Expected value of the product of dependent Normal random variables
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I have 3 independent Normal Random Variables: $A$, $B$ and $C$, each with mean=$0$ and Variance $1$.
Then I have $X=3A+5B$ and $Y=A-C$... because both of them are functions of $A$, we know they are not independent.
I have calculated the means and variance of both $X$ and $Y$, but now I need to calculate the $E[XY]$ and not sure how to approach the problem.
I'm inclined to use the law of iterated expectations ($E[XY] = E[E[XYmid A]]$)... since if you are given a value for $A$, then both $X$ and $Y$ become functions of $B$ and $C$ and therefore independent. Is this approach sound? Any other tips on how to calculate this Expectation? Thank you!
probability-distributions normal-distribution expected-value
edited Nov 17 at 18:45
StubbornAtom
4,89911137
asked Nov 17 at 18:29
Javier Cordero
83
add a comment |
up vote
1
down vote
favorite
I have 3 independent Normal Random Variables: $A$, $B$ and $C$, each with mean=$0$ and Variance $1$.
Then I have $X=3A+5B$ and $Y=A-C$... because both of them are functions of $A$, we know they are not independent.
I have calculated the means and variance of both $X$ and $Y$, but now I need to calculate the $E[XY]$ and not sure how to approach the problem.
I'm inclined to use the law of iterated expectations ($E[XY] = E[E[XYmid A]]$)... since if you are given a value for $A$, then both $X$ and $Y$ become functions of $B$ and $C$ and therefore independent. Is this approach sound? Any other tips on how to calculate this Expectation? Thank you!
probability-distributions normal-distribution expected-value
edited Nov 17 at 18:45
StubbornAtom
4,89911137
asked Nov 17 at 18:29
Javier Cordero
83
Actually you can just multiply $X$ and $Y$ and calculate the expectation using linearity and independence of $A,B,C$.
– StubbornAtom
Nov 17 at 18:42
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have 3 independent Normal Random Variables: $A$, $B$ and $C$, each with mean=$0$ and Variance $1$.
Then I have $X=3A+5B$ and $Y=A-C$... because both of them are functions of $A$, we know they are not independent.
I have calculated the means and variance of both $X$ and $Y$, but now I need to calculate the $E[XY]$ and not sure how to approach the problem.
I'm inclined to use the law of iterated expectations ($E[XY] = E[E[XYmid A]]$)... since if you are given a value for $A$, then both $X$ and $Y$ become functions of $B$ and $C$ and therefore independent. Is this approach sound? Any other tips on how to calculate this Expectation? Thank you!
probability-distributions normal-distribution expected-value
edited Nov 17 at 18:45
StubbornAtom
4,89911137
asked Nov 17 at 18:29
Javier Cordero
83
I have 3 independent Normal Random Variables: $A$, $B$ and $C$, each with mean=$0$ and Variance $1$.
Then I have $X=3A+5B$ and $Y=A-C$... because both of them are functions of $A$, we know they are not independent.
I have calculated the means and variance of both $X$ and $Y$, but now I need to calculate the $E[XY]$ and not sure how to approach the problem.
I'm inclined to use the law of iterated expectations ($E[XY] = E[E[XYmid A]]$)... since if you are given a value for $A$, then both $X$ and $Y$ become functions of $B$ and $C$ and therefore independent. Is this approach sound? Any other tips on how to calculate this Expectation? Thank you!
probability-distributions normal-distribution expected-value
probability-distributions normal-distribution expected-value
edited Nov 17 at 18:45
StubbornAtom
4,89911137
asked Nov 17 at 18:29
Javier Cordero
83
edited Nov 17 at 18:45
StubbornAtom
4,89911137
asked Nov 17 at 18:29
Javier Cordero
83
edited Nov 17 at 18:45
StubbornAtom
4,89911137
edited Nov 17 at 18:45
StubbornAtom
4,89911137
edited Nov 17 at 18:45
StubbornAtom
4,89911137
4,89911137
asked Nov 17 at 18:29
Javier Cordero
83
asked Nov 17 at 18:29
Javier Cordero
83
asked Nov 17 at 18:29
Javier Cordero
83
83
Actually you can just multiply $X$ and $Y$ and calculate the expectation using linearity and independence of $A,B,C$.
– StubbornAtom
Nov 17 at 18:42
add a comment |
Actually you can just multiply $X$ and $Y$ and calculate the expectation using linearity and independence of $A,B,C$.
