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 there are no two people with the same birthday, but when person X arrives there is a match. Find the PMF of X.

Answer:=(k−1)*P(X>k−1) or (k−1)365k(365k−1)(k−1)!Step-by-step explanation:First of all, we need to find PMFLet X = k represent the case in which there is no birthday match within (k-1) peopleHowever, there is a birthday match when kth person arrivesHence, there is 365^k possibilities in birthday arrangementsSupposing (k-1) dates are placed on specific days in a yearPick one of k-1 of them & make it the date of the kth person that arrives, then:The CDF is P(X≤k)=(1−(365k)k)/!365k, so the can obtain the PMF by  P(X=k) =P (X≤k) − P(X≤k−1)=(1−(365k)k!/365^k)−(1−(365k−1)(k−1)!/365^(k−1))= (k−1)/365^k * (365k−1) * (k−1)! =(k−1)*(1−P(X≤k−1)) =(k−1)*P(X>k−1)