Mixing
Transformation of random Variable that appends a constant
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I have a function $g(x)=(x_1, dots, x_n, 1)$ (=just appends one to the vector). Now, let's say $X$ is a random variable, then $g(X)$ must also be a random variable. But what have I done here? What is it's distribution of $g(X)$?
Something like $D_X times Dirac$, where $Dirac$ is the Dirac-Distribution and $D_X$ is the Distribution of $X$?
My probability-theory background is a bit weak, so I think I just want to make sure that the math checks out (in a rigorous sense, I need this in a proof!).
EDIT: this is not homework, just to be sure
Example: $X sim N(0,1)$, what's the distribution of $g(X)$?
probability probability-theory
asked Aug 2 at 17:55
Leander
1134
add a comment |Â
up vote
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I have a function $g(x)=(x_1, dots, x_n, 1)$ (=just appends one to the vector). Now, let's say $X$ is a random variable, then $g(X)$ must also be a random variable. But what have I done here? What is it's distribution of $g(X)$?
Something like $D_X times Dirac$, where $Dirac$ is the Dirac-Distribution and $D_X$ is the Distribution of $X$?
My probability-theory background is a bit weak, so I think I just want to make sure that the math checks out (in a rigorous sense, I need this in a proof!).
EDIT: this is not homework, just to be sure
Example: $X sim N(0,1)$, what's the distribution of $g(X)$?
probability probability-theory
asked Aug 2 at 17:55
Leander
1134
Your notation is a little confusing. When you say $D_X$ is the distribution of $X$, what do you mean? How does the Dirac distribution (?) fit in?
– herb steinberg
Aug 2 at 18:56
thanks, I added an example. is it clearer now? I mean $X sim D_X$ when I say $X$ is $D_X$ distributed. Well, it must be Dirac distributed, doesn't it? It's a coupling of $X$ with a constant, so i don't see how it can be modeled in another way.
– Leander
Aug 2 at 20:15
Is $X$ an n dimensional random variable, where each component is standard normally distributed and $g(X)$ is an n+1 dimensional vector with the first n components $X$ and the extra component 1? If so, you gt it right.
– herb steinberg
Aug 3 at 0:13
Thanks! That was essentially my question
– Leander
Aug 3 at 12:39
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have a function $g(x)=(x_1, dots, x_n, 1)$ (=just appends one to the vector). Now, let's say $X$ is a random variable, then $g(X)$ must also be a random variable. But what have I done here? What is it's distribution of $g(X)$?
Something like $D_X times Dirac$, where $Dirac$ is the Dirac-Distribution and $D_X$ is the Distribution of $X$?
My probability-theory background is a bit weak, so I think I just want to make sure that the math checks out (in a rigorous sense, I need this in a proof!).
EDIT: this is not homework, just to be sure
Example: $X sim N(0,1)$, what's the distribution of $g(X)$?
probability probability-theory
asked Aug 2 at 17:55
Leander
1134
I have a function $g(x)=(x_1, dots, x_n, 1)$ (=just appends one to the vector). Now, let's say $X$ is a random variable, then $g(X)$ must also be a random variable. But what have I done here? What is it's distribution of $g(X)$?
Something like $D_X times Dirac$, where $Dirac$ is the Dirac-Distribution and $D_X$ is the Distribution of $X$?
My probability-theory background is a bit weak, so I think I just want to make sure that the math checks out (in a rigorous sense, I need this in a proof!).
EDIT: this is not homework, just to be sure
Example: $X sim N(0,1)$, what's the distribution of $g(X)$?
probability probability-theory
asked Aug 2 at 17:55
Leander
1134
edited Aug 2 at 20:18
asked Aug 2 at 17:55
Leander
1134
asked Aug 2 at 17:55
Leander
1134
asked Aug 2 at 17:55
Leander
1134
1134
Your notation is a little confusing. When you say $D_X$ is the distribution of $X$, what do you mean? How does the Dirac distribution (?) fit in?
