- #1
DCG
- 6
- 0
Hi all,
I have the following quantity:
[itex]f = \int_{\mu}^{\infty} (1-F(x))a(x)dx[/itex]
I want to claim that by increasing the following quantity:
[itex]g = \int_{\mu}^{\infty} a(x)f(x)dx[/itex]
then [itex] f [/itex] can only increase. Can I differentiate [itex] f [/itex] with respect
to [itex] g [/itex]? Is the following correct?
[itex] \frac{\partial (\int_{\mu}^{\infty} (1-F(x))a(x)dx)}{\partial (\int_{\mu}^{\infty} a(x)f(x)dx)}[/itex]
[itex] = \frac{\frac{\partial (\int_{\mu}^{\infty} (1-F(x))a(x)dx)}{\partial x}}{\frac{\partial (\int_{\mu}^{\infty} a(x)f(x)dx)}{\partial x}}[/itex]
[itex] = \frac{(1-F(x))a(x)}{f(x)a(x)} [/itex]
[itex] = \frac{1-F(x)}{f(x)} [/itex]
I already know that [itex] a(x) > 0 [/itex] for [itex] x>\mu[/itex] and that [itex] \frac{1-F(x)}{f(x)} [/itex] is positive. Therefore [itex] f [/itex] increases when [itex] g [/itex] is increased. Does this resolve the problem?
Thank you for taking the time to read and answer!
I have the following quantity:
[itex]f = \int_{\mu}^{\infty} (1-F(x))a(x)dx[/itex]
I want to claim that by increasing the following quantity:
[itex]g = \int_{\mu}^{\infty} a(x)f(x)dx[/itex]
then [itex] f [/itex] can only increase. Can I differentiate [itex] f [/itex] with respect
to [itex] g [/itex]? Is the following correct?
[itex] \frac{\partial (\int_{\mu}^{\infty} (1-F(x))a(x)dx)}{\partial (\int_{\mu}^{\infty} a(x)f(x)dx)}[/itex]
[itex] = \frac{\frac{\partial (\int_{\mu}^{\infty} (1-F(x))a(x)dx)}{\partial x}}{\frac{\partial (\int_{\mu}^{\infty} a(x)f(x)dx)}{\partial x}}[/itex]
[itex] = \frac{(1-F(x))a(x)}{f(x)a(x)} [/itex]
[itex] = \frac{1-F(x)}{f(x)} [/itex]
I already know that [itex] a(x) > 0 [/itex] for [itex] x>\mu[/itex] and that [itex] \frac{1-F(x)}{f(x)} [/itex] is positive. Therefore [itex] f [/itex] increases when [itex] g [/itex] is increased. Does this resolve the problem?
Thank you for taking the time to read and answer!