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Let X1,X2,… be a sequence of independent random variables, uniformly distributed on [0,1]. Define Nn to...

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Basic Math

Let X1,X2,… be a sequence of independent random variables,uniformly distributed on [0,1]. Define Nn to be the smallest k suchthat X1+X2+?+Xn exceeds cn=n2+12n???, namely,

Nn=min{k?1:X1+X2+?+Xk>ck}

Does the limit

limn??P(Nn>n)

exist? If yes, enter its numerical value. If not, enter?999.

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