normal variates are krieg

Normally, I only really like word problems that force you to model the solution. This one, from a book I don't even like that much, was sufficiently cute to merit posting it along with its solution.

A random variable is normally distributed. If and , determine and .

The data appear pretty scant, but don't let that upset you, it's not too hard to think through. Notice that is normally distributed. In other words, (a linear transformation of the standardized normal distribution ), and so . If you multiply that term out, the expectation is .

Because the expectation is one (can be proven with calculus, but I'm leaving it out), the expectation of the first term is of course . The expectatation is obviously zero, because is standardized normal. And the last term is expectation of a constant, which is always that constant! So, all told, .

The next clue given is the value of the cdf at ten. Again, because is a normal variate, you can use this value to deduce further constraints on the values of and , which are the standard deviation and mean of , respectively. Since , do a reverse lookup on a table of values for . The value where is one. So solve for in and substitute its value in terms of into the expectation from earlier. Since , the problem boils down beautifully to middle school algebra from here. The roots of are eight and two. As such, it can be true either that (and thus that ) or that (and thus that ). And of course and (the variance).

At the bottom of it, everything is pretty basic. Problem is finding out how.