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Konrad Viltersten
science forum addict
Joined: 27 Jun 2005
Posts: 79
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 Posted: Wed Jun 28, 2006 12:16 pm Post subject:
Variance for a linear combination of three (normal) random variables
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Suppose there's a sum as follows:
X := a * W_1 + b * W_2 + c * W_3
How can we compute the variance of X? Well, it's rather
uncomplicated if one can be assured of independency.
Now, if one is assured that they _ARE_ dependent (as
for instance, being three states of a Wiener process at
three consecutive times), is there anything else to do but
sit down and cry loudly like a confused child?
Of, course, it's obvious that X ~ N(0,s^2) and that every
W_i ~ N(0,i). But how can we use it?!
--
Vänligen
Konrad
---------------------------------------------------
Sleep - thing used by ineffective people
as a substitute for coffee
Ambition - a poor excuse for not having
enough sence to be lazy
--------------------------------------------------- |
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Michael Jørgensen
science forum addict
Joined: 19 May 2005
Posts: 86
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| Posted: Wed Jun 28, 2006 12:23 pm Post subject:
Re: Variance for a linear combination of three (normal) random variables
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"Konrad Viltersten" <tmp1@viltersten.com> wrote in message
news:4gfa2vF1locgaU1@individual.net...
| Quote: | Suppose there's a sum as follows:
X := a * W_1 + b * W_2 + c * W_3
How can we compute the variance of X? Well, it's rather
uncomplicated if one can be assured of independency.
Now, if one is assured that they _ARE_ dependent (as
for instance, being three states of a Wiener process at
three consecutive times), is there anything else to do but
sit down and cry loudly like a confused child?
Of, course, it's obvious that X ~ N(0,s^2) and that every
W_i ~ N(0,i). But how can we use it?!
|
Well, if the W_i are not independent, then you need to know there
correlation/covariance.
Specifically, you need to calculate E{X^2}, and that involves terms like
E{W_1*W_2}, etc.
-Michael. |
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Konrad Viltersten
science forum addict
Joined: 27 Jun 2005
Posts: 79
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| Posted: Wed Jun 28, 2006 1:27 pm Post subject:
Re: Variance for a linear combination of three (normal) random variables
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| Quote: | Suppose there's a sum as follows:
X := a * W_1 + b * W_2 + c * W_3
How can we compute the variance of X? Well, it's rather
uncomplicated if one can be assured of independency.
Now, if one is assured that they _ARE_ dependent (as
for instance, being three states of a Wiener process at
three consecutive times), is there anything else to do but
sit down and cry loudly like a confused child?
Of, course, it's obvious that X ~ N(0,s^2) and that every
W_i ~ N(0,i). But how can we use it?!
Well, if the W_i are not independent, then you need to know
there correlation/covariance.
Specifically, you need to calculate E{X^2}, and that involves
terms like E{W_1*W_2}, etc.
|
Any suggestions how _THAT_ might be done for dependend
variables W? The only thing i know about them is that they're
part of a Wiener process, meaning they're N(0,t_i) but that
doesn't help me very greatly. Do i miss something obvious?
--
Vänligen
Konrad
---------------------------------------------------
Sleep - thing used by ineffective people
as a substitute for coffee
Ambition - a poor excuse for not having
enough sence to be lazy
--------------------------------------------------- |
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Randy Poe
science forum Guru
Joined: 24 Mar 2005
Posts: 2485
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| Posted: Wed Jun 28, 2006 1:42 pm Post subject:
Re: Variance for a linear combination of three (normal) random variables
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Konrad Viltersten wrote:
| Quote: | Suppose there's a sum as follows:
X := a * W_1 + b * W_2 + c * W_3
How can we compute the variance of X? Well, it's rather
uncomplicated if one can be assured of independency.
Now, if one is assured that they _ARE_ dependent (as
for instance, being three states of a Wiener process at
three consecutive times), is there anything else to do but
sit down and cry loudly like a confused child?
Of, course, it's obvious that X ~ N(0,s^2) and that every
W_i ~ N(0,i). But how can we use it?!
Well, if the W_i are not independent, then you need to know
there correlation/covariance.
Specifically, you need to calculate E{X^2}, and that involves
terms like E{W_1*W_2}, etc.
Any suggestions how _THAT_ might be done for dependend
variables W? The only thing i know about them is that they're
part of a Wiener process, meaning they're N(0,t_i) but that
doesn't help me very greatly. Do i miss something obvious?
|
Perhaps searching for "autocorrelation of Wiener process"
will find someone who has calculated the covariances
you need.
