Optimizing over an Integral
$begingroup$
I have to solve the following optimization problem:
$max_tau int_underline{epsilon}^bar{epsilon} tau(1-tau)^epsilon depsilon$
How does one solve these with the integral. I think I may be able to do it point-wise but I am not sure.
integration optimization maxima-minima
asked Jan 14 at 15:37
user52932user52932
93114
$endgroup$
add a comment |
$begingroup$
I have to solve the following optimization problem:
$max_tau int_underline{epsilon}^bar{epsilon} tau(1-tau)^epsilon depsilon$
How does one solve these with the integral. I think I may be able to do it point-wise but I am not sure.
integration optimization maxima-minima
asked Jan 14 at 15:37
user52932user52932
93114
$endgroup$
$begingroup$
Is $tau$ a number or a function?
$endgroup$
– Botond
Jan 14 at 15:44
$begingroup$
@Botond its a number.
$endgroup$
– user52932
Jan 14 at 21:40
$begingroup$
Then @lightxbulb's answer is a good way to do it.
$endgroup$
– Botond
Jan 14 at 21:44
$begingroup$
@Botond The issue I am running into is that when I do that, I am unable to express tau as a function of just b and a. That is I am unable to get a closed form expression. What does this mean?
$endgroup$
– user52932
Jan 14 at 21:55
$begingroup$
@user52932 You'll have to analyze a number of cases for different intervals of $a$ and $b$.
$endgroup$
– lightxbulb
Jan 14 at 22:09
add a comment |
$begingroup$
I have to solve the following optimization problem:
$max_tau int_underline{epsilon}^bar{epsilon} tau(1-tau)^epsilon depsilon$
How does one solve these with the integral. I think I may be able to do it point-wise but I am not sure.
integration optimization maxima-minima
asked Jan 14 at 15:37
user52932user52932
93114
$endgroup$
I have to solve the following optimization problem:
$max_tau int_underline{epsilon}^bar{epsilon} tau(1-tau)^epsilon depsilon$
How does one solve these with the integral. I think I may be able to do it point-wise but I am not sure.
integration optimization maxima-minima
integration optimization maxima-minima
asked Jan 14 at 15:37
user52932user52932
93114
asked Jan 14 at 15:37
user52932user52932
93114
asked Jan 14 at 15:37
user52932user52932
93114
asked Jan 14 at 15:37
user52932user52932
93114
asked Jan 14 at 15:37
user52932user52932
93114
93114
$begingroup$
Is $tau$ a number or a function?
$endgroup$
– Botond
Jan 14 at 15:44
$begingroup$
@Botond its a number.
$endgroup$
– user52932
Jan 14 at 21:40
$begingroup$
Then @lightxbulb's answer is a good way to do it.
$endgroup$
– Botond
Jan 14 at 21:44
$begingroup$
@Botond The issue I am running into is that when I do that, I am unable to express tau as a function of just b and a. That is I am unable to get a closed form expression. What does this mean?
$endgroup$
– user52932
Jan 14 at 21:55
$begingroup$
@user52932 You'll have to analyze a number of cases for different intervals of $a$ and $b$.
$endgroup$
– lightxbulb
Jan 14 at 22:09
add a comment |
$begingroup$
Is $tau$ a number or a function?
$endgroup$
– Botond
Jan 14 at 15:44
$begingroup$
@Botond its a number.
$endgroup$
– user52932
Jan 14 at 21:40
$begingroup$
Then @lightxbulb's answer is a good way to do it.
$endgroup$
– Botond
Jan 14 at 21:44
$begingroup$
@Botond The issue I am running into is that when I do that, I am unable to express tau as a function of just b and a. That is I am unable to get a closed form expression. What does this mean?
$endgroup$
– user52932
Jan 14 at 21:55
$begingroup$
@user52932 You'll have to analyze a number of cases for different intervals of $a$ and $b$.
$endgroup$
– lightxbulb
Jan 14 at 22:09
$begingroup$
Is $tau$ a number or a function?
$endgroup$
– Botond
Jan 14 at 15:44
$begingroup$
Is $tau$ a number or a function?
$endgroup$
– Botond
Jan 14 at 15:44
$begingroup$
@Botond its a number.
$endgroup$
– user52932
Jan 14 at 21:40
$begingroup$
@Botond its a number.
