The probability mass function (pmf) gives the probability that a discrete random variable takes on a certain value, usually denoted $P(X = x)$. The corollary for a continuous random variable is the [[probability density function]]. ## Properties From the axioms of probability 1. $0 <= P(X = x) <= 1$ 2. $\displaystyle \sum_x P(X = x) = P(S) = 1$ 3. $P(X = a \cup X=b) = P(X=a) + P(X=b)$ #check The sum across all values of the pmf must equal 1. You can use this to confirm that any derived pmf is valid. $\displaystyle \sum_x P(X = x) = 1$ ## Deriving a pmf A patient is waiting for a kidney transplant. The probability of the next donor being a match is $p$. Derive the pmf for this distribution. First, let's define the sample space ($S$) as {1, 01, 001, 0001, ... }. In other words, one outcomes is the next donor is a match ($1$), the next outcome is the next donor is not a match but the following donor is ($01$), and so on. Let $X$ be the number of donors tested until a match is found. We can derive the probability mass function by observing the probability of each outcome until we can define a general rule. The probability of a donor match is $p$ and the probability of a donor not being a match is $1-p$. $\displaylines{ P(X=1) = \text{\{1\}} = p \\ P(X=2) = \text{\{01\}} = (1-p) * p \\ P(X=3) = \text{\{001\}} = (1-p)^2 * p \\ P(X=4) = \text{\{0001\}} = (1-p)^3 * p \\ }$ We can see the pattern now, the $P(X=x)$ is given by $P(X=x) = (1-p)^{x-1} * p$ We can find this pmf in a table of common distributions and recognize that this is the pmf for the [[first success distribution]].