Hassett, chapter 7: continuous random variables
7.1 Defining a continuous random variable
Calculus! The probability density function $f(x)$ obeys $$ \begin{aligned} f(x) &≥ 0 \text{ everywhere} \\ \int f(x) \mathrm dx &=1\\ P(a≤X≤b) &= \int_a^b f(x)\mathrm dx \end{aligned} $$
The cumulative distribution function is $$\begin{aligned} F(x)&=\int_{-\infty}^x f(x) \mathrm dx \\ F'(x)&=f(x) \end{aligned}$$
7.2 Mode, median, and percentiles
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- mode
- This is where $f(x)$ has a maximum.
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- median
- The solution to $F(x)=\frac 12$, a.k.a. the 50th percentile.
7.3 Mean and variance
In the continuum limit of the discrete expectation value $$\begin{aligned} E[g(x)] &= \sum g(x) p(x) \end{aligned}$$
the continuous expectation value is $$\begin{aligned} E[g(x)] &= \int g(x) f(x)\mathrm dx \end{aligned}$$
So we have $$\begin{aligned} \text{linearity:}&& E(aX+b) &= aE(x)+b \\ \end{aligned}$$
and, as expected, $$\begin{aligned} V(X) &= E(X^2) - (E(X))^2 \end {aligned}$$
And that's it! That's the chapter.