Let X1,X2,… be a sequence of independent random variables, uniformly distributed on [0,1] . Define Nn to be the smallest k such that X1+X2+⋯+Xk exceeds cn=n2+n12−−−√ , namely,

Question

Nn = min{k≥1:X1+X2+⋯+Xk>cn}

Does the limit
limn→∞P(Nn>n)
exist? If yes, enter its numerical value. If not, enter −999 .

Answer 1

0.8413

Answer 2

I find (1 - phi(1)). anyone else?

Answer 3

I had the answer 0

Answer 4

is Nn a random variable?

Answer 5

it's Irwin–Hall distribution

Answer 6

Does anyone is sure about their answer?

Answer 7

now i am reconsidering and got phi(1)
p(Nn>n) = P(x1+..xn <cn) = P(Zn<1) central limit