- Randy |
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Konrad Viltersten
science forum addict
Joined: 27 Jun 2005
Posts: 79
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| Posted: Wed Jun 28, 2006 1:47 pm Post subject:
Re: Variance for a linear combination of three (normal) random variables
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| Quote: | Suppose there's a sum as follows:
X := a * W_1 + b * W_2 + c * W_3
How can we compute the variance of X? Well, it's rather
uncomplicated if one can be assured of independency.
Now, if one is assured that they _ARE_ dependent (as
for instance, being three states of a Wiener process at
three consecutive times), is there anything else to do but
sit down and cry loudly like a confused child?
Of, course, it's obvious that X ~ N(0,s^2) and that every
W_i ~ N(0,i). But how can we use it?!
Well, if the W_i are not independent, then you need to
know there correlation/covariance. Specifically, you
need to calculate E{X^2}, and that involves terms like
E{W_1*W_2}, etc.
|
Right after i sent my reply, i realized that i was unclear.
What i can do is to compute the covariance of W_ti and
W_tj to be sqrt(t_i * t_j).
Still, what i can't do, is obtaining the variance of the
linear combination, even if i have the covariance matrix...
Any suggestions?
--
Vänligen
Konrad
---------------------------------------------------
Sleep - thing used by ineffective people
as a substitute for coffee
Ambition - a poor excuse for not having
enough sence to be lazy
--------------------------------------------------- |
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| Back to top |
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Michael Jørgensen
science forum addict
Joined: 19 May 2005
Posts: 86
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| Posted: Wed Jun 28, 2006 1:51 pm Post subject:
Re: Variance for a linear combination of three (normal) random variables
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"Konrad Viltersten" <tmp1@viltersten.com> wrote in message
news:4gfe38F1mhbcjU1@individual.net...
| Quote: | Suppose there's a sum as follows:
X := a * W_1 + b * W_2 + c * W_3
How can we compute the variance of X? Well, it's rather
uncomplicated if one can be assured of independency.
Now, if one is assured that they _ARE_ dependent (as
for instance, being three states of a Wiener process at
three consecutive times), is there anything else to do but
sit down and cry loudly like a confused child?
Of, course, it's obvious that X ~ N(0,s^2) and that every
W_i ~ N(0,i). But how can we use it?!
Well, if the W_i are not independent, then you need to know
there correlation/covariance.
Specifically, you need to calculate E{X^2}, and that involves
terms like E{W_1*W_2}, etc.
Any suggestions how _THAT_ might be done for dependend
variables W? The only thing i know about them is that they're
part of a Wiener process, meaning they're N(0,t_i) but that
doesn't help me very greatly. Do i miss something obvious?
|
[begin rambling]
Don't know details about a Wiener process, but since there is a time
dependence, there must be an autocorrelation function.
Intuitively, if the values W_1 and W_2 are sampled at different times, say,
t_1 and t_2, then the correlation between them must depend on the time
difference t_2-t_1. If the difference is large, then the values are
practically uncorrelated, while if the difference is 0, then the values are
identical.
Actually, looking at http://mathworld.wolfram.com/WienerProcess.html it is
stated that the difference W_2-W_1 is a Gaussian. You seem to state that the
actual sample value W_1 is Gaussian. That does quite add up, in my mind...
[end rambling]
-Michael. |
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Stephen J. Herschkorn
science forum Guru
Joined: 24 Mar 2005
Posts: 641
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| Posted: Wed Jun 28, 2006 3:10 pm Post subject:
Re: Variance for a linear combination of three (normal) random variables
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Konrad Viltersten wrote:
| Quote: | Suppose there's a sum as follows:
X := a * W_1 + b * W_2 + c * W_3
How can we compute the variance of X? Well, it's rather
uncomplicated if one can be assured of independency.
Now, if one is assured that they _ARE_ dependent (as
for instance, being three states of a Wiener process at
three consecutive times), is there anything else to do but
sit down and cry loudly like a confused child?
Of, course, it's obvious that X ~ N(0,s^2) and that every
W_i ~ N(0,i). But how can we use it?!
Well, if the W_i are not independent, then you need to
know there correlation/covariance. Specifically, you
need to calculate E{X^2}, and that involves terms like
E{W_1*W_2}, etc.
Right after i sent my reply, i realized that i was unclear.
What i can do is to compute the covariance of W_ti and
W_tj to be sqrt(t_i * t_j).
Still, what i can't do, is obtaining the variance of the
linear combination, even if i have the covariance matrix...
Any suggestions?
|
For *any* finite collection (X_i) of random variables on the same
space, the bilinearity of covariance implies
Var(sum(i, a_i X_i)) = sum(i, (a_i)^2 Var(X_i)) + 2 sum((i,j): i<j,
a_i a_j Cov(X_i, X_j))
Suppose W is a Brownian motion and t1 < t2. To determine Cov(W(t1),
W(t2)), express W(t2) as W(t1) + [W(t2) - W(t1)] and exploit
independent increments.
