Introduction to Probability Final Exam
Due:August 8th Solve all the problems
1. (15 points) You have three coins, showing “Head” with probabilities p1, p2
and p3. You perform two different experiments:
1. You choose one coin at random and toss it repeatedly.
2. You repeatedly choose a coin at random and toss it.
In both cases calculate the average number of “Heads” among the first n
tosses, and the average time you have to wait for the first “Head”.
2. (15 points) Let X1,· · · , X4be four independent random variables, and
gi:R2→Rfunctions for i= 1,2. Show that Y1=g1(X1, X2) and
Y2=g2(X3, X4) are independent.
3. (20 points) Let Y1, Y2,· · · be a sequence of independent Bernoulli p. Let
X1, X2,· · · denote the time of the first, second, · · · success. (e.g. for the
outcome 0,1,1,0,0,1,0,1,· · · we have X1= 2,X2= 3, X3= 6,· · · )
1. Find the joint PMF of X1,· · · , Xn, i.e., P(X1=k1,· · · , Xn=kn).
2. Use (a) to calculate the PMF of Xn, the time of the n-th success.
3. Use (a) to calculate the PMF of the RVs Y1=X1, Y2=X2−
X1,· · · , Yn=Xn−Xn−1, and show that they are independent
geometric RVs.
4. (20 points) Suppose that every day can be described by either S(sunny)
or R(rainy). The probability for a day to be sunny is 4/8 if the preceding
day was rainy, 6/8 if the preceding day was sunny, but 7/8 if both of the
preceding days were sunny.
1. Let S={S, R}, and let Xn= the weather at nth day. Is Xna
Markov chain?
2. Define a Markov chain on the state space S={SS, SR, RS, RR}that
describes the above weather model, and determine the corresponding
transition matrix.
3. Given that the weekend(Saturday and Sunday) was sunny calculate
the probability for the next five days to be SSRS S.
5. (10 points) How often do you have to roll a fair die on the average until
you have seen all possible values? (Hint: Let Xndenote the number of
different values you have seen at time n.)
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