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Tight upper and lower bounds of the CDF of a summation of random variables
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I have this random variable
$$Y = sum_k=1^KX_k$$
where $X_k$ are i.i.d. random variables with CDF and PDF $F_X(x)$ and $f_X(x)$, respectively.
In my application, the CDF of $Y$ denoted by $F_Y(y)=textPrleft[Yleq yright] = textPrleft[sum_k=1^KX_kleq yright]$ is not easy to evaluate directly. So, I am trying to find a tight lower and upper bound, such that the exact CDF is very close to both.
One obvious options for the lower and upper bounds are
$$textPrleft[Kmin_k X_kleq yright]geq textPrleft[sum_k=1^KX_kleq yright] geq textPrleft[Kmax_k X_kleq yright]$$
I have done some simulations on the lower bound, and it is lose. What are other lower and upper bounds that I can use?
EDIT: In my application, $X_k=A_k/B_k$, where $A_k$ and $B_k$ are i.i.d exponential random variables with parameter 1, and the means of such random variables don't exist.
statistics probability-distributions upper-lower-bounds
asked Aug 3 at 0:20
BlackMath
827
add a comment |Â
up vote
1
down vote
favorite
I have this random variable
$$Y = sum_k=1^KX_k$$
where $X_k$ are i.i.d. random variables with CDF and PDF $F_X(x)$ and $f_X(x)$, respectively.
In my application, the CDF of $Y$ denoted by $F_Y(y)=textPrleft[Yleq yright] = textPrleft[sum_k=1^KX_kleq yright]$ is not easy to evaluate directly. So, I am trying to find a tight lower and upper bound, such that the exact CDF is very close to both.
One obvious options for the lower and upper bounds are
$$textPrleft[Kmin_k X_kleq yright]geq textPrleft[sum_k=1^KX_kleq yright] geq textPrleft[Kmax_k X_kleq yright]$$
I have done some simulations on the lower bound, and it is lose. What are other lower and upper bounds that I can use?
EDIT: In my application, $X_k=A_k/B_k$, where $A_k$ and $B_k$ are i.i.d exponential random variables with parameter 1, and the means of such random variables don't exist.
statistics probability-distributions upper-lower-bounds
asked Aug 3 at 0:20
BlackMath
827
1
Here’s a potentially useful paper (paywall)
– David M.
Aug 3 at 0:33
@DavidM. Thanks, but this method depends on the mean of the summation. In my case, the means don't exist. In my application $X_k=A_k/B_k$, where both $A_k$ and $B_k$ are exponential RVs with parameter 1.
– BlackMath
Aug 3 at 3:23
Ahhh gotcha I think it would be helpful if you edit your post and add that info
– David M.
Aug 3 at 3:27
@DavidM. Just did. Do you have another suggestion in light of the new information?
– BlackMath
Aug 3 at 3:31
Not off the top of my head—will post if I think of anything
– David M.
Aug 3 at 3:32
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have this random variable
$$Y = sum_k=1^KX_k$$
where $X_k$ are i.i.d. random variables with CDF and PDF $F_X(x)$ and $f_X(x)$, respectively.
In my application, the CDF of $Y$ denoted by $F_Y(y)=textPrleft[Yleq yright] = textPrleft[sum_k=1^KX_kleq yright]$ is not easy to evaluate directly. So, I am trying to find a tight lower and upper bound, such that the exact CDF is very close to both.
One obvious options for the lower and upper bounds are
$$textPrleft[Kmin_k X_kleq yright]geq textPrleft[sum_k=1^KX_kleq yright] geq textPrleft[Kmax_k X_kleq yright]$$
I have done some simulations on the lower bound, and it is lose. What are other lower and upper bounds that I can use?
EDIT: In my application, $X_k=A_k/B_k$, where $A_k$ and $B_k$ are i.i.d exponential random variables with parameter 1, and the means of such random variables don't exist.
statistics probability-distributions upper-lower-bounds
asked Aug 3 at 0:20
BlackMath
827
I have this random variable
$$Y = sum_k=1^KX_k$$
where $X_k$ are i.i.d. random variables with CDF and PDF $F_X(x)$ and $f_X(x)$, respectively.
In my application, the CDF of $Y$ denoted by $F_Y(y)=textPrleft[Yleq yright] = textPrleft[sum_k=1^KX_kleq yright]$ is not easy to evaluate directly. So, I am trying to find a tight lower and upper bound, such that the exact CDF is very close to both.