If this is homework, please cite sources of assitance on submitted work.
--
Stephen J. Herschkorn sjherschko@netscape.net
Math Tutor on the Internet and in Central New Jersey and Manhattan |
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C6L1V@shaw.ca
science forum Guru
Joined: 23 May 2005
Posts: 628
|
| Posted: Wed Jun 28, 2006 3:22 pm Post subject:
Re: Variance for a linear combination of three (normal) random variables
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|
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Konrad Viltersten wrote:
| Quote: | Suppose there's a sum as follows:
X := a * W_1 + b * W_2 + c * W_3
How can we compute the variance of X? Well, it's rather
uncomplicated if one can be assured of independency.
Now, if one is assured that they _ARE_ dependent (as
for instance, being three states of a Wiener process at
three consecutive times), is there anything else to do but
sit down and cry loudly like a confused child?
Of, course, it's obvious that X ~ N(0,s^2) and that every
W_i ~ N(0,i). But how can we use it?!
Well, if the W_i are not independent, then you need to
know there correlation/covariance. Specifically, you
need to calculate E{X^2}, and that involves terms like
E{W_1*W_2}, etc.
Right after i sent my reply, i realized that i was unclear.
What i can do is to compute the covariance of W_ti and
W_tj to be sqrt(t_i * t_j).
Still, what i can't do, is obtaining the variance of the
linear combination, even if i have the covariance matrix...
Any suggestions?
|
Sure. Use the simple formula from Introduction to Probability 101: if
(v_{ij}) is the variance/covariance matrix (v_{ii} = var(W_i) and
v_{ij} = Cov(W+i,W_j) for i =/= j) then for constants a_i we have
Var (sum a_i W_i) = sum_{i=1..n} a_i^2 v_{ii} + 2 sum_{1<=i < j <=n}
a_i a_j v_{ij}.
This is true for any multivariate distribution, not just for Wiener
processes; it results from the fact that (sum_i a_i W_i)^2 = sum_i
a_i^2 W_i^2 + 2 sum_{i < j} a_i a_j w_i W_j , together with E(W_i^2) Var(W_i) + (E W_i)^2 and E(W_i W_j) = Cov(W_i ,W_j) + E(W_i W_j) for i
< j.
R.G. Vickson
| Quote: |
--
Vänligen
Konrad
---------------------------------------------------
Sleep - thing used by ineffective people
as a substitute for coffee
Ambition - a poor excuse for not having
enough sence to be lazy
--------------------------------------------------- |
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Konrad Viltersten
science forum addict
Joined: 27 Jun 2005
Posts: 79
|
| Posted: Fri Jun 30, 2006 5:06 am Post subject:
Re: Variance for a linear combination of three (normal) random variables
|
|
|
| Quote: | Suppose there's a sum as follows:
X := a * W_1 + b * W_2 + c * W_3
How can we compute the variance of X? Well, it's rather
uncomplicated if one can be assured of independency.
Now, if one is assured that they _ARE_ dependent (as
for instance, being three states of a Wiener process at
three consecutive times), is there anything else to do but
sit down and cry loudly like a confused child?
Of, course, it's obvious that X ~ N(0,s^2) and that every
W_i ~ N(0,i). But how can we use it?!
Well, if the W_i are not independent, then you need to
know there correlation/covariance. Specifically, you
need to calculate E{X^2}, and that involves terms like
E{W_1*W_2}, etc.
Right after i sent my reply, i realized that i was unclear.
What i can do is to compute the covariance of W_ti and
W_tj to be sqrt(t_i * t_j).
Still, what i can't do, is obtaining the variance of the
linear combination, even if i have the covariance matrix...
Any suggestions?
Sure. Use the simple formula from Introduction to Probability 101: if
(v_{ij}) is the variance/covariance matrix (v_{ii} = var(W_i) and
v_{ij} = Cov(W+i,W_j) for i =/= j) then for constants a_i we have
Var (sum a_i W_i) = sum_{i=1..n} a_i^2 v_{ii} + 2 sum_{1<=i < j <=n}
a_i a_j v_{ij}.
This is true for any multivariate distribution, not just for Wiener
processes; it results from the fact that (sum_i a_i W_i)^2 = sum_i
a_i^2 W_i^2 + 2 sum_{i < j} a_i a_j w_i W_j , together with E(W_i^2) =
Var(W_i) + (E W_i)^2 and E(W_i W_j) = Cov(W_i ,W_j) + E(W_i W_j) for i
j.
|
Thanks. It helped. Thanks to everybody else too.
--
Vänligen
Konrad
---------------------------------------------------
Sleep - thing used by ineffective people
as a substitute for coffee
Ambition - a poor excuse for not having
enough sence to be lazy
--------------------------------------------------- |
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