Description
Necessary Minimum Background Test [45 pts]
While you are welcome to use online resources, such as Wolfram-Alpha, you should be able to solve
these problems by hand.
1 Multivariate Calculus [2 pts]
Consider y = x sin(z)e
−x
. What is the partial derivative of y with respect to x?
3
2 Linear Algebra [8 pts]
Consider the matrix X and the vectors y and z below:
X =
2 4
1 3
y =
1
3
z =
2
3
(a) What is the inner product y
T z?
(b) What is the product Xy?
(c) Is X invertible? If so, give the inverse; if not, explain why not.
(d) What is the rank of X?
4
3 Probability and Statistics [10 pts]
Consider a sample of data S obtained by flipping a coin five times. Xi
, i ∈ {1, . . . , 5} is a random
variable that takes a value 0 when the outcome of coin flip i turned up heads, and 1 when it turned
up tails. Assume that the outcome of each of the flips does not depend on the outcomes of any of
the other flips. The sample obtained S = (X1, X2, X3, X4, X5) = (1, 1, 0, 1, 0).
(a) What is the sample mean for this data?
(b) What is the unbiased sample variance ?
(c) What is the probability of observing this data assuming that a coin with an equal probability
of heads and tails was used? (i.e., The probability distribution of Xi
is P(Xi = 1) = 0.5,
P(Xi = 0) = 0.5.)
(d) Note the probability of this data sample would be greater if the value of the probability of
heads P(Xi = 1) was not 0.5 but some other value. What is the value that maximizes the
probability of the sample S? [Optional: Can you prove your answer is correct?]
(e) Given the following joint distribution between X and Y , what is P(X = T|Y = b)?
P(X, Y )
Y
a b c
X
T 0.2 0.1 0.2
F 0.05 0.15 0.3
5
4 Probability axioms [5 pts]
Let A and B be two discrete random variables. In general, are the following true or false? (Here
Ac denotes complement of the event A.)
(a) P(A ∪ B) = P(A ∩ (B ∩ Ac
))
(b) P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
(c) P(A) = P(A ∩ B) + P(Ac ∩ B)
(d) P(A|B) = P(B|A)
(e) P(A1 ∩ A2 ∩ A3) = P(A3|(A2 ∩ A1))P(A2|A1)P(A1)
6
5 Discrete and Continuous Distributions[5 pts]
Match the distribution name to its formula.
(a) Gaussian (i) p
x
(1 − p)
1−x
, when x ∈ {0, 1}; 0 otherwise
(b) Exponential (ii) 1
b−a when a ≤ x ≤ b; 0 otherwise
(c) Uniform (iii)
n
x
p
x
(1 − p)
n−x
(d) Bernoulli (iv) λe−λx when x ≥ 0; 0 otherwise
(e) Binomial (v) √
1
(2π)σ2
exp
−
1
2σ2 (x − µ)
2
