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Normal random variables (3rd)

  1. Mar 29, 2009 #1
    1. The problem statement, all variables and given/known data
    Let Z be a standard normal random variable and [tex]\alpha[/tex] be a given constant. Find the real number x that maximizes P(x < Z < x + [tex]\alpha[/tex])/


    2. Relevant equations



    3. 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.
     
  2. jcsd
  3. Mar 30, 2009 #2

    statdad

    User Avatar
    Homework Helper

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

    [tex]
    P(x \le X \le x + \alpha) = \Phi(x+\alpha) - \Phi(x)
    [/tex]

    If this is maximized then its derivative (with respect to [tex] x [/tex]) is zero. Find the derivative, and work with that.
     
  4. Mar 30, 2009 #3
    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!
     
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