$endgroup$
– user52932
Jan 14 at 21:40
$begingroup$
Then @lightxbulb's answer is a good way to do it.
$endgroup$
– Botond
Jan 14 at 21:44
$begingroup$
Then @lightxbulb's answer is a good way to do it.
$endgroup$
– Botond
Jan 14 at 21:44
$begingroup$
@Botond The issue I am running into is that when I do that, I am unable to express tau as a function of just b and a. That is I am unable to get a closed form expression. What does this mean?
$endgroup$
– user52932
Jan 14 at 21:55
$begingroup$
@Botond The issue I am running into is that when I do that, I am unable to express tau as a function of just b and a. That is I am unable to get a closed form expression. What does this mean?
$endgroup$
– user52932
Jan 14 at 21:55
$begingroup$
@user52932 You'll have to analyze a number of cases for different intervals of $a$ and $b$.
$endgroup$
– lightxbulb
Jan 14 at 22:09
$begingroup$
@user52932 You'll have to analyze a number of cases for different intervals of $a$ and $b$.
$endgroup$
– lightxbulb
Jan 14 at 22:09
add a comment |
1 Answer
1
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$begingroup$
Try integrating:
$$tauint_{a}^{b}{(1-tau)^xdx} =left. frac{tau(1-tau)^x}{log(1-tau)} right|_a^b$$
Then take the derivative with respect to $tau$ to find the extrema.
edited Jan 14 at 15:53
MPW
29.9k12056
answered Jan 14 at 15:47
lightxbulblightxbulb
75619
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add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Try integrating:
$$tauint_{a}^{b}{(1-tau)^xdx} =left. frac{tau(1-tau)^x}{log(1-tau)} right|_a^b$$
Then take the derivative with respect to $tau$ to find the extrema.
edited Jan 14 at 15:53
MPW
29.9k12056
answered Jan 14 at 15:47
lightxbulblightxbulb
75619
$endgroup$
add a comment |
$begingroup$
Try integrating:
$$tauint_{a}^{b}{(1-tau)^xdx} =left. frac{tau(1-tau)^x}{log(1-tau)} right|_a^b$$
Then take the derivative with respect to $tau$ to find the extrema.
edited Jan 14 at 15:53
MPW
29.9k12056
answered Jan 14 at 15:47
lightxbulblightxbulb
75619
$endgroup$
add a comment |
$begingroup$
Try integrating:
$$tauint_{a}^{b}{(1-tau)^xdx} =left. frac{tau(1-tau)^x}{log(1-tau)} right|_a^b$$
Then take the derivative with respect to $tau$ to find the extrema.
edited Jan 14 at 15:53
MPW
29.9k12056
answered Jan 14 at 15:47
lightxbulblightxbulb
75619
$endgroup$
Try integrating:
$$tauint_{a}^{b}{(1-tau)^xdx} =left. frac{tau(1-tau)^x}{log(1-tau)} right|_a^b$$
Then take the derivative with respect to $tau$ to find the extrema.
edited Jan 14 at 15:53
MPW
29.9k12056
answered Jan 14 at 15:47
lightxbulblightxbulb
75619
edited Jan 14 at 15:53
MPW
29.9k12056
edited Jan 14 at 15:53
MPW
29.9k12056
edited Jan 14 at 15:53
MPW
29.9k12056
29.9k12056
answered Jan 14 at 15:47
lightxbulblightxbulb
75619
answered Jan 14 at 15:47
lightxbulblightxbulb
75619
answered Jan 14 at 15:47
lightxbulblightxbulb
75619
75619
add a comment |
add a comment |
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$begingroup$
Is $tau$ a number or a function?
$endgroup$
– Botond
Jan 14 at 15:44
$begingroup$
@Botond its a number.
$endgroup$
– user52932
Jan 14 at 21:40
$begingroup$
Then @lightxbulb's answer is a good way to do it.
$endgroup$
– Botond
Jan 14 at 21:44
$begingroup$
@Botond The issue I am running into is that when I do that, I am unable to express tau as a function of just b and a. That is I am unable to get a closed form expression. What does this mean?
$endgroup$
– user52932
Jan 14 at 21:55
$begingroup$
@user52932 You'll have to analyze a number of cases for different intervals of $a$ and $b$.
$endgroup$
– lightxbulb
Jan 14 at 22:09