{{% hassett_navigation %}} [TOC] # 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 + mode : This is where $f(x)$ has a maximum. + 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.