Solution Found!
Let X equal the number of flips of a fair coin that are required to observe headstails
Chapter 2, Problem 2.3-17(choose chapter or problem)
Let X equal the number of flips of a fair coin that are required to observe headstails on consecutive flips. (a) Find the pmf of X. Hint: Draw a tree diagram. (b) Show that the mgf of X is M(t) = e2t /(et 2)2. (c) Use the mgf to find the values of (i) the mean and (ii) the variance of X. (d) Find the values of (i) P(X 3), (ii) P(X 5), and (iii) P(X = 3).
Questions & Answers
QUESTION:
Let X equal the number of flips of a fair coin that are required to observe headstails on consecutive flips. (a) Find the pmf of X. Hint: Draw a tree diagram. (b) Show that the mgf of X is M(t) = e2t /(et 2)2. (c) Use the mgf to find the values of (i) the mean and (ii) the variance of X. (d) Find the values of (i) P(X 3), (ii) P(X 5), and (iii) P(X = 3).
ANSWER:Step 1 of 5
Let X equal the number of flips of a fair coin that are required to observe the same face on consecutive flips.
When the coin is flipped for one time, it is not possible to observe the same face on consecutive flips. Therefore,
When the coin is flipped twice, there are two points in sample space having the same face on consecutive flips. Thus, the probability to get same face on consecutive flips is 
Therefore, 
When the coin is flipped for three times, the probability to get same face on consecutive flips is 
Therefore, 
Continuing in this manner, when the coin is flipped for 
Therefore,
Hence, the probability mass function for the random variable 
The tree diagram is shown below.
