# Interesting question regarding Weibull distribution, sample mean, and sample median

Consider a random sample from a Weibull distribution, $X_i \sim WEI(1,2)$. Find approximate values $a$ and $b$ such that for $n=35$:

(a) $P[a < \bar{X} < b]=0.95$

(b) $P[a < X_{18:35} < b] = 0.95$

Here $\bar{X}=\frac{X_1+\cdots+X_n}n$ (the sample mean) and $X_{18:35}$ is the sample median. Also the pdf of $X_i$ is $2xe^{-x^2}$. We need $P[\bar{X}<b]-P[\bar{X}<a]=0.95$. But is $(a,b)$ unique? If so, why? I think it's not unique.

• You are right, it is not unique. If you try to make the interval as symmetric as possible about the mean, you will get uniqueness. That is likely what is intended. Nov 30 '13 at 7:46
• For part (a), a=0.733 b = 1.0397... but why? This choice looks arbitrary to me... Nov 30 '13 at 7:48

You are right, $a$ and $b$ are not uniquely determined. But if we decide to make $a$ and $b$ as symmetric as possible about the mean, they are determined.
In this case, the mean is $\frac{\sqrt{\pi}}{2}$. The numbers you mentioned in a comment are in fact chosen to be symmetric about this.