How Do You Maximize the Probability Interval for a Standard Normal Variable?

[Proggy99]

Homework Statement

Let Z be a standard normal random variable and $$\alpha$$ be a given constant. Find the real number x that maximizes P(x < Z < x + $$\alpha$$)/

Homework Equations

The Attempt at a Solution

Looking at the standard normal tables, it seems obvious to me that x=0 gives the largest spread regardless of the value of the constant, but I am not sure how to find this computationally. Can anyone give me a hint on what direction to take? Thanks for any help.

I got most of the rest of the homework done, but the three I just posted really have me stumped.

 

[statdad]

Homework Statement

Let Z be a standard normal random variable and $$\alpha$$ be a given constant. Find the real number x that maximizes P(x < Z < x + $$\alpha$$)/

Homework Equations

The Attempt at a Solution

Looking at the standard normal tables, it seems obvious to me that x=0 gives the largest spread regardless of the value of the constant, but I am not sure how to find this computationally. Can anyone give me a hint on what direction to take? Thanks for any help.

I got most of the rest of the homework done, but the three I just posted really have me stumped.

Looking at the level of your other posts, I'm assuming you have a calculus background? If not - sorry.
Note that

$$P(x \le X \le x + \alpha) = \Phi(x+\alpha) - \Phi(x)$$

If this is maximized then its derivative (with respect to $$x$$) is zero. Find the derivative, and work with that.

 

[Proggy99]

Looking at the level of your other posts, I'm assuming you have a calculus background? If not - sorry.
Note that

$$P(x \le X \le x + \alpha) = \Phi(x+\alpha) - \Phi(x)$$

If this is maximized then its derivative (with respect to $$x$$) is zero. Find the derivative, and work with that.

I took the calculus series as a freshman in college 19 years ago. I am struggling through a few higher level courses in my pursuit of a 7-12 integrated mathematics teaching degree. This homework is for an advanced statistics and probability class through independent study at LSU. I have abstract/modern algebra after this and I will have my math prereqs completed. I bought a calculus book to refresh my memory on some things while working through these classes. Number Theory is what brought me to these boards in the first place!

I am going to go see if I can figure it out from what you gave me. Thanks for the hint, I hope!