Continuous random variables (Further)

Probability as area

• For a continuous variable 连续变量 — a height, a waiting time — no single exact value has a probability; there are infinitely many.
• Instead, probability is the area under a curve, the probability density function 概率密度函数 $f(x)$. The whole area is $1$.

The density function and expectation 期望

• $P(a < X < b) = \displaystyle\int_a^b f(x)\,dx$, and the total area $\int f = 1$.
• The mean of any function of $X$:

Probabilities are areas under f(x); the total area over the whole range is exactly 1.

The cumulative distribution function 累积分布函数

• The CDF accumulates probability from the left:

• It runs from $0$ to $1$, and differentiating recovers the density: $f(x) = F'(x)$.

The CDF $F(x)$ rises from 0 to 1; the median is read off where it reaches a height of 0.5

Worked example — the median 中位数

• The median $m$ splits the area in half: $F(m) = 0.5$.
• For $f(x) = \tfrac12 x$ on $[0, 2]$: $F(x) = \tfrac14 x^2$, so $\tfrac14 m^2 = 0.5$ gives $m = \sqrt2 \approx 1.41$.

Density is not probability

• Use cumulative distribution functions (CDFs) to find probabilities and percentiles, and the CDF of a related variable.

You've got it

Write from a median of 4

• A uniform density on the interval from 2 to 6 has total area 1.
• Find the median. A density value may be greater than 1.