What is a distribution?
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Instructor lecture
A distribution tells us which values a random variable can take and how probability is allocated.
It is a complete description of our uncertainty about the random variable X.
P(X ∈ A)
gives the chance that X lands somewhere in A.
X
the random variable being described
A
a set of values we care about
Same idea, two common forms
Discrete · probability mass
X = number of heads in two flips
pₓ(x) = P(X = x)
| Quantity | 0 heads | 1 head | 2 heads |
|---|---|---|---|
| Probability | 25% | 50% | 25% |
Individual values can have positive probability; the masses sum to 1.
Continuous · probability density
H = height
P(a ≤ H ≤ b)
= ∫ₐᵇ fₕ(h) dh
= ∫ₐᵇ fₕ(h) dh
Interval probabilities come from areas under the density.
At any one exact height, the probability is zero.
01
Support
Every value the random variable X can take.
02
Probability rule
Mass at discrete values or density over continuous ranges.
03
Total equals 1
All possible values together carry all of the probability.