Let X1 X2 be a sequence of independent random variables t

Let X1, X2, . . . be a sequence of independent random variables that are uniformly distributed between 0 and 1. For every n, we let Yn be the median of the values of X1, X2, . . . , X2n+1. [That is, we order X1, . . . , X2n+1 in increasing order and let Yn be the (n + 1)st element in this ordered sequence.] Show that that the sequence Yn converges to 1/2, in probability

Solution

If we see the trend,

For n=0,
Y0=X1
For n=1
Y1 = X2
For n=2
Y2 = X3 and so on ..
.
.
.
.
At n,
Yn = X(n+1)

Therefore sequence Yn = X1 ,X2 ,X3 ,........,X(n+1)
Also it is given that : X1, X2, . .,x(2n+1) be a sequence of independent random variables that are uniformly distributed between 0 and 1.
So, in probability its middle value (i.e median) will be X(n+1)= 1/2.

But,
Sequence Yn has its last value as X(n+1) which is equal to 1/2.
We conclude that sequence Yn converges to 1/2.

Proved