# Maximizing the probability with standard normal random variables

• Nov 30th 2010, 01:37 PM
Zennie
Maximizing the probability with standard normal random variables
Let $\displaystyle Z$ be a standard normal random variable and $\displaystyle \alpha$ be a given constant. Find the real number $\displaystyle x$ that maximizes $\displaystyle P( x < Z < x+\alpha)$.

Not sure where to even begin other than just looking at the probability function for a normal random variable. Any help or explanation is much appreciated.
• Nov 30th 2010, 01:51 PM
Plato
Quote:

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

I assume that $\displaystyle Z$ has $\displaystyle \mu=0~\&~\sigma=1$ and $\displaystyle \alpha>0$.
If that is the case, we want to choose $\displaystyle x$ so that $\displaystyle 0$ is the midpoint of the interval $\displaystyle [x,x+\alpha]$.

To see why, look at the graph.
• Nov 30th 2010, 08:47 PM
matheagle
Use the fundamental theorem of calculus and differentiate the integral of the normal density over that interval.

That leads to $\displaystyle e^{-(x+\alpha)^2/2}= e^{-x^2/2}$ making $\displaystyle x=-\alpha/2$

which makes sense, since this straddles the origin.