| 202 |
standard normal random variable.
Use this form when $X\lt0.$
|
\[
\Phi \left( -x \right) = 1- \Phi \left( x \right),\quad x \lt 0
\]
|
| 202 |
normal random variable
|
\[
\begin{array}{ll}
F_X(x)
&= Pr \left\{ X \leq x \right\} \\
&= Pr \left\{ \frac{X - \mu}{\sigma} \leq \frac{x - \mu}{\sigma} \right\} \\
&= \Phi \left( \frac{x - \mu}{\sigma} \right)
\end{array}
\]
|
| 187 |
The
probability density function
$f$ of a random variable $X$
is the function that satisfies
the three given conditions.
|
-
$ f \left( x \right) \geq 0 $
-
$ \int_{-\infty}^{\infty} f \left( x \right) dx = 1 $
-
$ \Pr{ \left\{ a \le X \le b \right \} } = \int_{a}^{b} f \left( x \right) dx $
|
| 190 |
The
density function
is the derivative of the
distribution function,
and the distribution function is the integral of the density function.
|
\[
\eqalign{
F \left( b \right)
&= \Pr{ \left\{ X \le b \right\} }\\
&= \Pr{ \left\{ -\infty \lt X \le b \right\} }\\
&= \int_{-\infty}^{b} f \left( x \right) dx
}
\]
|
| 190 |
Expectation of a Continuous Random Variable
|
\[
E \left[ X \right] = \int_{-\infty}^{\infty}{ x \cdot f \left( x \right)\ dx}
\]
|
| 190 |
Expectation of a Function of a Continuous Random Variable
|
\[
E \left[ g \left( X \right) \right] = \int_{-\infty}^{\infty} g \left( x \right) f \left( x \right) dx
\]
|
| 195 |
Variance for Continuous Random Variable
|
\[
\eqalign{
\Var \left( X \right) =
&\int_{-\infty}^{\infty}{x^2 \cdot f \left( x \right)\ dx}\\
&- \left[ \int_{-\infty}^{\infty}{x \cdot f \left( x \right)\ dx} \right]^2
}
\]
|
| 195 |
Uniform Random Variable
|
|
| 196 |
density function of a uniform random variable from a to b
|
\[
f \left( x \right) =
\left\{
\begin{array}{ll}
\frac{1}{b-a} &a \lt x \lt b\\
&\mathrm{else}
\end{array}
\right.
\]
|
Alex Notes