The question:
(Introduction to Probability, Blitzstein and Nwang, p.128)
People are arriving at a party one at a time. While waiting for more people to arrive they entertain themselves by comparing their birthdays. Let X be the number of people needed to obtain a birthday match, i.e., before person X arrives no two people have the same birthday, but when person X arrives there is a match. Find the PMF of X.
Has already been answered here at: Birthday problem: Let X be number of people needed for a match. Find the PMF of X.
My question/doubt is: This is my first lesson in RVs, so I have very less knowledge how to go about it. What I understood and did was: X is the minimum number of people needed to have atleast 1 birthday match. I let X=k be minimum people required. So, for k people, probability of getting atleast 1 birthday match is given as $P(X=k) = 1 - Probability \space of \space getting \space no \space birthday \space match$
$P(X=k) = 1 - \frac{\binom{365}k k!}{365^k}$
But in Birthday problem: Let X be number of people needed for a match. Find the PMF of X., they say it is CDF. I thought it to be PMF. I am not conceptually confident.
So my question is: How do we know what are we finding? CDF or PMF? We can verify what is CDF or PMF by checking for their respective properties. Like for CDF, we can check if it is increasing, right continuous and if it converges to 0 and 1 in limits. But this is post-facto treatment - i.e. we are given a function and then we check if it is CDF or PMF etc. But how do we know while solving the problem that what we'll get is (say) CDF or PMF?
(P.S. I apologise beforehand if I've violated any forum norms. I hope this is not a duplicate question as I am not asking for the solution. I seek conceptual guidance.
$X$ is the integer-valued random variable showing the number of people present when the first match appears. Its CDF is $\mathbb P(X \le k)$ and its PMF is $\mathbb P(X=k)$.
You know $$\frac{\binom{365}k k!}{365^k}=\frac{365!}{(365-k)! 365^k}$$ is the probability of no matches in the first $k$ people, so is $\mathbb P(X \gt k)$.
Its complement is therefore $$\mathbb P(X \le k)= 1-\frac{365!}{(365-k)! 365^k}$$ as the probability the first match occurs within the first $k$ people, which is the cumulative distribution function for $X$.
Taking the first difference gives $$\mathbb P(X = k) = \mathbb P(X \le k)-\mathbb P(X \le k-1) = \frac{(k-1)\,365!}{(365-k+1)! 365^k}$$ as the probability the first match occurs on precisely the $k$th person and is thus the probability mass function for $X$.