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Stat 110 Midterm Review, Fall 2011
Prof. Joe Blitzstein (Department of Statistics, Harvard University)
1 General Information
The midterm will be in class on Wednesday, October 12. There is no alternate time for the exam, so please be there and arrive on time! Cell phones must be o↵, so it is a good idea to bring a watch. No books, notes, or calculators are allowed, except that you may bring two sheets of standard-sized paper (8.5” x 11”) with whatever you want written on it (two-sided): notes, theorems, formulas, information about the important distributions, etc.
There will be 4 problems, weighted equally. Many of the parts can be done quickly if you have a good understanding of the ideas covered in class (e.g., seeing if you can use Bayes’ Rule or understand what independence means), and for many you can just write down an answer without needing to simplify. None will require long or messy calculations. They are not arranged in order of increasing di culty. Since it is a short exam, make sure not to spend too long on any one problem.
Suggestions for studying: review all the homeworks and read the solutions, study your lecture notes (and possibly relevant sections from either book), the strategic practice problems, and this handout. Solving practice problems (which means trying hard to work out the details yourself, not just trying for a minute and then looking at the solution!) is extremely important.
2 Topics
- Combinatorics: multiplication rule, tree diagrams, binomial coe cients, per- mutations and combinations, inclusion-exclusion, story proofs.
- Basic Probability: sample spaces, events, axioms of probability, equally likely outcomes, inclusion-exclusion, unions, intersections, and complements.
- Conditional Probability: definition and meaning, writing P (A 1 \ A 2 \ · · · \ An ) as a product, Bayes’ Rule, Law of Total Probability, thinking conditionally, independence vs. conditional independence.
- Random Variables: definition, meaning of X = x, stories, indicator r.v.s, prob- ability mass functions (PMFs), probability density functions (PDFs), cumula- tive distribution functions (CDFs), independence, Poisson approximation.
- Expected Value and Variance: definitions, linearity, standard deviation, Law of the Unconscious Statistician (LOTUS).
- Important Discrete Distributions: Bernoulli, Binomial, Geometric, Negative Binomial, Hypergeometric, Poisson.
- Important Continuous Distributions: Uniform, Normal.
- General Concepts: stories, symmetry, discrete vs. continuous, conditional prob- ability is the soul of statistics, checking simple and extreme cases.
- Important Examples: birthday problem, matching problem, Newton-Pepys problem, Monty Hall problem, testing for a rare disease, elk problem (capture- recapture), gambler’s ruin, Simpson’s paradox, St. Petersburg paradox.
4 Some Useful Formulas
4.1 De Morgan’s Laws
(A 1 [ A 2 · · · [ A (^) n ) c^ = A c 1 \ Ac 2 · · · \ Acn (A 1 \ A 2 · · · \ A (^) n ) c^ = A c 1 [ Ac 2 · · · [ Acn
4.2 Complements
P (Ac^ ) = 1 P (A)
4.3 Unions
P (A [ B) = P (A) + P (B) P (A \ B)
P (A 1 [ A 2 [ · · · [ An ) =
X^ n
i=
P (Ai ), if the Ai are disjoint
P (A 1 [ A 2 [ · · · [ An )
X^ n
i=
P (Ai )
P (A 1 [A 2 [· · ·[An ) =
X^ n
k=
( 1) k+^
X
i 1 <i 2 <···<i (^) k
P (Ai 1 \ Ai 2 \ · · · \ A (^) i (^) k )
(Inclusion-Exclusion)
4.4 Intersections
P (A \ B) = P (A)P (B|A) = P (B)P (A|B)
P (A 1 \ A 2 \ · · · \ An ) = P (A 1 )P (A 2 |A 1 )P (A 3 |A 1 , A 2 ) · · · P (An |A 1 ,... , A (^) n 1 )
4.5 Law of Total Probability
If E 1 , E 2 ,... , E (^) n are a partition of the sample space S (i.e., they are disjoint and their union is all of S) and P (E (^) j ) 6 = 0 for all j, then
P (B) =
X^ n
j=
P (B|E (^) j )P (E (^) j )
4.6 Bayes’ Rule
P (A|B) =
P (B|A)P (A)
P (B)
Often the denominator P (B) is then expanded by the Law of Total Probability.
