Hey can anyone please help me with this.
Find the probability that range of a random sample of size 4 from $U(0,1)$ is less than $\frac{1}{2}$.
i.e to find $ \mathbb{P}(Y_{4}-Y_{1})< 1/2$ where $Y_n$ is the nth order statistic.
An approach is to first get the joint distribution, then do the bivariate transformation and finally integrate but I am unable to solve it, Would really appreciate help.
The joint distribution of the $n+1$ intervals formed by $n$ points uniformly randomly chosen in an interval is invariant under permutations of the intervals. In particular, the joint distribution of the first and last intervals is the same as the joint distribution of the first and second intervals. (This can be seen by uniformly randomly choosing $n+1$ points on a circle, then uniformly randomly choosing one of the points as the point at which to cut the circle into an interval; this demonstrates the equivalence of the $n+1$ intervals formed by the $n$ points.)
Thus, the probability that the first and last interval together are greater than $\frac12$ is the probability that the first and second interval together are greater than $\frac12$, which is the probability that at most one of the $4$ points lies in the first half of $[0,1]$. This is
$$ \left(\frac12\right)^4+\binom41\left(\frac12\right)^4=\frac5{16}\;. $$