# Assume that (y) is normally distributed with mean (mu) and variance (sigma^{2}). Let (phi(cdot)) and (Phi(cdot)) be

## Question:

Assume that $$y$$ is normally distributed with mean $$\mu$$ and variance $$\sigma^{2}$$. Let $$\phi(\cdot)$$ and $$\Phi(\cdot)$$ be the standard normal density and distribution functions, respectively. Define $$\mathrm{h}(d)=\phi (d) /(1-\Phi(d))$$, a hazard rate. Let $$d$$ be a known constant and $$d_{s}=(d-\mu) / \sigma$$ be the standardized version.

a. Determine the density of $$y$$, conditional on $$\{y>d\}$$.

b. Show that $$\mathrm{E}(y \mid y>d)=\mu+\sigma \mathrm{h}\left(d_{s}\right)$$.

c. Show that $$\mathrm{E}(y \mid y \leq d)=\mu-\sigma \phi\left(d_{s}\right) / \Phi\left(d_{s}\right)$$.

d. Show that $$\operatorname{Var}(y \mid y>d)=\sigma^{2}\left(1-\delta\left(d_{s}\right)\right)$$, where $$\delta(d)=\mathrm{h}(d)$$ $$(\mathrm{h}(d)-d)$$.

e. Show that $$\mathrm{E} \max (y, d)=\left(\mu+\sigma \mathrm{h}\left(d_{s}\right)\right)\left(1-\Phi\left(d_{s}\right)\right)+d \Phi\left(d_{s}\right)$$.

f. Show that $$\mathrm{E} \min (y, d)=\mu+d-\left(\left(\mu+\sigma \mathrm{h}\left(d_{s}\right)\right)\left(1-\Phi\left(d_{s}\right)\right)+\right.$$ $$\left.d \Phi\left(d_{s}\right)\right)$$.

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