This is a continuation of our test of the platform. Before making this post we uploaded the content of our custom LaTeX style file to the preamble. So far it is working fine.
Consider independent flips of a coin with probability of landing heads. A transition occurs whenever the outcome on a flip differs from the flip preceding it. The expected number of transition is found.
The outcomes of the flips are represented by an i.i.d sequence of random variables . The number of transition is expressed as function of this sequence as follows. Define a new sequence of random variables in term of the . . Clearly, the sequence is Bernoulli. Further, if and only if a transition occurs on flip . Thus
Next,the expectation of is found with the help of linearity of expectation, the independence of the , and the fact that as follows where , . Thus the sequence is identically distributed .
It follow from the linearity of expectation and the fact that , that