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| -rw-r--r-- | sem6/prob/m2/noter.tex (renamed from sem6/prob/m2/noter.md) | 48 |
1 files changed, 22 insertions, 26 deletions
diff --git a/sem6/prob/m2/noter.md b/sem6/prob/m2/noter.tex index f8faec4..3eb2e4f 100644 --- a/sem6/prob/m2/noter.md +++ b/sem6/prob/m2/noter.tex @@ -1,13 +1,12 @@ -# Noter til probability m2 - -## Random variables +\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. Derfor er et random variabel egentlig en transformation mellem S og real tal. -*X er en descrete random variable hvis dens range er countable.* +\emph{X er en descrete random variable hvis dens range er countable.} For continues random variables the following is true: @@ -15,9 +14,8 @@ $$ P(X = x) = 0 $$ -## Functions beskriver ens random variable -### Cumulative Distribution function +\subsection{Cumulative Distribution Function} Her måler man prob for at ens random er mindre end et bestemt tal. @@ -33,9 +31,9 @@ $$ Ved discrete random variables vil denne være en slags trappe. -Kan sige at den er *continues from the right* eftersom man har $\leq$ i definition. +Kan sige at den er \emph{continues from the right} eftersom man har $\leq$ i definition. -### Probability Mass Function +\subsection{Probability Mass Function} Works only for discrete random variables. Is defines as the probability that $X = a$: @@ -47,56 +45,54 @@ $$ From here CDF can be found: $$ - F(a) = \sum_{all x \leq a} p(a) +F(a) = \sum_{all x \leq a} p(a) $$ - -### Probability Density Function +\subsection{Probability Density Function} Her finder man P i et evigt lille interval: Is the derivative of the CDF. $$ - F(a) = P(X \in (-\infty,a]) = \int_{-\infty}^a f(x) dx \\ - f(a) = \frac{d}{da} F(a) +F(a) = P(X \in (-\infty,a]) = \int_{-\infty}^a f(x) dx \\ +f(a) = \frac{d}{da} F(a) $$ The following must be true: $$ - \int_{-\infty}^{\infty} f(x) dx = 1 +\int_{-\infty}^{\infty} f(x) dx = 1 $$ -## Multiple random variables +\subsection{Multiple Random Variables} Have multiple random variables, which can be or is not correlated. Can define the joined CDF: $$ - F_{XY}(x,y) = P(X \leq x, Y \leq y) +F_{XY}(x,y) = P(X \leq x, Y \leq y) $$ -One can also find the probability of one of the variables. (The *marginal*) +One can also find the probability of one of the variables. (The \emph{marginal}) $$ - F_X(x) = P(X \leq x) = P(X \leq, Y < \intfy) = F(x, \infty) +F_X(x) = P(X \leq x) = P(X \leq, Y < \infty) = F(x, \infty) $$ 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 **independent**. +This is only possible if X and Y are \emph{independent}. $$ - F_XY(x,y) = F_X(x) \cdot F_Y(x) \\ - p(x,y) = p_X(x) \cdot p_Y(y) \\ - f(x,y) = f_X(x) \cdot f_Y(y) +F_{XY}(x,y) = F_X(x) \cdot F_Y(x) \\ +p(x,y) = p_X(x) \cdot p_Y(y) \\ +f(x,y) = f_X(x) \cdot f_Y(y) $$ -### Joined PMF +\subsection{Joined PMF} $$ - P_{XY}(x,y) = P(X = x, Y = y) +P_{XY}(x,y) = P(X = x, Y = y) $$ - vim: spell spelllang=da,en - +% vim: spell spelllang=da,en |