Mathematics, 20.02.2020 23:01 jeffffffff

LetX1; X2; : : : ;be a sequence of inde-pendent identically distributed random variables. Assume that there are no ties. Wesay that a record occurs at timenifXn>max(X1; : : : ; Xn1). That is, Xnis arecord if it is larger than each ofX1; : : : ; Xn1. Show(i)Pfa record occurs at timeng= 1=n.(ii)E[number of records by timen] =∑ni=11=i.(iii)Var[number of records by timen] =∑ni=1(i1)=i2.Hint for (ii){(iii): At each timei, think about de ning some 1/0 variableYi, whichindicates whether there is a record at timei. Fori < j, what isPfYi= 1jYj= 1g?

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LetX1; X2; : : : ;be a sequence of inde-pendent identically distributed random variables. Assume tha...