4.7 Expected Value and Variance
Expected value is linear: for any random variables X and Y and constant c,
E(X + Y ) = E(X) + E(Y ) E(cX) = cE(X)
It is not true in general that Var(X + Y ) = Var(X) + Var(Y ). For example, let X be Bernoulli(1/2) and Y = 1 X (note that Y is also Bernoulli(1/2)). Then Var(X) + Var(Y ) = 1/4 + 1/4 = 1/2, but Var(X + Y ) = Var(1) = 0 since X + Y is always equal to the constant 1. (We will see later exactly when the variance of the sum is the sum of the variances.)
Constants come out from variance as the constant squared:
Var(cX) = c 2 Var(X)
The variance of X is defined as E(X EX) 2 , but often it is easier to compute using the following: Var(X) = E(X 2 ) (EX) 2
4.8 Law of the Unconscious Statistician (LOTUS)
Let X be a discrete random variable and h be a real-valued function. Then Y = h(X) is a random variable. To compute E(Y ) using the definition of expected value, we would need to first find the PMF of Y , and then use E(Y ) =
P
y yP^ (Y^ =^ y). The Law of the Unconscious Statistician says we can use the PMF of X directly:
E(h(X)) =
X
x
h(x)P (X = x),
where the sum is over all possible values of X. Similarly, for X a continuous r.v. with PDF f (^) X , we can find the expected value of Y = h(X) using the PDF of X, without having to find the PDF of Y :
E(h(X)) =
Z 1
h(x)f (^) X (x)dx
- Two coins are in a hat. The coins look alike, but one coin is fair (with probability 1 /2 of Heads), while the other coin is biased, with probability 1/4 of Heads. One of the coins is randomly pulled from the hat, without knowing which of the two it is. Call the chosen coin “Coin C”.
(a) Coin C is tossed twice, showing Heads both times. Given this information, what is the probability that Coin C is the fair coin? (Simplify.)
(b) Are the events “first toss of Coin C is Heads” and “second toss of Coin C is Heads” independent? Explain briefly.
(c) Find the probability that in 10 flips of Coin C, there will be exactly 3 Heads. (The coin is equally likely to be either of the 2 coins; do not assume it already landed Heads twice as in (a). Do not simplify.)
- Five people have just won a $100 prize, and are deciding how to divide the $ up between them. Assume that whole dollars are used, not cents. Also, for example, giving $50 to the first person and $10 to the second is di↵erent from vice versa.
(a) How many ways are there to divide up the $100, such that each gets at least $10?
Hint: there are
(^) n+k 1 k
ways to put k indistinguishable balls into n distinguishable boxes; you can use this fact without deriving it.
(b) Assume that the $100 is randomly divided up, with all of the possible allocations counted in (a) equally likely. Find the expected amount of money that the first person receives (justify your reasoning).
(c) Let A (^) j be the event that the jth person receives more than the first person (for 2 j 5), when the $100 is randomly allocated as in (b). Are A 2 and A (^3) independent? (No explanation needed for this.) Express I (^) A 2 \A 3 and I (^) A 2 [A 3 in terms of I (^) A 2 and I (^) A 3 (where I (^) A is the indicator random variable of any event A).
- Alice flips a fair coin n times and Bob flips another fair coin n + 1 times, resulting in independent X ⇠ Bin(n, 12 ) and Y ⇠ Bin(n + 1, 12 ).
(a) Let V = min(X, Y ) be the smaller of X and Y , and let W = max(X, Y ) be the larger of X and Y. (If X = Y , then V = W = X = Y .) Find E(V ) + E(W ) in terms of n (simplify).
(b) Is it true that P (X < Y ) = P (n X < n + 1 Y )? Explain why or why not.
(c) Compute P (X < Y ) (simplify). Hint: use (b) and that X and Y are integers.
6 Stat 110 Midterm from 2008
- The gambler de M´er´e asked Pascal whether it is more likely to get at least one six in 4 rolls of a die, or to get at least one double-six in 24 rolls of a pair of dice. Continuing this pattern, suppose that a group of n fair dice is rolled 4 · 6 n ^1 times.
