Description
1 Counting (5 pts)
(a) Two cards are drawn from a deck of 52 cards without replacement:
(i) What is the probability that the second card is a heart, given that the first card is a
heart?
(ii) What is the probability that none of the cards are hearts, given that at most one card is
a heart?
(b) One card is selected from a deck of 52 cards and placed in a second deck containing 52 cards.
A card is then selected from the second deck.
(i) What is the probability that a card drawn from the second deck is an ace?
(ii) If the first card is placed into a deck of 54 cards containing two jokers, then what is the
probability that a card drawn from the second deck is an ace?
(iii) Given that an ace was drawn from the second deck in (ii), what is the conditional probability that an ace was transferred from the first deck?
2 Probability and conditional probability (4 pts)
(a) Suppose that 30 percent of computer owners use an Apple machine, 50 percent use a Windows
machine, and 20 percent use Linux. Suppose that 40 percent of Apple users have succumbed
to a computer virus, 76 percent of Windows users get the virus, and 55 percent of Linux users
get the virus. We select a person at random and learn that their system was infected with the
virus. What is the probability that the person is a Windows user?
(b) There are three cards. The first is green on both sides, the second is red on both sides, and the
third is green on one side and red on the other. Consider the scenario where a card is chosen
at random and one side is shown (also chosen at random). If the side shown is green, what is
the probability that the other side is also green?
3 Probability distributions (5 pts)
(a) Let X be a random variable with discrete pdf f(x) = x
8
if x = 1, 2, or 5 and zero otherwise.
(i) Sketch the graph of the discrete pdf f(x).
(ii) Find E[X] and V ar(X).
(iii) Find E[2X + 3].
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(b) The form of the Bernoulli(p) distribution is not symmetric between the two values of X. In some
situations, it will be more convenient to use an equivalent formulation for which x ∈ {−1, 1},
in which case the distribution can be written as:
P(x|p) =
1 − p
2
(1−x)/2
1 + p
2
(1+x)/2
Show that this distribution is normalized (i.e., sums to 1) and evaluate its mean and variance.
4 Independence (5 pts)
(a) Prove the following:
If A and B are independent events, then P(A|B) = P(A).
(b) Prove the following:
If A and B are conditionally independent given Z, that is, P(A, B|Z) = P(A|Z)P(B|Z), then
P(A|B, Z) = P(A|Z).
(c) A box contains the following four slips of paper, each having exactly the same dimensions:
(1) win prize 1, (2) win prize 2, (3) win prize 3, (4) win prizes 1, 2, and 3. One slip
will be randomly selected. Let A1 = win prize 1, A2 = win prize 2, and A3 = win prize
3. Show that A1, A2, and A3 are pairwise independent, but that the three events are not
mutually independent (i.e., P(A1 ∧ A2 ∧ A3) 6= P(A1)P(A2)P(A3)).
5 Expectation (5 pts)
(a) Let X1, …, Xn ∼ Bernoulli(p=0.5). Let Yn = max{X1, …, Xn}.
(i) Find E[Yn].
(ii) Plot E[Yn] as a function of n.
(iii) How is the distribution of the max (Yn) different from that of a single Bernoulli (Xi)?
(b) You and your friend are playing the following game: two dice are rolled; if the total showing is
divisible by 4, you pay your friend $12. If you want to make the game fair, how much should
she pay you when the total is not divisible by 4? A fair game is one in which your expected
winnings are $0.
6 Conditional Expectation (4 pts)
Consider the setting where you first roll a fair 6-sided die, and then you flip a fair coin the number
of times shown by the die. Let D refer to the outcome of the die roll (i.e., number of coin flips)
and let H refer to the number of heads observed after D coin flips.
(a) Suppose the outcome of rolling the fair 6-sided die is d. Determine E[H|d] and V ar(H|d) for
all possible values of d.
(b) Determine E[H] and V ar(H).
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7 Covariance and Correlation (6 pts)
(a) Show that if E[X|Y = y] = c for some constant c, then X and Y are uncorrelated.
(b) Show Cov(X, Y + Z) = Cov(X, Y ) + Cov(X, Z).
(c) Let X1 and X2 be quantitative and verbal scores on one aptitude exam and Y1 and Y2 be
corresponding scores on another exam. If Cov(X1, Y1) = 5, Cov(X1, Y2) = 1, Cov(X2, Y1) = 2,
and Cov(X2, Y2) = 8, what is the covariance between the two total scores X1 +X2 and Y1 +Y2?
8 Distance and Correlation Measures (5 pts)
(a) Show how Euclidean distance can be expressed as a function of cosine similarity when each
data vector has an L2 length of 1.
(b) Show how Euclidean distance can be expressed as a function of correlation when each data
point has been standardized by subtracting its mean and dividing by its standard deviation.
9 Linear Algebra (5 pts)
(a) Specify whether the following matrix has an inverse without trying to compute the inverse:
(Recall that a matrix A is invertible if and only if det(A) 6= 0.)
9 1 9 9 9
9 0 9 9 2
4 0 0 5 0
9 0 3 9 0
6 0 0 7 0
(b) Find eigenvalues and eigenvectors of the following matrix:
A =
0 1
−2 −3
10 Statistical Inference (6 pts)
Suppose you obtain N data points X = {x1, x2, …, xN } from a normal distribution whose variance
is δ
2 and mean is unknown.
(a) What is the maximum likelihood estimation of the normal distribution’s mean value µ?
(b) If the prior distribution for µ is a normal distribution with mean value of η and variance of λ
2
,
i.e., µ ∼ N(η, λ2
), what is the maximum a-posteriori estimation of µ?
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