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Given a random variable R, find its characteristic function
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Let $R$ be a random variable such that his density function is $f_R(r)=rcdot e^-rcdot u(r)$ for $rgeq 0$. Find $R$'s characteristic function $Phi(omega)$.
I got no idea where to start, any ideas?
P.S there is a clue: note that the integrand is "almost" a derivative by $omega$.
So what I tried is by definition $Phi_R(omega)=E[e^iomega R]=int_0^infty r^2cdot e^-rcdot e^iomega rcdot u(r)dr$
Because $u$ isn't given, I don't know how am I supposed to do that. I didn't entirely understand the clue. Is it that $fracddomegar^2cdot e^-rcdot e^iomega rcdot u(r)$ is "almost" $r^2cdot e^-rcdot e^iomega rcdot u(r)$?
I'd be happy for some help if anyone got any ideas.
calculus probability
asked Aug 3 at 2:47
Liav Cohen
11
 |Â
show 1 more comment
up vote
0
down vote
favorite
Let $R$ be a random variable such that his density function is $f_R(r)=rcdot e^-rcdot u(r)$ for $rgeq 0$. Find $R$'s characteristic function $Phi(omega)$.
I got no idea where to start, any ideas?
P.S there is a clue: note that the integrand is "almost" a derivative by $omega$.
So what I tried is by definition $Phi_R(omega)=E[e^iomega R]=int_0^infty r^2cdot e^-rcdot e^iomega rcdot u(r)dr$
Because $u$ isn't given, I don't know how am I supposed to do that. I didn't entirely understand the clue. Is it that $fracddomegar^2cdot e^-rcdot e^iomega rcdot u(r)$ is "almost" $r^2cdot e^-rcdot e^iomega rcdot u(r)$?
I'd be happy for some help if anyone got any ideas.
calculus probability
asked Aug 3 at 2:47
Liav Cohen
11
2
You could try starting with the definition of the characteristic function...
– Theoretical Economist
Aug 3 at 2:49
i tried but got stuck, Nothing fits into the integral
– Liav Cohen
Aug 3 at 2:57
2
What is the function $u$? Can you edit your post to include the integral you mention, and explain where you got stuck?
– David M.
Aug 3 at 4:08
$u$ isn't given. It's one of the problems I had
– Liav Cohen
Aug 3 at 7:14
Why do you have $r^2$ in the integrand?
– zoli
Aug 3 at 13:33
 |Â
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $R$ be a random variable such that his density function is $f_R(r)=rcdot e^-rcdot u(r)$ for $rgeq 0$. Find $R$'s characteristic function $Phi(omega)$.
I got no idea where to start, any ideas?
P.S there is a clue: note that the integrand is "almost" a derivative by $omega$.
So what I tried is by definition $Phi_R(omega)=E[e^iomega R]=int_0^infty r^2cdot e^-rcdot e^iomega rcdot u(r)dr$
Because $u$ isn't given, I don't know how am I supposed to do that. I didn't entirely understand the clue. Is it that $fracddomegar^2cdot e^-rcdot e^iomega rcdot u(r)$ is "almost" $r^2cdot e^-rcdot e^iomega rcdot u(r)$?
I'd be happy for some help if anyone got any ideas.
calculus probability
asked Aug 3 at 2:47
Liav Cohen
11
Let $R$ be a random variable such that his density function is $f_R(r)=rcdot e^-rcdot u(r)$ for $rgeq 0$. Find $R$'s characteristic function $Phi(omega)$.
I got no idea where to start, any ideas?
P.S there is a clue: note that the integrand is "almost" a derivative by $omega$.
So what I tried is by definition $Phi_R(omega)=E[e^iomega R]=int_0^infty r^2cdot e^-rcdot e^iomega rcdot u(r)dr$
Because $u$ isn't given, I don't know how am I supposed to do that. I didn't entirely understand the clue. Is it that $fracddomegar^2cdot e^-rcdot e^iomega rcdot u(r)$ is "almost" $r^2cdot e^-rcdot e^iomega rcdot u(r)$?
I'd be happy for some help if anyone got any ideas.
calculus probability
asked Aug 3 at 2:47
Liav Cohen
11
edited Aug 3 at 7:20
asked Aug 3 at 2:47
Liav Cohen
11
asked Aug 3 at 2:47
Liav Cohen
11
asked Aug 3 at 2:47
Liav Cohen
11
11
2
You could try starting with the definition of the characteristic function...
– Theoretical Economist
Aug 3 at 2:49
i tried but got stuck, Nothing fits into the integral
– Liav Cohen
Aug 3 at 2:57
2
What is the function $u$? Can you edit your post to include the integral you mention, and explain where you got stuck?
– David M.
Aug 3 at 4:08
$u$ isn't given. It's one of the problems I had
– Liav Cohen
Aug 3 at 7:14
Why do you have $r^2$ in the integrand?
– zoli
Aug 3 at 13:33
 |Â
show 1 more comment
2
You could try starting with the definition of the characteristic function...
– Theoretical Economist
Aug 3 at 2:49
i tried but got stuck, Nothing fits into the integral
– Liav Cohen
Aug 3 at 2:57
2
What is the function $u$? Can you edit your post to include the integral you mention, and explain where you got stuck?
– David M.
Aug 3 at 4:08
$u$ isn't given. It's one of the problems I had
– Liav Cohen
Aug 3 at 7:14
Why do you have $r^2$ in the integrand?
– zoli
Aug 3 at 13:33
2
2
You could try starting with the definition of the characteristic function...
– Theoretical Economist
Aug 3 at 2:49
You could try starting with the definition of the characteristic function...
– Theoretical Economist
Aug 3 at 2:49
i tried but got stuck, Nothing fits into the integral
– Liav Cohen
Aug 3 at 2:57
i tried but got stuck, Nothing fits into the integral
– Liav Cohen
Aug 3 at 2:57
2
2
What is the function $u$? Can you edit your post to include the integral you mention, and explain where you got stuck?
– David M.
Aug 3 at 4:08
What is the function $u$? Can you edit your post to include the integral you mention, and explain where you got stuck?
– David M.
Aug 3 at 4:08
$u$ isn't given. It's one of the problems I had
– Liav Cohen
Aug 3 at 7:14
$u$ isn't given. It's one of the problems I had
– Liav Cohen
Aug 3 at 7:14
Why do you have $r^2$ in the integrand?
– zoli
Aug 3 at 13:33
Why do you have $r^2$ in the integrand?
– zoli
Aug 3 at 13:33
 |Â
show 1 more comment
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2
You could try starting with the definition of the characteristic function...
– Theoretical Economist
Aug 3 at 2:49
i tried but got stuck, Nothing fits into the integral
– Liav Cohen
Aug 3 at 2:57
2
What is the function $u$? Can you edit your post to include the integral you mention, and explain where you got stuck?
– David M.
Aug 3 at 4:08
$u$ isn't given. It's one of the problems I had
– Liav Cohen
Aug 3 at 7:14
Why do you have $r^2$ in the integrand?
– zoli
Aug 3 at 13:33