– StubbornAtom
Nov 17 at 18:42
Actually you can just multiply $X$ and $Y$ and calculate the expectation using linearity and independence of $A,B,C$.
– StubbornAtom
Nov 17 at 18:42
Actually you can just multiply $X$ and $Y$ and calculate the expectation using linearity and independence of $A,B,C$.
– StubbornAtom
Nov 17 at 18:42
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
accepted
Hint: $XY$ is a product of $X$ and $Y$. You already have the definition of $X$ and $Y$ through i.i.d variables $A, B$ and $C$. Substitute, use linearity and remember, what is the definition of variance and how independent condition influences the calculation of expectation
answered Nov 17 at 18:35
Makina
1,006113
You Rock. Thanks for the tip, including the variance tip to get the result of the E of the square of one of the RV. BTW - I got the same answer using the iterated expectations approach I described in my question, but it was definitely easier using this way, and it's always good to get the same answers via 2 different methods! :)
– Javier Cordero
Nov 17 at 19:41
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
Hint: $XY$ is a product of $X$ and $Y$. You already have the definition of $X$ and $Y$ through i.i.d variables $A, B$ and $C$. Substitute, use linearity and remember, what is the definition of variance and how independent condition influences the calculation of expectation
answered Nov 17 at 18:35
Makina
1,006113
You Rock. Thanks for the tip, including the variance tip to get the result of the E of the square of one of the RV. BTW - I got the same answer using the iterated expectations approach I described in my question, but it was definitely easier using this way, and it's always good to get the same answers via 2 different methods! :)
– Javier Cordero
Nov 17 at 19:41
add a comment |
up vote
0
down vote
accepted
Hint: $XY$ is a product of $X$ and $Y$. You already have the definition of $X$ and $Y$ through i.i.d variables $A, B$ and $C$. Substitute, use linearity and remember, what is the definition of variance and how independent condition influences the calculation of expectation
answered Nov 17 at 18:35
Makina
1,006113
You Rock. Thanks for the tip, including the variance tip to get the result of the E of the square of one of the RV. BTW - I got the same answer using the iterated expectations approach I described in my question, but it was definitely easier using this way, and it's always good to get the same answers via 2 different methods! :)
– Javier Cordero
Nov 17 at 19:41
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
Hint: $XY$ is a product of $X$ and $Y$. You already have the definition of $X$ and $Y$ through i.i.d variables $A, B$ and $C$. Substitute, use linearity and remember, what is the definition of variance and how independent condition influences the calculation of expectation
answered Nov 17 at 18:35
Makina
1,006113
Hint: $XY$ is a product of $X$ and $Y$. You already have the definition of $X$ and $Y$ through i.i.d variables $A, B$ and $C$. Substitute, use linearity and remember, what is the definition of variance and how independent condition influences the calculation of expectation
answered Nov 17 at 18:35
Makina
1,006113
answered Nov 17 at 18:35
Makina
1,006113
answered Nov 17 at 18:35
Makina
1,006113
answered Nov 17 at 18:35
Makina
1,006113
1,006113
You Rock. Thanks for the tip, including the variance tip to get the result of the E of the square of one of the RV. BTW - I got the same answer using the iterated expectations approach I described in my question, but it was definitely easier using this way, and it's always good to get the same answers via 2 different methods! :)
– Javier Cordero
Nov 17 at 19:41
add a comment |
You Rock. Thanks for the tip, including the variance tip to get the result of the E of the square of one of the RV. BTW - I got the same answer using the iterated expectations approach I described in my question, but it was definitely easier using this way, and it's always good to get the same answers via 2 different methods! :)
– Javier Cordero
Nov 17 at 19:41
You Rock. Thanks for the tip, including the variance tip to get the result of the E of the square of one of the RV. BTW - I got the same answer using the iterated expectations approach I described in my question, but it was definitely easier using this way, and it's always good to get the same answers via 2 different methods! :)
– Javier Cordero
Nov 17 at 19:41
You Rock. Thanks for the tip, including the variance tip to get the result of the E of the square of one of the RV. BTW - I got the same answer using the iterated expectations approach I described in my question, but it was definitely easier using this way, and it's always good to get the same answers via 2 different methods! :)
– Javier Cordero
Nov 17 at 19:41
add a comment |
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Actually you can just multiply $X$ and $Y$ and calculate the expectation using linearity and independence of $A,B,C$.
– StubbornAtom
Nov 17 at 18:42