The value of u which maximises P(10.9<X<12.1) where X~N(u,2.25) isOriginally Posted by guess
(10.9+12.1)/2=11.5. Now this seems obviouse to me, but let me know if
you need something more.
The basic idea is that the normal density is symmetric about the mean, andOriginally Posted by guess
is decreasing as you move away from the mean, which means that if the
mean is not in the centre of the interval (a,b) that P(a<x<b) can be
increased by moving the mean in the direction of the mid point of the interval.
Alternatively you could write:
changing the variable of integration in each of the integrals:
Which can now be differentiated (using the fundamental theorem of calculus)
So the stationary points of as varies are the solutions of:
which are solutions of: , which if is , and this clearly corresponds to a maximum.
Now I could fill in a few more of the details of the argument, but I won't do
that here. I will just observe that the blindingly obvious can have a relativly