(a) Find the expected number of times that “all sixes” is achieved (i.e., how often among the 4 · 6 n ^1 rolls it happens that all n dice land 6 simultaneously). (Simplify.)
(b) Give a simple but accurate approximation of the probability of having at least one occurrence of “all sixes”, for n large (in terms of e but not n).
(c) de M´er´e finds it tedious to re-roll so many dice. So after one normal roll of the n dice, in going from one roll to the next, with probability 6/7 he leaves the dice in the same configuration and with probability 1/7 he re-rolls. For example, if n = 3 and the 7th roll is (3, 1 , 4), then 6/7 of the time the 8th roll remains (3, 1 , 4) and 1 /7 of the time the 8th roll is a new random outcome. Does the expected number of times that “all sixes” is achieved stay the same, increase, or decrease (compared with (a))? Give a short but clear explanation.
- (a) Let X 1 , X 2 ,... be independent N (0, 4) r.v.s., and let J be the smallest value of j such that X (^) j > 4 (i.e., the index of the first X (^) j exceeding 4). In terms of the standard Normal CDF , find E(J) (simplify).
(b) Let f and g be PDFs with f (x) > 0 and g(x) > 0 for all x. Let X be a random
variable with PDF f. Find the expected value of the ratio g f ((XX)) (simplify).
(c) Define F (x) = e e^ x
. This is a CDF (called the Gumbel distribution) and is a continuous, strictly increasing function. Let X have CDF F , and define W = F (X). What are the mean and variance of W (simplify)?
- (a) Find E(2 X^ ) for X ⇠ Pois( ) (simplify).
(b) Let X and Y be independent Pois( ) r.v.s, and T = X + Y. Later in the course, we will show that T ⇠ Pois(2 ); here you may use this fact. Find the conditional distribution of X given T = n, i.e., find the conditional PMF P (X = k|T = n) (simplify). Which “important distribution” is this conditional distribution, if any?
(c) Again let X and Y be Pois( ) r.v.s, and T = X + Y , but now assume now that X and Y are not independent, and in fact X = Y. Prove or disprove the claim that T ⇠ Pois(2 ) in this scenario.
- Consider four nonstandard dice (the Efron dice), whose sides are labeled as follows (the 6 sides on each die are equally likely).
A: 4, 4 , 4 , 4 , 0 , 0 B: 3, 3 , 3 , 3 , 3 , 3 C: 6, 6 , 2 , 2 , 2 , 2 D: 5, 5 , 5 , 1 , 1 , 1
These four dice are each rolled once. Let A be the result for die A, B be the result for die B, etc.
(a) Find P (A > B), P (B > C), P (C > D), and P (D > A).
(b) Is the event A > B independent of the event B > C? Is the event B > C independent of the event C > D? Explain.
- A discrete distribution has the memoryless property if for X a random variable with that distribution, P (X j + k|X j) = P (X k) for all nonnegative integers j, k.
(a) If X has a memoryless distribution with CDF F and PMF p (^) i = P (X = i), find an expression for P (X j + k) in terms of F (j), F (k), p (^) j , p (^) k.
(b) Name one important discrete distribution we have studied so far which has the memoryless property. Justify your answer with a clear interpretation in words or with a computation.
8 Stat 110 Midterm from 2010
- A family has two children. The genders of the first-born and second-born are independent (with boy and girl equally likely), and which seasons the children were born in are independent, with all 4 seasons equally likely.
(a) Find the probability that both children are girls, given that a randomly chosen one of the two is a girl who was born in winter (simplify).
(b) Find the probability that both children are girls, given that at least one of the two is a girl who was born in winter (simplify).
- In each day that the “Mass Cash” lottery is run in Massachusetts, 5 of the integers from 1 to 35 are chosen (randomly and without replacement).
(a) When playing this lottery, find the probability of guessing exactly 3 numbers right, given that you guess at least 1 of the numbers right (leave your answer in terms of binomial coe cients).
(b) Find an exact expression for the expected number of days needed so that all of the
5
possible lottery outcomes will have occurred (leave your answer as a sum, which can involve binomial coe cients).
(c) Approximate the probability that after 50 days of the lottery, every number from 1 to 35 has been picked at least once (don’t simplify, but your answer shouldn’t involve a sum).
