}(p)$ is a typical notation indicating that random variable $X$ follows a Bernoulli distribution.   That is that: $X$ is the indicator of success in a single succeed or fail trial, which has the given expected rate for success, $p$.   Some authors may use "$\mathcal{Bern}$", "$\mathcal{Bernoulli}$", or simply "$\mathcal{B}$".

Where $f_X()$ is the probability mass function for $X$, then:

$$X\sim\mathcal{Ber}(p) ~\iff~ f_X(x) = p^x\,(1-p)^{1-x}~\mathbf 1_{x\in\{0,1\}}\\ f_X(0)=1-p\\ f_X(1)=p\qquad$$

For a single coin toss, you will also need to indicate what are you counting as a success, and what value is the success rate, $p$.   Some of this has to be done with words.   In short:

Let $X$ be the count for obtaining heads in the toss of an unbiassed coin, so $X\sim\mathcal {Ber}(1/2)$.

Answer 2

How about this notation:

$X\sim \mbox{Bernoulli}(p)$

(Where $X$ is the random variable denoting the number of successes ($0$ or $1$) from the Bernoulli trial.)