A random variable with normal distribution with unknown mean u and variance 2.25. with a suitable value of u. find the greatest value of P(10.9<X<12.1)

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- Jul 4th 2006, 12:25 AMguessProbabilty ans statistics
A random variable with normal distribution with unknown mean u and variance 2.25. with a suitable value of u. find the greatest value of P(10.9<X<12.1)

- Jul 4th 2006, 01:12 AMCaptainBlackQuote:

Originally 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.

RonL - Jul 4th 2006, 01:31 AMguess
ya... u got the ans.. but i does not really know the reason behind it... can you try to explain?

- Jul 4th 2006, 02:19 AMCaptainBlackQuote:

Originally 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:

where

Then:

,

changing the variable of integration in each of the integrals:

The rewriting:

Which can now be differentiated (using the fundamental theorem of calculus)

to give:

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

complicated proof :eek:

RonL