10.1.1 PDFs and CDFs
Consider the random process $\big\{X(t), t \in J \big\}$. For any $t_0 \in J$, $X(t_0)$ is a random variable, so we can write its CDF
\begin{align}%\label{}
\nonumber F_{X(t_0)}(x)=P\big(X(t_0) \leq x\big).
\end{align}
If $t_1,t_2 \in J$, then we can find the joint CDF of $X(t_1)$ and $X(t_2)$ by
\begin{align}%\label{}
\nonumber F_{X(t_1)X(t_2)}(x_1,x_2)=P\big(X(t_1) \leq x_1, X(t_2) \leq x_2\big).
\end{align}
More generally for $t_1,t_2,\cdots,t_n \in J$, we can write
\begin{align}%\label{}
&F_{X(t_1)X(t_2)\cdots X(t_n)}(x_1,x_2,\cdots, x_n)=P\big(X(t_1) \leq x_1, X(t_2) \leq x_2, \cdots, X(t_n) \leq x_n\big).
\end{align}
Similarly, we can write joint PDFs or PMFs depending on whether $X(t)$ is continuous-valued (the $X(t_i)$'s are continuous random variables) or discrete-valued (the $X(t_i)$'s are discrete random variables).
Example
Consider the random process $\big\{X_n, n=0,1,2,\cdots\}$, in which $X_i$'s are i.i.d. standard normal random variables.
- Write down $f_{X_{\large{n}}}(x)$ for $n=0,1,2,\cdots$.
- Write down $f_{X_{\large{m}} X_{\large{n}}}(x_1,x_2)$ for $m \neq n$.
- Solution
-
- Since $X_n \sim N(0,1)$, we have
\begin{align}%\label{}
f_{X_n}(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \quad \textrm{ for all }x \in \mathbb{R}.
\end{align}
- If $m \neq n$, then $X_m$ and $X_n$ are independent (because of the i.i.d. assumption), so
\begin{align}%\label{}
f_{X_m X_n}(x_1,x_2)&=f_{X_m}(x_1) f_{X_n}(x_2)\\
&=\frac{1}{\sqrt{2\pi}}e^{-\frac{x_1^2}{2}} \cdot \frac{1}{\sqrt{2\pi}}e^{-\frac{x_2^2}{2}}\\
&=\frac{1}{2\pi} \exp\left\{-\frac{x_1^2+x_2^2}{2}\right\}, \quad \textrm{ for all }x_1,x_2 \in \mathbb{R}.
\end{align}
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Practical uncertainty: Useful Ideas in Decision-Making, Risk, Randomness, & AI
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