# Thread: Maximizing the probability with standard normal random variables

1. ## 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.

2. 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.

3. 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.