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18.440: Probability and Random Variables
Final Exam
Thursday, December 16th, 2010
You will have 180 minutes to complete this test. Each of the problems is worth the same fraction of your exam grade. No calculators, notes, or other study aids are permitted. If a question calls for a numerical answer, you don’t need to multiply everything out. (For example, it’s okay to write something like (0.9)7!/(3! 2!) as your answer.) Don’t forget to write your name on the top of every page. Please show your work and explain your answers—we will not award full credit for the correct numerical answer without proper explanation. Good luck!
Problem 2: Suppose you have three (biased) coins in a bag such that, when flipped:
- Coin 1 comes up heads with probability 1/3.
- Coin 2 comes up heads with probability 2/3.
- Coin 3 comes up heads with probability 1.
When you draw a coin from the bag, each of the three coins is equally likely.
(a) Suppose you draw a random coin from the bag and flip it, and it comes up heads. What is the probability that it was Coin 1?
(b) Suppose you draw a random coin from the bag and flip it twice. (Note that you are flipping the same coin both times.) If it comes up heads on the first flip, what is the probability that it comes up heads on the second flip as well?
(c) Suppose you draw a random coin from the bag and repeatedly flip it until it comes up heads. (Note that you are flipping the same coin each time.) Let F be the number of times that you flip the coin (including the last one). What is the probability mass function of F?
(d) Compute E[F ] and Var(F ).
Problem 3: An company claims that it has a magic spell that improves the test scores of students who are doing badly in school. It has large amounts of data that show that, when it casts its spell on students who score in the bottom 1% on a standardized test, their scores improve significantly (on average) when they take a similar test a second time. In this question, we’ll evaluate whether this necessarily means that their spell works. We’ll do this by seeing if we can explain this data assuming the spell doesn’t do anything.^2 Suppose that a student’s test score is a combination of skill and random chance. The student’s skill won’t change when he takes the test twice, but the random part of his score will. Formally, let S, R 1 , and R 2 be independent random variables such that:
- S is uniform on [0, 2].
- R 1 is uniform on [0, 1].
- R 2 is uniform on [0, 1].
If you pick a random student and give him two tests, his score on the first test will be given by T 1 = S + R 1 , and his score on the second test will be given by T 2 = S + R 2.
(a) For a ∈ [0, 1], compute the probability that T 1 ≤ a. (We only care about students who did badly, so we won’t bother dealing with the case when a > 1.) Find the value of a such that P (T 1 ≤ a) = 0.01.
(b) For a, b ∈ [0, 1], compute P (R 1 ≤ b | T 1 ≤ a).
(c) For a ∈ [0, 1], use your answer to part (b) to compute the probability density function and expected value of R 1 given that T 1 ≤ a.
(d) Given that T 1 ≤ a (for a ∈ [0, 1]), what is the expected value of T 2 − T 1? In particular, what is the expected change in a randomly chosen student’s score, given that he scored in the bottom 1% on the first exam?
(e) Briefly explain in words why the company’s spell appears to improve students in the bottom 1% even though it doesn’t do anything.
(^2) The general phenomenon we’re studying here is sometimes called “reversion to the mean” and is a widespread cause of misinterpretations of statistical data.
Problem 5: Let X 1 ,... , X 100 and Y 1 ,... , Y 200 be independent random variables such that:
- Xi equals 1, 2, or 3, each with probability 1/3.
- Yj is a geometric random variable with parameter 1/4.
(Remember that you have a formula sheet in the back of the exam that may be helpful for parts of this problem. Also, remember that you don’t need to compute everything out—an answer like “(0.01+0.02)/(0.03+2)” is fine.)
(a) Compute the expectation, variance, and moment generating function of each Xi.
(b) Let Z =
i=1 Xi,^ W^ =^
j=1 Yj^ ,^ and^ T^ = 3Z^ −^2 W.^ Compute the expectation, variance, and moment generating function of T.
(c) Approximate T by an appropriate normal random variable and use this to estimate the probability that T > 0. (Write your answer in terms of Φ(x), the CDF of a standard normal.) Note that the variables X 1 ,... , X 100 , Y 1 ,... , Y 200 are not all identically distributed (since there are two different types of variables, the Xs and the Y s). In your answer, explain briefly why you can still approximate T using a normal.
(d) Let A =
i=1 Xi^ and let^ B^ =^
i=50 Xi. Compute the covariance Cov(A, B) and the correlation ρ(A, B).
Problem 6: A math professor is trying to bake a cake for his friends.
- With probability 1/3, he gets it right. In this case, he spends a number of hours that is an exponential random variable with parameter 0.5 and finishes the cake.
- With probability 2/3, he drops the cake and has to start over. In this case, he spends a number of hours that is an exponential random variable with parameter 1, and then he starts again from the beginning.
Assume that the exponential variables in each round are independent. Let H be the total number of hours it takes him to bake the cake.
(a) Let
Y =
1 if he succeeds on his first try 2 if he drops the cake on his first try
Compute E[H | Y = 1] and E[H | Y = 2].
(b) Use your answer to part (a) to compute E[H].
(c) Compute E[H^2 | Y = 1] and E[H^2 | Y = 2].
(d) Use your answer to part (c) to compute Var(H).
1 Random Variables
Bernoulli variable with parameter p:
- Probability mass function: Pr[X = 0] = 1 − p, Pr[X = 1] = p
- Expectation: p
- Variance: p(1 − p)
- Moment generating Function: M (t) = pet^ + 1 − p
Binomial variable with parameters n and p:
- Probability mass function: Pr[X = i] =
(n i
) pi(1 − p)n−i^ for i = 0, 1 , 2 ,... , n
- Expectation: np
- Variance: np(1 − p)
- Moment generating Function: M (t) = (pet^ + 1 − p)n
Poisson variable with parameter λ:
- Probability mass function: Pr[X = i] = e−λ λ
i i! , for^ i^ = 0,^1 ,^2 ,...
- Expectation: λ
- Variance: λ
- Moment generating Function: M (t) = eλ(e t−1)
Geometric variable with parameter p:
- Probability mass function: Pr[X = i] = p(1 − p)i−^1 , for i = 1, 2 ,...
- Expectation: 1 /p
- Variance: (1 − p)/p^2
- Moment generating Function: M (t) = pe
t 1 −(1−p)et
Uniform variable over the interval [a, b]:
- Probability density function: f (x) = 1/(b − a) for x ∈ [a, b], 0 otherwise
- Expectation: (a + b)/ 2
- Variance: (b − a)^2 / 12
- Moment generating Function: M (t) = e
tb−eta t(b−a)
- Problem 1 /
- Problem 2 /
- Problem 3 /
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- Problem 7 /
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