Tail probability distribution for sums of variables belonging to three different distributions
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Assume we have given a natural number $N$, two sets $C$ and $D$ with cardinality $c,(N-c)$ respectively and random variables $e_{ij}$, where $i,j in {1,2,...,N}$.
Moreover these $e_{ij}$'s follow three different distributions depending on their indices:
$$ e_{ij} sim distr1 quad text{if $(i,j)in Dtimes D$} \
quad quad quad quad sim distr2 quad text{if $(i,j)in Ctimes D$ or $D times C$} \
quad sim distr3 quad text{if $(i,j)in Ctimes C$}
$$
For simplicity one can assume they are normal and all are independent.
I am interested in calculating the tail probability
$$mathbb{P}(sum_{i,j}e_{ij}-frac{1}{2}sum_{i=1}^N mid sum_{j=1}^{N-1}(e_{ij}-e_{ji} )mid > x)$$
My idea was to divide the terms depending to the 3 different cases: i.e.
$$mathbb{P}(sum_{i,j}e_{ij}-frac{1}{2}sum_{i=1}^n mid sum_{j=1}^{n-1}(e_{ij}-e_{ji} )mid > x) \= mathbb{P}(sum_{i,j}e_{ij}-frac{1}{2}(sum_{i in C} midsum_{j in C} sum_{j in D}(e_{ij}-e_{ji} ) mid + sum_{i in D} mid sum_{j in C}sum_{j in D}(e_{ij}-e_{ji} )mid )>x$$
And I could also split the first sum, which is summing over all random variables into
$$sum_{i,j}e_{ij}=sum_{i,j in C} e_{ij}+ (sum_{iin C,j in D} e_{ij} + sum_{iin D ,j in C} e_{ij}) + sum_{i,j in D} e_{ij}$$
But now I don't know how to proceed...
Any help is appreciated
probability probability-theory probability-distributions
asked Nov 3 at 12:48
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377
add a comment |
up vote
2
down vote
favorite
Assume we have given a natural number $N$, two sets $C$ and $D$ with cardinality $c,(N-c)$ respectively and random variables $e_{ij}$, where $i,j in {1,2,...,N}$.
Moreover these $e_{ij}$'s follow three different distributions depending on their indices:
$$ e_{ij} sim distr1 quad text{if $(i,j)in Dtimes D$} \
quad quad quad quad sim distr2 quad text{if $(i,j)in Ctimes D$ or $D times C$} \
quad sim distr3 quad text{if $(i,j)in Ctimes C$}
$$
For simplicity one can assume they are normal and all are independent.
I am interested in calculating the tail probability
$$mathbb{P}(sum_{i,j}e_{ij}-frac{1}{2}sum_{i=1}^N mid sum_{j=1}^{N-1}(e_{ij}-e_{ji} )mid > x)$$
My idea was to divide the terms depending to the 3 different cases: i.e.
$$mathbb{P}(sum_{i,j}e_{ij}-frac{1}{2}sum_{i=1}^n mid sum_{j=1}^{n-1}(e_{ij}-e_{ji} )mid > x) \= mathbb{P}(sum_{i,j}e_{ij}-frac{1}{2}(sum_{i in C} midsum_{j in C} sum_{j in D}(e_{ij}-e_{ji} ) mid + sum_{i in D} mid sum_{j in C}sum_{j in D}(e_{ij}-e_{ji} )mid )>x$$
And I could also split the first sum, which is summing over all random variables into
$$sum_{i,j}e_{ij}=sum_{i,j in C} e_{ij}+ (sum_{iin C,j in D} e_{ij} + sum_{iin D ,j in C} e_{ij}) + sum_{i,j in D} e_{ij}$$
But now I don't know how to proceed...
Any help is appreciated
probability probability-theory probability-distributions
asked Nov 3 at 12:48
Alisat
377
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Assume we have given a natural number $N$, two sets $C$ and $D$ with cardinality $c,(N-c)$ respectively and random variables $e_{ij}$, where $i,j in {1,2,...,N}$.
