Suppose X1 X2 are independent with density expx Let Mnmaxx1

Suppose X1, X2, ... are independent with density exp(-x). Let Mn=max{x1, ...,xn}.

(a) Show that if a < 1 <b, then P(Mn<_ a ln n) converges towards 0 and P(Mn <_ b ln n) converges toward 1.

Solution

X1, x2 ... are independent with density of exponential distribution.

Mn = Max of xi

Consider a <1<b

n^a<n<n^b

Take log

a ln n < 1 < b ln n

Mn < = a ln n

shows that prob for this as n tends to infinity is almost negligible =0

Similarly P(Mn <= bnlnn ) implies as n is large this is definite and prob tends to 1.