



Exercise 1. Customers at a coffee shop ask for hot, denote it X 0, or cold, denote it Xn 1, beverages according to a Bernoulli process with parameter p. Let N denote the first time that a customer wa...
1) Customers at a coffee shop ask for hot, say Xn 0, or cold, say X 1, beverages according to a Bernoulli process with parameter p. Let N denote the first time that a customer wants the same kind as their predecessor. (a) Find the pmf of N. (b) What is the probability that XN+ 1? (c) If cold drinks take 30 seconds to prepare and hot drinks take 60 seconds. What is the expected time taken to serve the...
. 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...
Exercise 2. Let Xn, n EN, be a Bernoulli process uith parameter p = 1/2. Define N = min(n > 1:X,メ } For any n 2 1, define Yn = XN4n-2. Show that P(Yn = 1) = 1/2, but Yn, n E N is not a Bernoulli process
Exercise 2. Let Xn, n EN, be a Bernoulli process uith parameter p = 1/2. Define N = min(n > 1:X,メ } For any n 2 1, define Yn = XN4n-2. Show...