One obvious options for the lower and upper bounds are
$$textPrleft[Kmin_k X_kleq yright]geq textPrleft[sum_k=1^KX_kleq yright] geq textPrleft[Kmax_k X_kleq yright]$$
I have done some simulations on the lower bound, and it is lose. What are other lower and upper bounds that I can use?
EDIT: In my application, $X_k=A_k/B_k$, where $A_k$ and $B_k$ are i.i.d exponential random variables with parameter 1, and the means of such random variables don't exist.
statistics probability-distributions upper-lower-bounds
asked Aug 3 at 0:20
BlackMath
827
edited Aug 3 at 3:30
asked Aug 3 at 0:20
BlackMath
827
asked Aug 3 at 0:20
BlackMath
827
asked Aug 3 at 0:20
BlackMath
827
827
1
Here’s a potentially useful paper (paywall)
– David M.
Aug 3 at 0:33
@DavidM. Thanks, but this method depends on the mean of the summation. In my case, the means don't exist. In my application $X_k=A_k/B_k$, where both $A_k$ and $B_k$ are exponential RVs with parameter 1.
– BlackMath
Aug 3 at 3:23
Ahhh gotcha I think it would be helpful if you edit your post and add that info
– David M.
Aug 3 at 3:27
@DavidM. Just did. Do you have another suggestion in light of the new information?
– BlackMath
Aug 3 at 3:31
Not off the top of my head—will post if I think of anything
– David M.
Aug 3 at 3:32
add a comment |Â
1
Here’s a potentially useful paper (paywall)
– David M.
Aug 3 at 0:33
@DavidM. Thanks, but this method depends on the mean of the summation. In my case, the means don't exist. In my application $X_k=A_k/B_k$, where both $A_k$ and $B_k$ are exponential RVs with parameter 1.
– BlackMath
Aug 3 at 3:23
Ahhh gotcha I think it would be helpful if you edit your post and add that info
– David M.
Aug 3 at 3:27
@DavidM. Just did. Do you have another suggestion in light of the new information?
– BlackMath
Aug 3 at 3:31
Not off the top of my head—will post if I think of anything
– David M.
Aug 3 at 3:32
1
1
Here’s a potentially useful paper (paywall)
– David M.
Aug 3 at 0:33
Here’s a potentially useful paper (paywall)
– David M.
Aug 3 at 0:33
@DavidM. Thanks, but this method depends on the mean of the summation. In my case, the means don't exist. In my application $X_k=A_k/B_k$, where both $A_k$ and $B_k$ are exponential RVs with parameter 1.
– BlackMath
Aug 3 at 3:23
@DavidM. Thanks, but this method depends on the mean of the summation. In my case, the means don't exist. In my application $X_k=A_k/B_k$, where both $A_k$ and $B_k$ are exponential RVs with parameter 1.
– BlackMath
Aug 3 at 3:23
Ahhh gotcha I think it would be helpful if you edit your post and add that info
– David M.
Aug 3 at 3:27
Ahhh gotcha I think it would be helpful if you edit your post and add that info
– David M.
Aug 3 at 3:27
@DavidM. Just did. Do you have another suggestion in light of the new information?
– BlackMath
Aug 3 at 3:31
@DavidM. Just did. Do you have another suggestion in light of the new information?
– BlackMath
Aug 3 at 3:31
Not off the top of my head—will post if I think of anything
– David M.
Aug 3 at 3:32
Not off the top of my head—will post if I think of anything
– David M.
Aug 3 at 3:32
add a comment |Â
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1
Here’s a potentially useful paper (paywall)
– David M.
Aug 3 at 0:33
@DavidM. Thanks, but this method depends on the mean of the summation. In my case, the means don't exist. In my application $X_k=A_k/B_k$, where both $A_k$ and $B_k$ are exponential RVs with parameter 1.
– BlackMath
Aug 3 at 3:23
Ahhh gotcha I think it would be helpful if you edit your post and add that info
– David M.
Aug 3 at 3:27
@DavidM. Just did. Do you have another suggestion in light of the new information?
– BlackMath
Aug 3 at 3:31
Not off the top of my head—will post if I think of anything
– David M.
Aug 3 at 3:32