Polynomial function of random variable
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let $x$ be a random variable with pdf $p(x)$, e.g., $ prob(A)=int_A p(x)dx$.
Define a random variable $y=f(x)$.
If $f$ and $p$ are polynomial functions, what we can tell about pdf of random variable $y$. Can we describe the pdf of $y$ in terms of the pdf of $x$?
probability polynomials probability-distributions random-variables
asked Aug 10 at 17:57
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up vote
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down vote
favorite
let $x$ be a random variable with pdf $p(x)$, e.g., $ prob(A)=int_A p(x)dx$.
Define a random variable $y=f(x)$.
If $f$ and $p$ are polynomial functions, what we can tell about pdf of random variable $y$. Can we describe the pdf of $y$ in terms of the pdf of $x$?
probability polynomials probability-distributions random-variables
asked Aug 10 at 17:57
M.A
163
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
let $x$ be a random variable with pdf $p(x)$, e.g., $ prob(A)=int_A p(x)dx$.
Define a random variable $y=f(x)$.
If $f$ and $p$ are polynomial functions, what we can tell about pdf of random variable $y$. Can we describe the pdf of $y$ in terms of the pdf of $x$?
probability polynomials probability-distributions random-variables
asked Aug 10 at 17:57
M.A
163
let $x$ be a random variable with pdf $p(x)$, e.g., $ prob(A)=int_A p(x)dx$.
Define a random variable $y=f(x)$.
If $f$ and $p$ are polynomial functions, what we can tell about pdf of random variable $y$. Can we describe the pdf of $y$ in terms of the pdf of $x$?
probability polynomials probability-distributions random-variables
asked Aug 10 at 17:57
M.A
163
asked Aug 10 at 17:57
M.A
163
asked Aug 10 at 17:57
M.A
163
asked Aug 10 at 17:57
M.A
163
163
add a comment |Â
add a comment |Â
1 Answer
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This is called a change of variable. One of the complications is that for a given $y$, there may be multiple values of $x$ with $f(x)=y$
You can say that the density function for $Y$ is $$sum_x:f(x)=y dfracp(x)f'(x)$$
and if $f(x)$ is invertible then this is $$dfracp(f^-1(y))$$
Typically this would not be a polynomial when $f(x)$ is of degree greater than $1$. Indeed it may be infinite if $f'(x)=0$ at a point where $p(x)>0$. Note that in any case $p(x)$ cannot be a polynomial function over the whole of $mathbb R$, as it would not integrate to $1$, so presumably is restricted to some finite interval
answered Aug 11 at 9:12
Henry
93.3k470148
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
This is called a change of variable. One of the complications is that for a given $y$, there may be multiple values of $x$ with $f(x)=y$
You can say that the density function for $Y$ is $$sum_x:f(x)=y dfracp(x)f'(x)$$
and if $f(x)$ is invertible then this is $$dfracp(f^-1(y))$$
Typically this would not be a polynomial when $f(x)$ is of degree greater than $1$. Indeed it may be infinite if $f'(x)=0$ at a point where $p(x)>0$. Note that in any case $p(x)$ cannot be a polynomial function over the whole of $mathbb R$, as it would not integrate to $1$, so presumably is restricted to some finite interval
answered Aug 11 at 9:12
Henry
93.3k470148
add a comment |Â
up vote
0
down vote
This is called a change of variable. One of the complications is that for a given $y$, there may be multiple values of $x$ with $f(x)=y$
You can say that the density function for $Y$ is $$sum_x:f(x)=y dfracp(x)f'(x)$$
and if $f(x)$ is invertible then this is $$dfracp(f^-1(y))$$
Typically this would not be a polynomial when $f(x)$ is of degree greater than $1$. Indeed it may be infinite if $f'(x)=0$ at a point where $p(x)>0$. Note that in any case $p(x)$ cannot be a polynomial function over the whole of $mathbb R$, as it would not integrate to $1$, so presumably is restricted to some finite interval
answered Aug 11 at 9:12
Henry
93.3k470148
add a comment |Â
up vote
0
down vote
up vote
0
down vote
This is called a change of variable. One of the complications is that for a given $y$, there may be multiple values of $x$ with $f(x)=y$
You can say that the density function for $Y$ is $$sum_x:f(x)=y dfracp(x)f'(x)$$
and if $f(x)$ is invertible then this is $$dfracp(f^-1(y))$$
Typically this would not be a polynomial when $f(x)$ is of degree greater than $1$. Indeed it may be infinite if $f'(x)=0$ at a point where $p(x)>0$. Note that in any case $p(x)$ cannot be a polynomial function over the whole of $mathbb R$, as it would not integrate to $1$, so presumably is restricted to some finite interval
answered Aug 11 at 9:12
Henry
93.3k470148
This is called a change of variable. One of the complications is that for a given $y$, there may be multiple values of $x$ with $f(x)=y$
You can say that the density function for $Y$ is $$sum_x:f(x)=y dfracp(x)f'(x)$$
and if $f(x)$ is invertible then this is $$dfracp(f^-1(y))$$
Typically this would not be a polynomial when $f(x)$ is of degree greater than $1$. Indeed it may be infinite if $f'(x)=0$ at a point where $p(x)>0$. Note that in any case $p(x)$ cannot be a polynomial function over the whole of $mathbb R$, as it would not integrate to $1$, so presumably is restricted to some finite interval
answered Aug 11 at 9:12
Henry
93.3k470148
answered Aug 11 at 9:12
Henry
93.3k470148
answered Aug 11 at 9:12
Henry
93.3k470148
answered Aug 11 at 9:12
Henry
93.3k470148
93.3k470148
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