Let Z be the standard normal random variable, and define the function: L(z) = E[ Z | Zz ]

• a) Evaluate the following two limits: lim_{z-> - infinity} and lim_{z-> + infinity} of L(z) for both
• b) Suppose you are running a business, and you estimate your profit to be normally distributed with mean = 1000 and standard deviation = 400. But, if your profits are negative then you will declare bankruptcy and your profit will instead be zero. What will your expected profit be? Give your answer in terms of the L(z) function and the cdf of the standard normal (z), and do not try to evaluate these functions. (For example, your final answer could be that the expected profit is 2*L(4) + (1) ).