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Where $\phi(z)$ and $\Phi(z)$ represent the standard normal pdf and cdf respectively.

1) Is the function

$$f(z)=\frac{\phi(z)}{1-\Phi(z)}$$

increasing for all values of $z$? If so, how can I show it?

2) Is the limit as $z\rightarrow\infty$ using L'Hôpital's rule

$$\lim_{z\rightarrow\infty}f(z) = \frac{\phi'(z)}{-\phi(z)}=\frac{-z\phi(z)}{-\phi(z)}=z=\infty \text{?}$$

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  • $\begingroup$ If you have two questions, like here, you should not ask them together like here. $\endgroup$ Jan 28, 2015 at 18:46

1 Answer 1

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The ratio yoy have given, is known as the inverse Mills ratio, see: https://en.wikipedia.org/wiki/Inverse_Mills_ratio From that article, we have the representation, if $X$ is a random variable with the standard normal distribution (expectation zero, variance one) that $$ E(X | x > \alpha) = \frac{\phi(\alpha)}{1-\Phi(\alpha)} = f(\alpha) $$ and from that interpretation it follows that $f$ is indeed increasing. You can also try to prove it directly using the usual differentiation rules.

As for your question 2) I think you solved it correctly.

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  • $\begingroup$ Regarding 1) I am not sure you can simply differentiate it to show the differentiation is positive without further getting an upper bound for the Mills ratio. Can you show your result if you have a direct way of doing this? $\endgroup$
    – Hans
    Apr 1, 2016 at 9:22

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