Statistics:minimizing an interval for a standard normal distribution

[k3k3]

Homework Statement

Generalize: For arbitrary 0 < p < 1, show that the method giving a and b produces the minimum length interval.

Hint: It might be helpful to use local extrema for the inverse function of the distribution function.

Homework Equations

The method is is talking about is locating the z scores using (1-p)/2 and [1-(1-p)/2]

The Attempt at a Solution

Let a be the area on the tail end of the distribution not included in p
Let b be the other end so that

a+b=1-p and b=1-p-a

Then the points A and B are the end points of the interval containing p.

B-A = (F^-1)(p+a)-(F^-1)(a)

This is where I am stuck. I know f(y). So d/dy(F(y))=f(y) and then (f^-1)'(y)=1/(f'(f^-1)(y))

I am not sure how to proceed.

 

[Ray Vickson]

Homework Statement

Generalize: For arbitrary 0 < p < 1, show that the method giving a and b produces the minimum length interval.

Hint: It might be helpful to use local extrema for the inverse function of the distribution function.

Homework Equations

The method is is talking about is locating the z scores using (1-p)/2 and [1-(1-p)/2]

The Attempt at a Solution

Let a be the area on the tail end of the distribution not included in p
Let b be the other end so that

a+b=1-p and b=1-p-a

Then the points A and B are the end points of the interval containing p.

B-A = (F^-1)(p+a)-(F^-1)(a)

This is where I am stuck. I know f(y). So d/dy(F(y))=f(y) and then (f^-1)'(y)=1/(f'(f^-1)(y))

I am not sure how to proceed.

It is helpful to assume the normal distribution is standard (mean = 0, variance = 1). Do you see why you can do that without loss of generality? Now, if F(z) is the standard normal cdf, you want to minimize b-a, subject to the constraint F(b) - F(a) = p. Hint: Lagrange multipliers.

RGV