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Version 11 |
| [this entry is currently being revised, so hold off on corrections until |
[this entry is currently being revised, so hold off on corrections until |
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| Let $F: \mathbb{R}\to \mathbb{R}$. Then $F$ is a \emph{distribution function} if |
Let $F: \mathbb{R}\to \mathbb{R}$. Then $F$ is a \emph{distribution function} if |
| \begin{enumerate} |
\begin{enumerate} |
| \item |
\item |
| $F$ is nondecreasing, |
$F$ is nondecreasing, |
| \item |
\item |
| $F$ is continuous from the right, |
$F$ is continuous from the right, |
| \item |
\item |
| $\lim_{x \rightarrow -\infty} F(x) = 0$, and $\lim_{x \rightarrow \infty} F(x) = 1$. |
$\lim_{x \rightarrow -\infty} F(x) = 0$, and $\lim_{x \rightarrow \infty} F(x) = 1$. |
| \end{enumerate} |
\end{enumerate} |
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| As an example, suppose that $\Omega = \mathbb{R}$ and that $\mathcal{B}$ |
As an example, suppose that $\Omega = \mathbb{R}$ and that $\mathcal{B}$ |
| is the $\sigma$-algebra of Borel subsets of $\mathbb{R}$. |
is the $\sigma$-algebra of Borel subsets of $\mathbb{R}$. |
| Let $P$ be a probability measure on $(\Omega, \mathcal{B})$. |
Let $P$ be a probability measure on $(\Omega, \mathcal{B})$. |
| Define $F$ by |
Define $F$ by |
| $$ |
$$ |
| F(x) = P((-\infty, x]). |
F(x) = P((-\infty, x]). |
| $$ |
$$ |
| This particular $F$ is called the \emph{distribution function} of $P$. It is |
This particular $F$ is called the \emph{distribution function} of $P$. It is |
| easy to verify that 1,2, and 3 hold for this $F$. |
easy to verify that 1,2, and 3 hold for this $F$. |
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| In fact, every distribution function is the distribution function of some |
In fact, every distribution function is the distribution function of some |
| probability measure on the Borel subsets of $\mathbb{R}$. To see this, |
probability measure on the Borel subsets of $\mathbb{R}$. To see this, |
| suppose that $F$ is a distribution function. We can define $P$ on a single half-open |
suppose that $F$ is a distribution function. We can define $P$ on a single half-open |
| interval by |
interval by |
| $$ |
$$ |
| P((a,b]) = F(b) - F(a) |
P((a,b]) = F(b) - F(a) |
| $$ |
$$ |
| and extend $P$ to unions of disjoint intervals by |
and extend $P$ to unions of disjoint intervals by |
| $$ |
$$ |
| P( \cup_{i=1}^\infty (a_i, b_i])= \sum_{i=1}^\infty P((a_i, b_i]). |
P( \cup_{i=1}^\infty (a_i, b_i])= \sum_{i=1}^\infty P((a_i, b_i]). |
| $$ |
$$ |
| and then further extend $P$ to all the Borel subsets of $\mathbb{R}$. |
and then further extend $P$ to all the Borel subsets of $\mathbb{R}$. |
| It is clear that the distribution function of $P$ is $F$. |
It is clear that the distribution function of $P$ is $F$. |
|
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| \subsection{Random Variables} |
\subsection{Random Variables} |
|
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| Suppose that $(\Omega, \mathcal{B}, P)$ is a probability space and |
Suppose that $(\Omega, \mathcal{B}, P)$ is a probability space and |
| $X: \Omega \to \mathbb{R}$ is a random variable. Then there is an |
$X: \Omega \to \mathbb{R}$ is a random variable. Then there is an |
| \emph{induced} probability measure $P_X$ on $\mathbb{R}$ defined as |
\emph{induced} probability measure $P_X$ on $\mathbb{R}$ defined as |
| follows: \\ |
follows: \\ |
| $$ |
$$ |
| P_X(E) = P(X^{-1}(E)) |
P_X(E) = P(X^{-1}(E)) |
| $$ |
$$ |
| for every Borel subset $E$ of $\mathbb{R}$. $P_X$ is called the |
for every Borel subset $E$ of $\mathbb{R}$. $P_X$ is called the |
| \emph{distribution} of $X$. The \emph{distribution function} |
\emph{distribution} of $X$. The \emph{distribution function} |
| of $X$ is |
of $X$ is |
| $$ |
$$ |
| F_X(x) = P(\omega | X(\omega) \leq x). |
F_X(x) = P(\omega | X(\omega) \leq x). |
| $$ |
$$ |
| The distribution function of $X$ is also known as the law of $X$. |
|
| Claim: $F_X$ = the distribution function of $P_X$. |
Claim: $F_X$ = the distribution function of $P_X$. |
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|
| \begin{eqnarray*} |
\begin{eqnarray*} |
| F_X(x) &=& P(\omega | X(\omega) \leq x) \\ |
F_X(x) &=& P(\omega | X(\omega) \leq x) \\ |
| &=& P(X^{-1}((-\infty, x]) \\ |
&=& P(X^{-1}((-\infty, x]) \\ |
| &=& P_X((-\infty, x]) \\ |
&=& P_X((-\infty, x]) \\ |
| &=& F(x). |
&=& F(x). |
| \end{eqnarray*} |
\end{eqnarray*} |
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| \subsection{Density Functions} |
\subsection{Density Functions} |
|
|
| Suppose that $f: \mathbb{R} \to \mathbb{R}$ is a nonnegative function |
Suppose that $f: \mathbb{R} \to \mathbb{R}$ is a nonnegative function |
| such that |
such that |
| $$ |
$$ |
| \int_{-\infty}^\infty f(t)dt=1. |
\int_{-\infty}^\infty f(t)dt=1. |
| $$ |
$$ |
| Then one can define $F: \mathbb{R} \to \mathbb{R}$ by |
Then one can define $F: \mathbb{R} \to \mathbb{R}$ by |
| $$ |
$$ |
| F(x) = \int_{-\infty}^x f(t)dt. |
F(x) = \int_{-\infty}^x f(t)dt. |
| $$ |
$$ |
| Then it is clear that $F$ satisfies the conditions 1,2,and 3 so $F$ |
Then it is clear that $F$ satisfies the conditions 1,2,and 3 so $F$ |
| is a distribution function. The function $f$ is called a density function |
is a distribution function. The function $f$ is called a density function |
| for the distribution $F$. |
for the distribution $F$. |
|
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| If $X$ is a discrete random variable with density function $f$ and distribution |
If $X$ is a discrete random variable with density function $f$ and distribution |
| function $F$ then |
function $F$ then |
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| $$F(x)=\sum_{x_j\leq x} f(x_j).$$ |
$$F(x)=\sum_{x_j\leq x} f(x_j).$$ |
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