let X1,X2, . . ., Xn be independent random variables all with uniform distribution on [0,1]. Let 0<=a<=b

show P(a<=(X1*X2*...*Xn)^(1/Sqr[n]))<=b) tends to a limit as n tends to infinity and find an expression for it?

i have some problems with my solution and am not sure it is right ..... thanks for any help

Let Y=(X1*X2*...*Xn)^(1/Sqr[n]))

c.d.f FY(y) = P({Y<=y})= P({(X1*X2*...*Xn)^(1/Sqr[n]))<=y})

=P({X1^(n/Sqr[n])<=y}) = P({X1^(Sqr[n])<=y})

=P({X1<=Sqr[y] to the (Sqr[n])th root})

=

0 if (Sqr[y] to the (Sqr[n])th root)<=0

Sqr[y] to the (Sqr[n])th root if 0<= Sqr[y] to the (Sqr[n])th root <=1

1 if 1<= Sqr[y] to the (Sqr[n])th root

but not sure if this is right and then when considering looking at this function as n tends to infinity i got that it tends to 1 for all y eeeeek ?????

thanks for any help