Mixing
Showing uniqueness of solution in constrained maximization
$begingroup$
Let $Phi()$ be the CDF for standard normal distribution, and $phi()$ be the PDF for the standard normal distribution. $sigma >0$ and $k ge 0$. We maximize the below equation over $x$.
Maximize
$ f(x) = x[ (k - frac{x}{2 sigma}) Phi(k - frac{x}{2 sigma}) + phi(k - frac{x}{2 sigma})] + (1-x) [(k - frac{x}{2 sigma} - frac{1}{2 sigma})Phi(k - frac{x}{2 sigma} - frac{1}{2 sigma}) + phi(k - frac{x}{2 sigma} - frac{1}{2 sigma})]$
s.t.
$0 le x le 1$.
I'd like to either show that the solution to the above maximization problem is unique or find the sufficient condition (on $sigma$ and $k$) such that the solution is unique.
Note: I have already proved that there exists an interior solution. Also, for some values of $k$ and $sigma$, $f(x)$ is not concave (at least in some values of $x in [0, 1]$). (This is partially why I've been having a difficult time solving/proving this.)
calculus probability-distributions
asked Jan 28 at 4:57
cxu228cxu228
11
$endgroup$
add a comment |
$begingroup$
Let $Phi()$ be the CDF for standard normal distribution, and $phi()$ be the PDF for the standard normal distribution. $sigma >0$ and $k ge 0$. We maximize the below equation over $x$.
Maximize
$ f(x) = x[ (k - frac{x}{2 sigma}) Phi(k - frac{x}{2 sigma}) + phi(k - frac{x}{2 sigma})] + (1-x) [(k - frac{x}{2 sigma} - frac{1}{2 sigma})Phi(k - frac{x}{2 sigma} - frac{1}{2 sigma}) + phi(k - frac{x}{2 sigma} - frac{1}{2 sigma})]$
s.t.
$0 le x le 1$.
I'd like to either show that the solution to the above maximization problem is unique or find the sufficient condition (on $sigma$ and $k$) such that the solution is unique.
Note: I have already proved that there exists an interior solution. Also, for some values of $k$ and $sigma$, $f(x)$ is not concave (at least in some values of $x in [0, 1]$). (This is partially why I've been having a difficult time solving/proving this.)
calculus probability-distributions
asked Jan 28 at 4:57
cxu228cxu228
11
$endgroup$
add a comment |
$begingroup$
Let $Phi()$ be the CDF for standard normal distribution, and $phi()$ be the PDF for the standard normal distribution. $sigma >0$ and $k ge 0$. We maximize the below equation over $x$.
Maximize
$ f(x) = x[ (k - frac{x}{2 sigma}) Phi(k - frac{x}{2 sigma}) + phi(k - frac{x}{2 sigma})] + (1-x) [(k - frac{x}{2 sigma} - frac{1}{2 sigma})Phi(k - frac{x}{2 sigma} - frac{1}{2 sigma}) + phi(k - frac{x}{2 sigma} - frac{1}{2 sigma})]$
s.t.
$0 le x le 1$.
I'd like to either show that the solution to the above maximization problem is unique or find the sufficient condition (on $sigma$ and $k$) such that the solution is unique.
Note: I have already proved that there exists an interior solution. Also, for some values of $k$ and $sigma$, $f(x)$ is not concave (at least in some values of $x in [0, 1]$). (This is partially why I've been having a difficult time solving/proving this.)
calculus probability-distributions
asked Jan 28 at 4:57
cxu228cxu228
11
$endgroup$
Let $Phi()$ be the CDF for standard normal distribution, and $phi()$ be the PDF for the standard normal distribution. $sigma >0$ and $k ge 0$. We maximize the below equation over $x$.
Maximize
$ f(x) = x[ (k - frac{x}{2 sigma}) Phi(k - frac{x}{2 sigma}) + phi(k - frac{x}{2 sigma})] + (1-x) [(k - frac{x}{2 sigma} - frac{1}{2 sigma})Phi(k - frac{x}{2 sigma} - frac{1}{2 sigma}) + phi(k - frac{x}{2 sigma} - frac{1}{2 sigma})]$
s.t.
$0 le x le 1$.
I'd like to either show that the solution to the above maximization problem is unique or find the sufficient condition (on $sigma$ and $k$) such that the solution is unique.
Note: I have already proved that there exists an interior solution. Also, for some values of $k$ and $sigma$, $f(x)$ is not concave (at least in some values of $x in [0, 1]$). (This is partially why I've been having a difficult time solving/proving this.)
calculus probability-distributions
calculus probability-distributions
asked Jan 28 at 4:57
cxu228cxu228
11
asked Jan 28 at 4:57
cxu228cxu228
11
asked Jan 28 at 4:57
cxu228cxu228
11
asked Jan 28 at 4:57
cxu228cxu228
11
asked Jan 28 at 4:57
cxu228cxu228
11
11
add a comment |
add a comment |
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