. The two questions that follow concern the following variant of the Bernoulli pro- cess: Fix k 2 1. At each (integer) time n 2 1 the process takes the value Xn, where Xn are i.i.d. random variab...
. The two questions that follow concern the following variant of the Bernoulli pro- cess: Fix k 2 1. At each (integer) time n 2 1 the process takes the value Xn, where Xn are i.i.d. random variables each with the uniform distribution on 12,,. (4) (a) What is the PMF for the random variable N defined as the smallest N 2 2 so that XN X1 (b) Is N a stopping time? (c) What is the probability that XN+1 3? (d) what is the probability that XN+1 ¢ {X1,XN)? (e) What is the PMF of the random variable N' defined as the smallest N' > 1 so that X1, Xv, and XN+N are all distinct? Are N and N' independent?
. The two questions that follow concern the following variant of the Bernoulli pro- cess: Fix k 2 1. At each (integer) time n 2 1 the process takes the value Xn, where Xn are i.i.d. random variables each with the uniform distribution on 12,,. (4) (a) What is the PMF for the random variable N defined as the smallest N 2 2 so that XN X1 (b) Is N a stopping time? (c) What is the probability that XN+1 3? (d) what is the probability that XN+1 ¢ {X1,XN)? (e) What is the PMF of the random variable N' defined as the smallest N' > 1 so that X1, Xv, and XN+N are all distinct? Are N and N' independent?