Approximate values of $operatornameE[sqrt X]$ and $operatornameVar[sqrt X]$ for $X$ Poisson distributed with parameter $lambdatoinfty$ [closed]
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Assume that $X$ has a Poisson distribution with rate parameter $lambda$. If $Y = sqrt X$, using moment-generating functions or otherwise, show that $$operatornameE[Y] approx sqrtlambda - frac 1 8 sqrtlambda$$ and $$operatornameVar[Y] approx frac14$$
A suggestion is to use MGFs but I've got no idea how to go from there as I keep getting jammed.
taylor-expansion poisson-distribution moment-generating-functions
edited Sep 7 at 6:53
Did
243k23209444
asked Sep 7 at 5:03
Jim
63
closed as off-topic by heropup, Jendrik Stelzner, José Carlos Santos, Adrian Keister, amWhy Sep 8 at 0:33
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." Рheropup, Jendrik Stelzner, Jos̩ Carlos Santos, Adrian Keister, amWhy
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Assume that $X$ has a Poisson distribution with rate parameter $lambda$. If $Y = sqrt X$, using moment-generating functions or otherwise, show that $$operatornameE[Y] approx sqrtlambda - frac 1 8 sqrtlambda$$ and $$operatornameVar[Y] approx frac14$$
A suggestion is to use MGFs but I've got no idea how to go from there as I keep getting jammed.
taylor-expansion poisson-distribution moment-generating-functions
edited Sep 7 at 6:53
Did
243k23209444
asked Sep 7 at 5:03
Jim
63
closed as off-topic by heropup, Jendrik Stelzner, José Carlos Santos, Adrian Keister, amWhy Sep 8 at 0:33
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." Рheropup, Jendrik Stelzner, Jos̩ Carlos Santos, Adrian Keister, amWhy
In what sense are you using "show" here? Since you don't specify the order or accuracy of the approximation, we can provide heuristic arguments for it, but there's nothing to "show" in a formal sense.
– joriki
Sep 7 at 5:22
I don't have a clue sorry, I didn't write the question but you're trying to show that E[X] and Var[X] can be approximated to those values
– Jim
Sep 7 at 5:40
If you didn't write the question, you should clearly mark it as a quote (e.g. using markdown's blockquote feature).
– joriki
Sep 7 at 6:01
Fixed that up sorry
– Jim
Sep 7 at 6:02
1
No worries thanks for that, I probably should have read that up before I posted
– Jim
Sep 7 at 6:12
 |Â
show 3 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Assume that $X$ has a Poisson distribution with rate parameter $lambda$. If $Y = sqrt X$, using moment-generating functions or otherwise, show that $$operatornameE[Y] approx sqrtlambda - frac 1 8 sqrtlambda$$ and $$operatornameVar[Y] approx frac14$$
A suggestion is to use MGFs but I've got no idea how to go from there as I keep getting jammed.
taylor-expansion poisson-distribution moment-generating-functions
edited Sep 7 at 6:53
Did
243k23209444
asked Sep 7 at 5:03
Jim
63
Assume that $X$ has a Poisson distribution with rate parameter $lambda$. If $Y = sqrt X$, using moment-generating functions or otherwise, show that $$operatornameE[Y] approx sqrtlambda - frac 1 8 sqrtlambda$$ and $$operatornameVar[Y] approx frac14$$
A suggestion is to use MGFs but I've got no idea how to go from there as I keep getting jammed.
taylor-expansion poisson-distribution moment-generating-functions
taylor-expansion poisson-distribution moment-generating-functions
edited Sep 7 at 6:53
Did
243k23209444
asked Sep 7 at 5:03
Jim
63
edited Sep 7 at 6:53
Did
243k23209444
asked Sep 7 at 5:03
Jim
63
edited Sep 7 at 6:53
Did
243k23209444
edited Sep 7 at 6:53
Did
243k23209444
edited Sep 7 at 6:53
Did
243k23209444
243k23209444
asked Sep 7 at 5:03
Jim
63
asked Sep 7 at 5:03
Jim
63
asked Sep 7 at 5:03
Jim
63
63
closed as off-topic by heropup, Jendrik Stelzner, José Carlos Santos, Adrian Keister, amWhy Sep 8 at 0:33
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." Рheropup, Jendrik Stelzner, Jos̩ Carlos Santos, Adrian Keister, amWhy
closed as off-topic by heropup, Jendrik Stelzner, José Carlos Santos, Adrian Keister, amWhy Sep 8 at 0:33
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." Рheropup, Jendrik Stelzner, Jos̩ Carlos Santos, Adrian Keister, amWhy
In what sense are you using "show" here? Since you don't specify the order or accuracy of the approximation, we can provide heuristic arguments for it, but there's nothing to "show" in a formal sense.
