Discrete Random Variable vs Continuous Random Variable

A discrete random variable is one whose possible values you could write out in a list, usually because it comes from counting something. If $X$ is the number of heads in two fair coin flips, then $X$ takes the values 0, 1, and 2 with probabilities $0.25$, $0.50$, and $0.25$. Every discrete probability distribution satisfies $\sum P(x_i) = 1$ (the probabilities sum to 1), and here $0.25 + 0.50 + 0.25 = 1$. Counts are discrete; measurements like time and weight are continuous.

A continuous random variable comes from measuring rather than counting, so between any two of its values there are infinitely many more. That is why the probability of any single exact value is 0, and probability instead means the area under a density curve over an interval. If your wait for a bus is equally likely anywhere between 0 and 10 minutes, the density height is $1/10 = 0.1$, so $P(X < 3) = 3(0.1) = 0.30$. The normal distribution is the continuous model you will use most in this course.