Discrete positive moment problem
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My teacher claims that, given all factorial moments $E((X)_r) = E(prod_{i=0}^{r-1}(X-i))$ of a positive discrete random variable $X$ it is possible to deduce the law of said variable.
The first thing I found out is that the ordinary moments can be deduced from the factorial moments and vice-versa, meaning that this is the same as the moment problem only in the discrete, positive case.
Indeed, we can clearly get $E((X)_r)$ from the $E(X^i)$ by expanding the polynomial and using linearity, and the other way can be obtained through an induction: $E(X^i)=E((X)_{i+1}-sum_{jleq i-1}alpha_jX^j)$.
I have not been able to go any further, I really feel like we need some extra hypothesis about the values taken by the variable other than the fact that they're countable and in $mathbb{R}^+$.
probability-theory random-variables moment-problem
asked Dec 10 '18 at 20:06
John DoJohn Do
373115
$endgroup$
add a comment |
$begingroup$
My teacher claims that, given all factorial moments $E((X)_r) = E(prod_{i=0}^{r-1}(X-i))$ of a positive discrete random variable $X$ it is possible to deduce the law of said variable.
The first thing I found out is that the ordinary moments can be deduced from the factorial moments and vice-versa, meaning that this is the same as the moment problem only in the discrete, positive case.
Indeed, we can clearly get $E((X)_r)$ from the $E(X^i)$ by expanding the polynomial and using linearity, and the other way can be obtained through an induction: $E(X^i)=E((X)_{i+1}-sum_{jleq i-1}alpha_jX^j)$.
I have not been able to go any further, I really feel like we need some extra hypothesis about the values taken by the variable other than the fact that they're countable and in $mathbb{R}^+$.
probability-theory random-variables moment-problem
asked Dec 10 '18 at 20:06
John DoJohn Do
373115
$endgroup$
add a comment |
$begingroup$
My teacher claims that, given all factorial moments $E((X)_r) = E(prod_{i=0}^{r-1}(X-i))$ of a positive discrete random variable $X$ it is possible to deduce the law of said variable.
The first thing I found out is that the ordinary moments can be deduced from the factorial moments and vice-versa, meaning that this is the same as the moment problem only in the discrete, positive case.
Indeed, we can clearly get $E((X)_r)$ from the $E(X^i)$ by expanding the polynomial and using linearity, and the other way can be obtained through an induction: $E(X^i)=E((X)_{i+1}-sum_{jleq i-1}alpha_jX^j)$.
I have not been able to go any further, I really feel like we need some extra hypothesis about the values taken by the variable other than the fact that they're countable and in $mathbb{R}^+$.
probability-theory random-variables moment-problem
asked Dec 10 '18 at 20:06
John DoJohn Do
373115
$endgroup$
My teacher claims that, given all factorial moments $E((X)_r) = E(prod_{i=0}^{r-1}(X-i))$ of a positive discrete random variable $X$ it is possible to deduce the law of said variable.
The first thing I found out is that the ordinary moments can be deduced from the factorial moments and vice-versa, meaning that this is the same as the moment problem only in the discrete, positive case.
Indeed, we can clearly get $E((X)_r)$ from the $E(X^i)$ by expanding the polynomial and using linearity, and the other way can be obtained through an induction: $E(X^i)=E((X)_{i+1}-sum_{jleq i-1}alpha_jX^j)$.
I have not been able to go any further, I really feel like we need some extra hypothesis about the values taken by the variable other than the fact that they're countable and in $mathbb{R}^+$.
probability-theory random-variables moment-problem
probability-theory random-variables moment-problem
asked Dec 10 '18 at 20:06
John DoJohn Do
373115
asked Dec 10 '18 at 20:06
John DoJohn Do
373115
asked Dec 10 '18 at 20:06
John DoJohn Do
373115
asked Dec 10 '18 at 20:06
John DoJohn Do
373115
asked Dec 10 '18 at 20:06
John DoJohn Do
373115
373115
add a comment |
add a comment |
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