– joriki
Sep 7 at 5:22
I don't have a clue sorry, I didn't write the question but you're trying to show that E[X] and Var[X] can be approximated to those values
– Jim
Sep 7 at 5:40
If you didn't write the question, you should clearly mark it as a quote (e.g. using markdown's blockquote feature).
– joriki
Sep 7 at 6:01
Fixed that up sorry
– Jim
Sep 7 at 6:02
1
No worries thanks for that, I probably should have read that up before I posted
– Jim
Sep 7 at 6:12
 |Â
show 3 more comments
In what sense are you using "show" here? Since you don't specify the order or accuracy of the approximation, we can provide heuristic arguments for it, but there's nothing to "show" in a formal sense.
– joriki
Sep 7 at 5:22
I don't have a clue sorry, I didn't write the question but you're trying to show that E[X] and Var[X] can be approximated to those values
– Jim
Sep 7 at 5:40
If you didn't write the question, you should clearly mark it as a quote (e.g. using markdown's blockquote feature).
– joriki
Sep 7 at 6:01
Fixed that up sorry
– Jim
Sep 7 at 6:02
1
No worries thanks for that, I probably should have read that up before I posted
– Jim
Sep 7 at 6:12
In what sense are you using "show" here? Since you don't specify the order or accuracy of the approximation, we can provide heuristic arguments for it, but there's nothing to "show" in a formal sense.
– joriki
Sep 7 at 5:22
In what sense are you using "show" here? Since you don't specify the order or accuracy of the approximation, we can provide heuristic arguments for it, but there's nothing to "show" in a formal sense.
– joriki
Sep 7 at 5:22
I don't have a clue sorry, I didn't write the question but you're trying to show that E[X] and Var[X] can be approximated to those values
– Jim
Sep 7 at 5:40
I don't have a clue sorry, I didn't write the question but you're trying to show that E[X] and Var[X] can be approximated to those values
– Jim
Sep 7 at 5:40
If you didn't write the question, you should clearly mark it as a quote (e.g. using markdown's blockquote feature).
– joriki
Sep 7 at 6:01
If you didn't write the question, you should clearly mark it as a quote (e.g. using markdown's blockquote feature).
– joriki
Sep 7 at 6:01
Fixed that up sorry
– Jim
Sep 7 at 6:02
Fixed that up sorry
– Jim
Sep 7 at 6:02
1
1
No worries thanks for that, I probably should have read that up before I posted
– Jim
Sep 7 at 6:12
No worries thanks for that, I probably should have read that up before I posted
– Jim
Sep 7 at 6:12
 |Â
show 3 more comments
1 Answer
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0
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This is an exercise in the delta method. Since $dY/dX=1/(2sqrtX)$, $operatornameVarYapproxfraclambda(2sqrtlambda)^2=frac14$. Then $E(Y)=sqrtlambdasqrt1-frac14lambda$ gives the desired result.
answered Sep 7 at 5:36
J.G.
14.8k11626
You're a legend cheers J.G. what formula did you use to find E[X]?
– Jim
Sep 7 at 5:45
@JamesSilber The Poisson distribution has mean & variance both equal to $lambda$.
– J.G.
Sep 7 at 5:48
I know that I meant E[Y] sorry, because why do you multiply sqrt(lambda) by sqrt(1-1/4*lambda)
– Jim
Sep 7 at 5:49
Well, $E^2Y=E(Y^2)-operatornameVarYapproxlambda-frac14$, which we square-root by first taking out the factor of $lambda$.
– J.G.
Sep 7 at 5:51
Got it thanks you're a life saver
– Jim
Sep 7 at 5:56
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
This is an exercise in the delta method. Since $dY/dX=1/(2sqrtX)$, $operatornameVarYapproxfraclambda(2sqrtlambda)^2=frac14$. Then $E(Y)=sqrtlambdasqrt1-frac14lambda$ gives the desired result.
answered Sep 7 at 5:36
J.G.
14.8k11626
You're a legend cheers J.G. what formula did you use to find E[X]?
– Jim
Sep 7 at 5:45
@JamesSilber The Poisson distribution has mean & variance both equal to $lambda$.
– J.G.
Sep 7 at 5:48
I know that I meant E[Y] sorry, because why do you multiply sqrt(lambda) by sqrt(1-1/4*lambda)
– Jim
Sep 7 at 5:49
Well, $E^2Y=E(Y^2)-operatornameVarYapproxlambda-frac14$, which we square-root by first taking out the factor of $lambda$.
– J.G.
Sep 7 at 5:51
Got it thanks you're a life saver
– Jim
Sep 7 at 5:56
add a comment |Â
up vote
0
down vote
accepted
This is an exercise in the delta method. Since $dY/dX=1/(2sqrtX)$, $operatornameVarYapproxfraclambda(2sqrtlambda)^2=frac14$. Then $E(Y)=sqrtlambdasqrt1-frac14lambda$ gives the desired result.
answered Sep 7 at 5:36
J.G.
