Let X1X2X3Xn be independent Bernoulli random variables with

Let X₁,X₂,X₃,...,Xn be independent Bernoulli random variables with parameter p. These form a list of 0’s and 1’s of length n. Let Z be the number of times that a 0 is followed by a 1 in the sequence, find EZ.

Solution

since Z is the number of time that a 0 is followed by a 1 in the senquences

and we know that in a bernoulli random variables with parameter p

we have p= 1 and q = 0

so

expected value for Z will be 0.5