X is a continuous random variable if there exists a nonnegative function f defined for all real x \in (-\inf, \inf), having the property that for any set \{ x \le a \} of real numbers.
P(x \le a) = \int_{ x \le a } f(x) dx
The function f is the probability density function of the random variable X.
Properties: \boxed{ \text{Cont.: } \begin{cases} f(x) &\ge 0 \qquad \text{ for all } x \in R(X) \\~\\ \int_{R(X)}^a f(x) dx &= 1 \\~\\ P\{ a < X < b \} &= \int_a^b f(x) dx \\~\\ E(X) &= \int_{R(X)} x f(x) dx = \mu \\~\\ Var(X) &= E(X^2) - \mu^2 \\~\\ f(X &= a) \ne \text{Probability!} \end{cases} }
Note: For continuous random variables, equal-to inequality (e.g., \le and <) are interchangeable.
Q: A continuous r.v. X has the pdf function:
f(x) = \frac{1}{2}x + c \qquad 0 \le x \le 1
Find the value of:
A:
We know \int_{R(X)} f(x) dx = 1; so:
\begin{align*} 1 &= \int_0^1 ( \frac{1}{2} x + c) dx \\~\\ 1 &= \left[ \frac{1}{4} x^2 + c \right]_0^1 \\~\\ 1 &= \frac{1}{4} + c \\~\\ c &= \frac{3}{4} \end{align*}
Exponential distribution arises when looking at amount of time until some specific event occurs.
Properties: \boxed{ \text{Exp.: } \begin{cases} f(x) &= \lambda e^{- \lambda x} \text{ where } x \ge 0 \\~\\ E(x) &= \frac{1}{x} \\~\\ Var(x) &= \frac{1}{x^2} \\~\\ \end{cases} }
Properties: \boxed{ \text{Normal: } \begin{cases} f(x) &= \frac{1}{ \sigma \sqrt{2 \pi} } e^{ - \frac{1}{2\sigma^2}(X-\mu)^2} \text{ where } -\inf < x < \inf \\~\\ \text{Mean} &= \mu \text{ (location)} \\~\\ \text{SD} &= \sigma \text{ (shape)} \\~\\ \end{cases} }
Standard Normal: Z ~ N(0,1)
\boxed{ \text{Percentile as Stand. Norm.: } Z_\alpha = 100( 1 - \alpha )^\text{th} \text{percentile} }
\boxed{ \text{Transformation: } \\~\\ \begin{align*} \frac{X - \mu}{\sigma} &= Z \sim N(0,1) \\~\\ \sigma Z + \mu &= X \sim N(\mu, \sigma) \\~\\ \end{align*} } \\ \small\textit{ (if $X \sim N(\mu, \sigma)$) }
\boxed{ \bar x \pm k \times SD = (\bar x - k \times SD, \bar x + k \times SD) }
If the distribution is bell-shaped (symmetric and unimodal):
Z-Score: Distance between an observation and the mean in units of SD.
\boxed{ \text{Z-Score: } z = \frac{x - \bar x}{s} } \\ \small\textit{if $z < 0$, $x < \bar x$} \\ \textit{if $z > 0$, $x > \bar x$} \\
\boxed{ \text{Outlier: } |z| > 3 }