14.8k11626
You're a legend cheers J.G. what formula did you use to find E[X]?
– Jim
Sep 7 at 5:45
@JamesSilber The Poisson distribution has mean & variance both equal to $lambda$.
– J.G.
Sep 7 at 5:48
I know that I meant E[Y] sorry, because why do you multiply sqrt(lambda) by sqrt(1-1/4*lambda)
– Jim
Sep 7 at 5:49
Well, $E^2Y=E(Y^2)-operatornameVarYapproxlambda-frac14$, which we square-root by first taking out the factor of $lambda$.
– J.G.
Sep 7 at 5:51
Got it thanks you're a life saver
– Jim
Sep 7 at 5:56
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
This is an exercise in the delta method. Since $dY/dX=1/(2sqrtX)$, $operatornameVarYapproxfraclambda(2sqrtlambda)^2=frac14$. Then $E(Y)=sqrtlambdasqrt1-frac14lambda$ gives the desired result.
answered Sep 7 at 5:36
J.G.
14.8k11626
This is an exercise in the delta method. Since $dY/dX=1/(2sqrtX)$, $operatornameVarYapproxfraclambda(2sqrtlambda)^2=frac14$. Then $E(Y)=sqrtlambdasqrt1-frac14lambda$ gives the desired result.
answered Sep 7 at 5:36
J.G.
14.8k11626
answered Sep 7 at 5:36
J.G.
14.8k11626
answered Sep 7 at 5:36
J.G.
14.8k11626
answered Sep 7 at 5:36
J.G.
14.8k11626
14.8k11626
You're a legend cheers J.G. what formula did you use to find E[X]?
– Jim
Sep 7 at 5:45
@JamesSilber The Poisson distribution has mean & variance both equal to $lambda$.
– J.G.
Sep 7 at 5:48
I know that I meant E[Y] sorry, because why do you multiply sqrt(lambda) by sqrt(1-1/4*lambda)
– Jim
Sep 7 at 5:49
Well, $E^2Y=E(Y^2)-operatornameVarYapproxlambda-frac14$, which we square-root by first taking out the factor of $lambda$.
– J.G.
Sep 7 at 5:51
Got it thanks you're a life saver
– Jim
Sep 7 at 5:56
add a comment |Â
You're a legend cheers J.G. what formula did you use to find E[X]?
– Jim
Sep 7 at 5:45
@JamesSilber The Poisson distribution has mean & variance both equal to $lambda$.
– J.G.
Sep 7 at 5:48
I know that I meant E[Y] sorry, because why do you multiply sqrt(lambda) by sqrt(1-1/4*lambda)
– Jim
Sep 7 at 5:49
Well, $E^2Y=E(Y^2)-operatornameVarYapproxlambda-frac14$, which we square-root by first taking out the factor of $lambda$.
– J.G.
Sep 7 at 5:51
Got it thanks you're a life saver
– Jim
Sep 7 at 5:56
You're a legend cheers J.G. what formula did you use to find E[X]?
– Jim
Sep 7 at 5:45
You're a legend cheers J.G. what formula did you use to find E[X]?
– Jim
Sep 7 at 5:45
@JamesSilber The Poisson distribution has mean & variance both equal to $lambda$.
– J.G.
Sep 7 at 5:48
@JamesSilber The Poisson distribution has mean & variance both equal to $lambda$.
– J.G.
Sep 7 at 5:48
I know that I meant E[Y] sorry, because why do you multiply sqrt(lambda) by sqrt(1-1/4*lambda)
– Jim
Sep 7 at 5:49
I know that I meant E[Y] sorry, because why do you multiply sqrt(lambda) by sqrt(1-1/4*lambda)
– Jim
Sep 7 at 5:49
Well, $E^2Y=E(Y^2)-operatornameVarYapproxlambda-frac14$, which we square-root by first taking out the factor of $lambda$.
– J.G.
Sep 7 at 5:51
Well, $E^2Y=E(Y^2)-operatornameVarYapproxlambda-frac14$, which we square-root by first taking out the factor of $lambda$.
– J.G.
Sep 7 at 5:51
Got it thanks you're a life saver
– Jim
Sep 7 at 5:56
Got it thanks you're a life saver
– Jim
Sep 7 at 5:56
add a comment |Â
In what sense are you using "show" here? Since you don't specify the order or accuracy of the approximation, we can provide heuristic arguments for it, but there's nothing to "show" in a formal sense.
– joriki
Sep 7 at 5:22
I don't have a clue sorry, I didn't write the question but you're trying to show that E[X] and Var[X] can be approximated to those values
– Jim
Sep 7 at 5:40
If you didn't write the question, you should clearly mark it as a quote (e.g. using markdown's blockquote feature).
– joriki
Sep 7 at 6:01
Fixed that up sorry
– Jim
Sep 7 at 6:02
1
No worries thanks for that, I probably should have read that up before I posted
– Jim
Sep 7 at 6:12