Maximizing the probability with standard normal random variables

**Zennie** · November 30th 2010, 01:37 PM

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

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.

**Plato** · November 30th 2010, 01:51 PM

Originally Posted by Zennie

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

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

To see why, look at the graph.

**matheagle** · November 30th 2010, 08:47 PM

Use the fundamental theorem of calculus and differentiate the integral of the normal density over that interval.

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

which makes sense, since this straddles the origin.

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