– herb steinberg
Aug 2 at 18:56
thanks, I added an example. is it clearer now? I mean $X sim D_X$ when I say $X$ is $D_X$ distributed. Well, it must be Dirac distributed, doesn't it? It's a coupling of $X$ with a constant, so i don't see how it can be modeled in another way.
– Leander
Aug 2 at 20:15
Is $X$ an n dimensional random variable, where each component is standard normally distributed and $g(X)$ is an n+1 dimensional vector with the first n components $X$ and the extra component 1? If so, you gt it right.
– herb steinberg
Aug 3 at 0:13
Thanks! That was essentially my question
– Leander
Aug 3 at 12:39
add a comment |Â
Your notation is a little confusing. When you say $D_X$ is the distribution of $X$, what do you mean? How does the Dirac distribution (?) fit in?
– herb steinberg
Aug 2 at 18:56
thanks, I added an example. is it clearer now? I mean $X sim D_X$ when I say $X$ is $D_X$ distributed. Well, it must be Dirac distributed, doesn't it? It's a coupling of $X$ with a constant, so i don't see how it can be modeled in another way.
– Leander
Aug 2 at 20:15
Is $X$ an n dimensional random variable, where each component is standard normally distributed and $g(X)$ is an n+1 dimensional vector with the first n components $X$ and the extra component 1? If so, you gt it right.
– herb steinberg
Aug 3 at 0:13
Thanks! That was essentially my question
– Leander
Aug 3 at 12:39
Your notation is a little confusing. When you say $D_X$ is the distribution of $X$, what do you mean? How does the Dirac distribution (?) fit in?
– herb steinberg
Aug 2 at 18:56
Your notation is a little confusing. When you say $D_X$ is the distribution of $X$, what do you mean? How does the Dirac distribution (?) fit in?
– herb steinberg
Aug 2 at 18:56
thanks, I added an example. is it clearer now? I mean $X sim D_X$ when I say $X$ is $D_X$ distributed. Well, it must be Dirac distributed, doesn't it? It's a coupling of $X$ with a constant, so i don't see how it can be modeled in another way.
– Leander
Aug 2 at 20:15
thanks, I added an example. is it clearer now? I mean $X sim D_X$ when I say $X$ is $D_X$ distributed. Well, it must be Dirac distributed, doesn't it? It's a coupling of $X$ with a constant, so i don't see how it can be modeled in another way.
– Leander
Aug 2 at 20:15
Is $X$ an n dimensional random variable, where each component is standard normally distributed and $g(X)$ is an n+1 dimensional vector with the first n components $X$ and the extra component 1? If so, you gt it right.
– herb steinberg
Aug 3 at 0:13
Is $X$ an n dimensional random variable, where each component is standard normally distributed and $g(X)$ is an n+1 dimensional vector with the first n components $X$ and the extra component 1? If so, you gt it right.
– herb steinberg
Aug 3 at 0:13
Thanks! That was essentially my question
– Leander
Aug 3 at 12:39
Thanks! That was essentially my question
– Leander
Aug 3 at 12:39
add a comment |Â
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Your notation is a little confusing. When you say $D_X$ is the distribution of $X$, what do you mean? How does the Dirac distribution (?) fit in?
– herb steinberg
Aug 2 at 18:56
thanks, I added an example. is it clearer now? I mean $X sim D_X$ when I say $X$ is $D_X$ distributed. Well, it must be Dirac distributed, doesn't it? It's a coupling of $X$ with a constant, so i don't see how it can be modeled in another way.
– Leander
Aug 2 at 20:15
Is $X$ an n dimensional random variable, where each component is standard normally distributed and $g(X)$ is an n+1 dimensional vector with the first n components $X$ and the extra component 1? If so, you gt it right.
– herb steinberg
Aug 3 at 0:13
Thanks! That was essentially my question
– Leander
Aug 3 at 12:39