1 Basic Calculus [10 pts]
The following questions test your basic skills in computing the derivatives of univariate functions,
as well as applying the concept of convexity to determine the properties of the functions.
(a) (3 pts) Find all extrema of the function f(x) = ln(2 x2). For each extremum, state if it is
a maximum or a minimum.
(b) (3 pts) Show that f(x) = ln 1
1+e x is concave.
(c) (4 pts) Show that f(x) = e x2
is neither convex nor concave.
2 Continuous Random Variables [10 pts]
(a) (2 pts) Given a continuous random variable X with probability density function f(X), what
are the expressions for the mean and variance of this variable?
(b) (2 pts) Can the value of the probability density function (PDF) f(X) exceed 1? Why or why
(c) (2 pts) Consider a random variable X that follows the uniform distribution between a and b,
i.e. its PDF is equal to a constant c on this interval, and 0 otherwise. Derive c in terms of a
(d) (2 pts) Derive the expected value of X in terms of a and b. Show all your steps.
(e) (2 pts) Derive the cumulative distribution function F(X) on the interval a X b.
3 Discrete Random Variables [10 pts]
(a) (2 pts) Two students taking a Machine Learning class became project partners. They are
trying to decide what operating system to use for the project. Suppose each student has a
laptop, which could be one of three types: Mac OS, Windows, or Linux. If the distribution of
laptops among students follows the PDF shown below, what is the probability that the two
teammates have dierent laptops?
Mac OS 0.6
Suppose we have three discrete random variables x, y and z that take values 0 or 1 according
to the distribution below.
x = 0
z = 0 z = 1
y = 0 0 1
y = 1 1
x = 1
z = 0 z = 1
y = 0 1
y = 1 0 1
(b) (2 pts) Find the joint distribution of y and z
(c) (2 pts) Find the marginal distributions of y and z
(d) (2 pts) Find the conditional distribution of x given that y = 0.
(e) (2 pts) Are y and z independent? Explain.
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