A random variable can take on any of n possible values x1, ... , xn with respective probabilities p(xi), i = 1, ... , n. We shall attempt to determine the value of X by asking a series of questions, each of which can be answered “yes” or “no.” For instance, we may ask “Is X = x1?” or “Is X equal to either x1 or x2 or x3?” and so on. What can you say about the average number of such questions that you will need to ask to determine the value of X?
The average number of questions to determine the value of is
We have to find the average number of questions that you will need to ask to determine the value of .
Mark that the answer to a particular question is i.e.
which means for . So, the average number of questions is
To calculate, multiply it by .
Subtracting that from the expression in , we have that
Using the formula localid="1648133543693" to obtain that the final answer. That is
A certain person goes for a run each morning. When he leaves his house for his run, he is equally likely to go out either the front or the back door, and similarly, when he returns, he is equally likely to go to either the front or the back door. The runner owns 5 pairs of running shoes, which he takes off after the run at whichever door he happens to be. If there are no shoes at the door from which he leaves to go running, he runs barefooted. We are interested in determining the proportion of time that he runs barefooted. (a) Set this problem up as a Markov chain. Give the states and the transition probabilities. (b) Determine the proportion of days that he runs barefooted.
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