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.
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