Question

Let X1, X2 · · · X10 be i.i.d. random variables. And all of them have the distribution uniform[0, 2]. Please calculate the variance of the 2nd largest number among these X’s.

Answer 1

Y = X/2

if X = Unif(0,2)

then

Y follow Unif(0,1)

here Y follow Unif(0,1)

n= 10

j= 9 for 2nd largest number

hence

Var(Y₉) = 9 * (10 - 9 + 1)/(11^2 * 12)

= 0.01239669

hence

Var(X₉) = Var(2Y₉) = 4* 0.01239669 = 0.04958677