Mixing
Basic probability, Mass function, Density, CDF, Distribution, Random Variables.
$begingroup$
Hi I'm hoping this will clear up any confusion on basic probability.
Take a probability space $(mathbb{P},mathcal{F},Omega)$ :
$Omega$ is some set of elements $omega in Omega$
$mathcal{F}$ is a sigma algebra on that set. In which elements of $mathcal{F}$ are thought of as 'events' or outcomes of an experiment.
$mathbb{P}$ is a probability measure on that sigma algebra.
Then a random variable $X$ is just a measurable function $X: Omega to mathcal{X}$ (usually $mathbb{R}$).
Then
The CDF of $X$ is F(x)=$mathbb{P}({ omega in Omega :X(omega) leq x })$
If $X$ is discrete the mass function can be defined as $f(x)=mathbb{P}({ omega in Omega :X(omega) = x })$
(HERE is where my confusion starts creeping in!) - If X is continuous (and $mathcal{X}=mathbb{R}$) the density function is the function f(x) such that for $Asubset mathcal{X} $ measurable
$$mathbb{P}({ omega in Omega :X(omega) in A })=int_{A}f(x)dx $$
- (HERE is my main confusion, it surrounds the distribution $mu$) - We can also refer to a random variable X having the 'distribution' $mu$ where $mu$ is a measure on $mathcal{X}$. Now this $mu$ basically tells us everything we need to know about X correct? How does it relate to the other functions named above? Specifically in the continuous case, it relates to $X$ through the Radon-Nikodym derivative correct? In the case we have a general random variable not continuous or discrete what role is the $mu$ playing?
probability probability-theory probability-distributions radon-nikodym
asked Jan 31 at 19:12
MontyMonty
361113
$endgroup$
add a comment |
$begingroup$
Hi I'm hoping this will clear up any confusion on basic probability.
Take a probability space $(mathbb{P},mathcal{F},Omega)$ :
$Omega$ is some set of elements $omega in Omega$
$mathcal{F}$ is a sigma algebra on that set. In which elements of $mathcal{F}$ are thought of as 'events' or outcomes of an experiment.
$mathbb{P}$ is a probability measure on that sigma algebra.
Then a random variable $X$ is just a measurable function $X: Omega to mathcal{X}$ (usually $mathbb{R}$).
Then
The CDF of $X$ is F(x)=$mathbb{P}({ omega in Omega :X(omega) leq x })$
If $X$ is discrete the mass function can be defined as $f(x)=mathbb{P}({ omega in Omega :X(omega) = x })$
(HERE is where my confusion starts creeping in!) - If X is continuous (and $mathcal{X}=mathbb{R}$) the density function is the function f(x) such that for $Asubset mathcal{X} $ measurable
$$mathbb{P}({ omega in Omega :X(omega) in A })=int_{A}f(x)dx $$
- (HERE is my main confusion, it surrounds the distribution $mu$) - We can also refer to a random variable X having the 'distribution' $mu$ where $mu$ is a measure on $mathcal{X}$. Now this $mu$ basically tells us everything we need to know about X correct? How does it relate to the other functions named above? Specifically in the continuous case, it relates to $X$ through the Radon-Nikodym derivative correct? In the case we have a general random variable not continuous or discrete what role is the $mu$ playing?
probability probability-theory probability-distributions radon-nikodym
asked Jan 31 at 19:12
MontyMonty
361113
$endgroup$
$begingroup$
The CDF makes no sense as you wrote it unless $mathcal{X} = mathbf{R}.$ In the case where $X$ is absolutely continuous, $F(x) = int_{-infty}^x f,$ for any density function $f$ and it can be shown that $F'(x) = f(x)$ a.e.
$endgroup$
– Will M.
Jan 31 at 19:37
add a comment |
$begingroup$
Hi I'm hoping this will clear up any confusion on basic probability.
Take a probability space $(mathbb{P},mathcal{F},Omega)$ :
$Omega$ is some set of elements $omega in Omega$
$mathcal{F}$ is a sigma algebra on that set. In which elements of $mathcal{F}$ are thought of as 'events' or outcomes of an experiment.
$mathbb{P}$ is a probability measure on that sigma algebra.
Then a random variable $X$ is just a measurable function $X: Omega to mathcal{X}$ (usually $mathbb{R}$).