Moreover these $e_{ij}$'s follow three different distributions depending on their indices:
$$ e_{ij} sim distr1 quad text{if $(i,j)in Dtimes D$} \
quad quad quad quad sim distr2 quad text{if $(i,j)in Ctimes D$ or $D times C$} \
quad sim distr3 quad text{if $(i,j)in Ctimes C$}
$$
For simplicity one can assume they are normal and all are independent.
I am interested in calculating the tail probability
$$mathbb{P}(sum_{i,j}e_{ij}-frac{1}{2}sum_{i=1}^N mid sum_{j=1}^{N-1}(e_{ij}-e_{ji} )mid > x)$$
My idea was to divide the terms depending to the 3 different cases: i.e.
$$mathbb{P}(sum_{i,j}e_{ij}-frac{1}{2}sum_{i=1}^n mid sum_{j=1}^{n-1}(e_{ij}-e_{ji} )mid > x) \= mathbb{P}(sum_{i,j}e_{ij}-frac{1}{2}(sum_{i in C} midsum_{j in C} sum_{j in D}(e_{ij}-e_{ji} ) mid + sum_{i in D} mid sum_{j in C}sum_{j in D}(e_{ij}-e_{ji} )mid )>x$$
And I could also split the first sum, which is summing over all random variables into
$$sum_{i,j}e_{ij}=sum_{i,j in C} e_{ij}+ (sum_{iin C,j in D} e_{ij} + sum_{iin D ,j in C} e_{ij}) + sum_{i,j in D} e_{ij}$$
But now I don't know how to proceed...
Any help is appreciated
probability probability-theory probability-distributions
asked Nov 3 at 12:48
Alisat
377
Assume we have given a natural number $N$, two sets $C$ and $D$ with cardinality $c,(N-c)$ respectively and random variables $e_{ij}$, where $i,j in {1,2,...,N}$.
Moreover these $e_{ij}$'s follow three different distributions depending on their indices:
$$ e_{ij} sim distr1 quad text{if $(i,j)in Dtimes D$} \
quad quad quad quad sim distr2 quad text{if $(i,j)in Ctimes D$ or $D times C$} \
quad sim distr3 quad text{if $(i,j)in Ctimes C$}
$$
For simplicity one can assume they are normal and all are independent.
I am interested in calculating the tail probability
$$mathbb{P}(sum_{i,j}e_{ij}-frac{1}{2}sum_{i=1}^N mid sum_{j=1}^{N-1}(e_{ij}-e_{ji} )mid > x)$$
My idea was to divide the terms depending to the 3 different cases: i.e.
$$mathbb{P}(sum_{i,j}e_{ij}-frac{1}{2}sum_{i=1}^n mid sum_{j=1}^{n-1}(e_{ij}-e_{ji} )mid > x) \= mathbb{P}(sum_{i,j}e_{ij}-frac{1}{2}(sum_{i in C} midsum_{j in C} sum_{j in D}(e_{ij}-e_{ji} ) mid + sum_{i in D} mid sum_{j in C}sum_{j in D}(e_{ij}-e_{ji} )mid )>x$$
And I could also split the first sum, which is summing over all random variables into
$$sum_{i,j}e_{ij}=sum_{i,j in C} e_{ij}+ (sum_{iin C,j in D} e_{ij} + sum_{iin D ,j in C} e_{ij}) + sum_{i,j in D} e_{ij}$$
But now I don't know how to proceed...
Any help is appreciated
probability probability-theory probability-distributions
probability probability-theory probability-distributions
asked Nov 3 at 12:48
Alisat
377
asked Nov 3 at 12:48
Alisat
377
edited Nov 27 at 14:07
asked Nov 3 at 12:48
Alisat
377
asked Nov 3 at 12:48
Alisat
377
asked Nov 3 at 12:48
Alisat
377
377
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