diff options
Diffstat (limited to 'sem6/prob/m2/noter.tex')
| -rw-r--r-- | sem6/prob/m2/noter.tex | 48 |
1 files changed, 31 insertions, 17 deletions
diff --git a/sem6/prob/m2/noter.tex b/sem6/prob/m2/noter.tex index 3eb2e4f..c35f52b 100644 --- a/sem6/prob/m2/noter.tex +++ b/sem6/prob/m2/noter.tex @@ -1,6 +1,5 @@ \title{Noter til probability m2} -\section{Random Variables} Her mapper man fra et sample space S til en variabel. Her kalder man variablen et stort tal R eller sådan noget. @@ -15,7 +14,7 @@ P(X = x) = 0 $$ -\subsection{Cumulative Distribution Function} +\section{Cumulative Distribution Function} Her måler man prob for at ens random er mindre end et bestemt tal. @@ -33,7 +32,7 @@ Ved discrete random variables vil denne være en slags trappe. Kan sige at den er \emph{continues from the right} eftersom man har $\leq$ i definition. -\subsection{Probability Mass Function} +\section{Probability Mass Function} Works only for discrete random variables. Is defines as the probability that $X = a$: @@ -49,15 +48,17 @@ F(a) = \sum_{all x \leq a} p(a) $$ -\subsection{Probability Density Function} +\section{Probability Density Function} Her finder man P i et evigt lille interval: -Is the derivative of the CDF. +In the following formula PDF is $f$. +\begin{equation*} + \begin{split} + F(a) = P(X \in (-\infty,a]) = \int_{-\infty}^a f(x) dx \\ + f(a) = \frac{d}{da} F(a) + \end{split} +\end{equation*} -$$ -F(a) = P(X \in (-\infty,a]) = \int_{-\infty}^a f(x) dx \\ -f(a) = \frac{d}{da} F(a) -$$ The following must be true: @@ -65,7 +66,7 @@ $$ \int_{-\infty}^{\infty} f(x) dx = 1 $$ -\subsection{Multiple Random Variables} +\section{Multiple Random Variables} Have multiple random variables, which can be or is not correlated. Can define the joined CDF: @@ -81,15 +82,28 @@ F_X(x) = P(X \leq x) = P(X \leq, Y < \infty) = F(x, \infty) $$ One can not go from marginal to the joined, as they do not contain enough information. -This is only possible if X and Y are \emph{independent}. -$$ -F_{XY}(x,y) = F_X(x) \cdot F_Y(x) \\ -p(x,y) = p_X(x) \cdot p_Y(y) \\ -f(x,y) = f_X(x) \cdot f_Y(y) -$$ +However if two random variables, and $A$ and $B$ are two sets of real numbers: +\[ + P(X \in A, Y \in B) = P(X \in A) P(Y \in B)\,. +\] +% This is only possible if X and Y are \emph{independent}. +% \begin{align*} +% F_{XY}(x,y) &= F_X(x) \cdot F_Y(x) \\ +% p(x,y) &= p_X(x) \cdot p_Y(y) \\ +% f(x,y) &= f_X(x) \cdot f_Y(y) +% \end{align*} + +\section{Conditional PDF} + +If $X$ and $Y$ have a joint PDF, then the conditional PDF of X given that $Y=y$ is +\[ + F_{X|Y}(x|y) = \frac {f(x, y)} {f_Y(y)} +\] + +There is also one for PMF not listed here. -\subsection{Joined PMF} +\section{Joined PMF} $$ P_{XY}(x,y) = P(X = x, Y = y) |