How to prove that exponential of an expected value of a variable is less than the expected value of the...
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I am trying to prove the following:
$e^{E(x)} le E(e^x)$ for a discrete random variable x.
I am stuck on how to proceed. None of the usual rules for expected value seem to apply for something like $f(E(x))$. Can I some help? Thanks!!
expected-value
asked Nov 24 '18 at 3:10
William DengWilliam Deng
32
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closed as off-topic by max_zorn, NCh, amWhy, Shailesh, John B Dec 1 '18 at 0:44
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." – max_zorn, NCh, amWhy, Shailesh, John B
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I am trying to prove the following:
$e^{E(x)} le E(e^x)$ for a discrete random variable x.
I am stuck on how to proceed. None of the usual rules for expected value seem to apply for something like $f(E(x))$. Can I some help? Thanks!!
expected-value
asked Nov 24 '18 at 3:10
William DengWilliam Deng
32
$endgroup$
closed as off-topic by max_zorn, NCh, amWhy, Shailesh, John B Dec 1 '18 at 0:44
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." – max_zorn, NCh, amWhy, Shailesh, John B
If this question can be reworded to fit the rules in the help center, please edit the question.
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Have you ever heard of Jensen's inequality( en.wikipedia.org/wiki/Jensen%27s_inequality)?
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– Seewoo Lee
Nov 24 '18 at 3:16
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THANKS!! THIS IS JUST WHAT I NEEDED!
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– William Deng
Nov 24 '18 at 3:21
add a comment |
$begingroup$
I am trying to prove the following:
$e^{E(x)} le E(e^x)$ for a discrete random variable x.
I am stuck on how to proceed. None of the usual rules for expected value seem to apply for something like $f(E(x))$. Can I some help? Thanks!!
expected-value
asked Nov 24 '18 at 3:10
William DengWilliam Deng
32
$endgroup$
I am trying to prove the following:
$e^{E(x)} le E(e^x)$ for a discrete random variable x.
I am stuck on how to proceed. None of the usual rules for expected value seem to apply for something like $f(E(x))$. Can I some help? Thanks!!
expected-value
expected-value
asked Nov 24 '18 at 3:10
William DengWilliam Deng
32
asked Nov 24 '18 at 3:10
William DengWilliam Deng
32
asked Nov 24 '18 at 3:10
William DengWilliam Deng
32
asked Nov 24 '18 at 3:10
William DengWilliam Deng
32
asked Nov 24 '18 at 3:10
William DengWilliam Deng
32
32
closed as off-topic by max_zorn, NCh, amWhy, Shailesh, John B Dec 1 '18 at 0:44
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." – max_zorn, NCh, amWhy, Shailesh, John B
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by max_zorn, NCh, amWhy, Shailesh, John B Dec 1 '18 at 0:44
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." – max_zorn, NCh, amWhy, Shailesh, John B
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Have you ever heard of Jensen's inequality( en.wikipedia.org/wiki/Jensen%27s_inequality)?
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– Seewoo Lee
Nov 24 '18 at 3:16
$begingroup$
THANKS!! THIS IS JUST WHAT I NEEDED!
$endgroup$
– William Deng
Nov 24 '18 at 3:21
add a comment |
$begingroup$
Have you ever heard of Jensen's inequality( en.wikipedia.org/wiki/Jensen%27s_inequality)?
$endgroup$
– Seewoo Lee
Nov 24 '18 at 3:16
$begingroup$
THANKS!! THIS IS JUST WHAT I NEEDED!
$endgroup$
– William Deng
Nov 24 '18 at 3:21
$begingroup$
Have you ever heard of Jensen's inequality( en.wikipedia.org/wiki/Jensen%27s_inequality)?
$endgroup$
– Seewoo Lee
Nov 24 '18 at 3:16
$begingroup$
Have you ever heard of Jensen's inequality( en.wikipedia.org/wiki/Jensen%27s_inequality)?
$endgroup$
– Seewoo Lee
Nov 24 '18 at 3:16
$begingroup$
THANKS!! THIS IS JUST WHAT I NEEDED!
$endgroup$
– William Deng
Nov 24 '18 at 3:21
$begingroup$
THANKS!! THIS IS JUST WHAT I NEEDED!
$endgroup$
– William Deng
Nov 24 '18 at 3:21
add a comment |
1 Answer
1
active
oldest
votes
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Exponent is a convex function. Use, as suggested in the comments, Jensen's inequality
answered Nov 24 '18 at 3:18
MakinaMakina
1,158316
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$begingroup$
Thanks! Just what I needed.
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– William Deng
Nov 24 '18 at 3:29
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Exponent is a convex function. Use, as suggested in the comments, Jensen's inequality
answered Nov 24 '18 at 3:18
MakinaMakina
1,158316
$endgroup$
$begingroup$
Thanks! Just what I needed.
$endgroup$
– William Deng
Nov 24 '18 at 3:29
add a comment |
$begingroup$
Exponent is a convex function. Use, as suggested in the comments, Jensen's inequality
answered Nov 24 '18 at 3:18
MakinaMakina
1,158316
$endgroup$
$begingroup$
Thanks! Just what I needed.
$endgroup$
– William Deng
Nov 24 '18 at 3:29
add a comment |
$begingroup$
Exponent is a convex function. Use, as suggested in the comments, Jensen's inequality
answered Nov 24 '18 at 3:18
MakinaMakina
1,158316
$endgroup$
Exponent is a convex function. Use, as suggested in the comments, Jensen's inequality
answered Nov 24 '18 at 3:18
MakinaMakina
1,158316
answered Nov 24 '18 at 3:18
MakinaMakina
1,158316
answered Nov 24 '18 at 3:18
MakinaMakina
1,158316
answered Nov 24 '18 at 3:18
MakinaMakina
1,158316
1,158316
$begingroup$
Thanks! Just what I needed.
$endgroup$
– William Deng
Nov 24 '18 at 3:29
add a comment |
$begingroup$
Thanks! Just what I needed.
$endgroup$
– William Deng
Nov 24 '18 at 3:29
$begingroup$
Thanks! Just what I needed.
$endgroup$
– William Deng
Nov 24 '18 at 3:29
$begingroup$
Thanks! Just what I needed.
$endgroup$
– William Deng
Nov 24 '18 at 3:29
add a comment |
$begingroup$
Have you ever heard of Jensen's inequality( en.wikipedia.org/wiki/Jensen%27s_inequality)?
$endgroup$
– Seewoo Lee
Nov 24 '18 at 3:16
$begingroup$
THANKS!! THIS IS JUST WHAT I NEEDED!
$endgroup$
– William Deng
Nov 24 '18 at 3:21