Continuous random variables

The curve that tells probabilities

• A continuous random variable 连续型随机变量 can take any value in a range — like the exact time a bus arrives, or the precise height of a person.
• Unlike discrete variables, you can't list probabilities for each value. Instead, a probability density function 概率密度函数 (pdf) describes the shape, and probabilities are areas 面积 under the curve.

The probability density function

• A continuous random variable can take any value in a range, described by a probability density function $f(x)$:

For a continuous variable, probability is the area under the density f(x)

Finding probabilities

• A probability is an area under $f$: $P(a < X < b) = \displaystyle\int_a^b f(x)\,dx$.

Probabilities are areas under f(x); the total area over all x must equal 1.

Finding the mean and variance

• The mean is $E(X) = \displaystyle\int_{-\infty}^{\infty} x\,f(x)\,dx$.
• The variance is $\text{Var}(X) = E(X^2) - (E(X))^2$, where $E(X^2) = \displaystyle\int_{-\infty}^{\infty} x^2\,f(x)\,dx$.

The cumulative distribution function 累积分布函数

• The cdf $F(x) = P(X \leq x) = \displaystyle\int_{-\infty}^x f(t)\,dt$.
• $F(x)$ is the area under $f$ up to $x$. It always increases from 0 to 1.
• Use the density function to find the median and other percentiles.

You've got it

Write from a continuous mean of 3

• A uniform density is 1/4 on the interval from 1 to 5.
• Check that it is a pdf and find the mean.