Discrete random variables
A discrete random variable takes values from a finite or countable set.
Flip a coin five times and let X be the number of heads. Then X ∈ {0,1,2,3,4,5}.
Random Variables
Turning randomness into measurable quantities
A random variable is a rule that turns the outcome of a random experiment into a number we can analyze.
random outcomeω ∈ Ω
→random variableX
→numerical valueX(ω)
The essential idea: randomness happens in the outcome space; the random variable gives us something numerical to measure.
One Concept · One Example
▶Random Variables and Probability Measures: Discrete and Continuous Cases
Presented by Charlotte Moser
Watch on YouTube ↗What to notice
- A random outcome does not have to be a number.
- A random variable assigns a number to that outcome.
- The values can form a discrete or continuous distribution.
Two ways probability can be organized
Continuous random variables
A continuous random variable takes values over a continuum and is described by a density pX(x).
Probability is area: P(a ≤ X ≤ b) is the integral of the density over [a,b].
Density is not probability. For a continuous variable, P(X = x) = 0.
From outcomes to numbers
Flip a fair coin five times. The complete sequence is the outcome ω. Our random variable X keeps only one piece of information: how many heads occurred?
The random experiment
HTTHH
ω = (H, T, T, H, H)→X(ω) = 3
Five flips produced 3 heads, so X = 3.
Different outcomes can map to the same value of X. For example, H · H · T · T · H and T · H · H · H · T both map to 3.
Build the PMF
Run repeated experiments to build an empirical histogram.
A PMF assigns a probability to each possible value of a discrete random variable.
What changes when the possible values are continuous?
We can no longer assign positive probability to every individual value. Instead, probability is measured by area under a density curve.
Probability under a density curve
Let X represent the height of a randomly selected adult. We use a smooth illustrative bell-shaped distribution only to build mathematical intuition.
Illustrative probability density, not empirical population data. Drag either blue boundary; hover over the curve to inspect density.
Same idea, two kinds of distributions
Discrete
P(X = x) is represented directly by the height of a PMF bar.
bars → PMFContinuous
P(a ≤ X ≤ b) is represented by area under a PDF.
smooth curve + shaded area → PDFRandom variables turn random outcomes into numbers. Discrete random variables are described by probability mass functions; continuous random variables are described by probability density functions.
random experimentoutcome ω
→mappingX(ω)
→resultdistribution