08 PDF vs PMF

PMF (Probability Mass Function):

The PMF is used for discrete random variables.

• It gives the probability that a discrete random variable takes a specific value.
• The PMF of a function $P(X=x)$, which tells you the probability that the random variable $X$ equals some specific value $x$.

PDF (Probability Density Function)

The PDF is used for continuous random variables.

• Unlike the PMF, the PDF does not give the probability of a specific outcome. Instead, it gives the density of probabilities at each point.
• For a continuous variable, the probability of any exact value occurring is technically 0. Instead, you use the PDF to find the probability of the variable falling within a certain range.
• How to use:
  • To find the probability that a continuous random variable $X$ lies within an interval $[a,b]$, you integrate the PDF over that interval:
  $$P(a\le X\le b)={\int }_a^b{f}_X(x)dx$$
  • So, for continuous variables like the height of people, the time taken for a task, or the temperature, the PDF gives you a curve, and you calculate probabilities over intervals.