Then
The CDF of $X$ is F(x)=$mathbb{P}({ omega in Omega :X(omega) leq x })$
If $X$ is discrete the mass function can be defined as $f(x)=mathbb{P}({ omega in Omega :X(omega) = x })$
(HERE is where my confusion starts creeping in!) - If X is continuous (and $mathcal{X}=mathbb{R}$) the density function is the function f(x) such that for $Asubset mathcal{X} $ measurable
$$mathbb{P}({ omega in Omega :X(omega) in A })=int_{A}f(x)dx $$
- (HERE is my main confusion, it surrounds the distribution $mu$) - We can also refer to a random variable X having the 'distribution' $mu$ where $mu$ is a measure on $mathcal{X}$. Now this $mu$ basically tells us everything we need to know about X correct? How does it relate to the other functions named above? Specifically in the continuous case, it relates to $X$ through the Radon-Nikodym derivative correct? In the case we have a general random variable not continuous or discrete what role is the $mu$ playing?
probability probability-theory probability-distributions radon-nikodym
asked Jan 31 at 19:12
MontyMonty
361113
$endgroup$
Hi I'm hoping this will clear up any confusion on basic probability.
Take a probability space $(mathbb{P},mathcal{F},Omega)$ :
$Omega$ is some set of elements $omega in Omega$
$mathcal{F}$ is a sigma algebra on that set. In which elements of $mathcal{F}$ are thought of as 'events' or outcomes of an experiment.
$mathbb{P}$ is a probability measure on that sigma algebra.
Then a random variable $X$ is just a measurable function $X: Omega to mathcal{X}$ (usually $mathbb{R}$).
Then
The CDF of $X$ is F(x)=$mathbb{P}({ omega in Omega :X(omega) leq x })$
If $X$ is discrete the mass function can be defined as $f(x)=mathbb{P}({ omega in Omega :X(omega) = x })$
(HERE is where my confusion starts creeping in!) - If X is continuous (and $mathcal{X}=mathbb{R}$) the density function is the function f(x) such that for $Asubset mathcal{X} $ measurable
$$mathbb{P}({ omega in Omega :X(omega) in A })=int_{A}f(x)dx $$
- (HERE is my main confusion, it surrounds the distribution $mu$) - We can also refer to a random variable X having the 'distribution' $mu$ where $mu$ is a measure on $mathcal{X}$. Now this $mu$ basically tells us everything we need to know about X correct? How does it relate to the other functions named above? Specifically in the continuous case, it relates to $X$ through the Radon-Nikodym derivative correct? In the case we have a general random variable not continuous or discrete what role is the $mu$ playing?
probability probability-theory probability-distributions radon-nikodym
probability probability-theory probability-distributions radon-nikodym
asked Jan 31 at 19:12
MontyMonty
361113
asked Jan 31 at 19:12
MontyMonty
361113
asked Jan 31 at 19:12
MontyMonty
361113
asked Jan 31 at 19:12
MontyMonty
361113
asked Jan 31 at 19:12
MontyMonty
361113
361113
$begingroup$
The CDF makes no sense as you wrote it unless $mathcal{X} = mathbf{R}.$ In the case where $X$ is absolutely continuous, $F(x) = int_{-infty}^x f,$ for any density function $f$ and it can be shown that $F'(x) = f(x)$ a.e.
$endgroup$
– Will M.
Jan 31 at 19:37
add a comment |
$begingroup$
The CDF makes no sense as you wrote it unless $mathcal{X} = mathbf{R}.$ In the case where $X$ is absolutely continuous, $F(x) = int_{-infty}^x f,$ for any density function $f$ and it can be shown that $F'(x) = f(x)$ a.e.
$endgroup$
– Will M.
Jan 31 at 19:37
$begingroup$
The CDF makes no sense as you wrote it unless $mathcal{X} = mathbf{R}.$ In the case where $X$ is absolutely continuous, $F(x) = int_{-infty}^x f,$ for any density function $f$ and it can be shown that $F'(x) = f(x)$ a.e.
$endgroup$
– Will M.
Jan 31 at 19:37
$begingroup$
The CDF makes no sense as you wrote it unless $mathcal{X} = mathbf{R}.$ In the case where $X$ is absolutely continuous, $F(x) = int_{-infty}^x f,$ for any density function $f$ and it can be shown that $F'(x) = f(x)$ a.e.
$endgroup$
– Will M.
Jan 31 at 19:37
add a comment |
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$begingroup$
The CDF makes no sense as you wrote it unless $mathcal{X} = mathbf{R}.$ In the case where $X$ is absolutely continuous, $F(x) = int_{-infty}^x f,$ for any density function $f$ and it can be shown that $F'(x) = f(x)$ a.e.
$endgroup$
– Will M.
Jan 31 at 19:37