pmf for coin toss

Question

I am currently studying Statistical Inference class on Coursera. In one of the assignments, the following question comes up.

Let $x=0$ represent a 'heads' outcome and $x=1$ represent a 'tails' outcome of a coin toss. If $p$ is the probability of 'heads' which of the following represents the PMF of the coin toss? The variable x is either 0 (heads) or 1 (tails).

1: $p^{1-x}(1-p)^x$

2: $p^x(1-p)^{1-x}$

while solving it gave hint to choose option when head exponent is 1 so correct option was 1

if it said p is probability of tail which following represent the PMF of the coin toss will answer change

pmf ---- is it for independent event only

Answer 1

PMF of the coin toss means $P_X(x) = P(X = x)$. Note that capitals are always used for the random variables and small characters are specific values. There are only two possible values for $X$, i.e $0, 1$. So, $P_X(0)=P(X=0)$ which is $p$ by definition. So, $P_X(1)=P(X=1)$ should be $1-p$.

So, the function you'll have should satisfy these conditions, and that option is $1$, as in the hint of the question. You only need to substitute $x = 0$ and $x = 1$ to see if it's true.