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MAT9004 Mathematics Foundation For Data Science

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Question:

where x ∈ [−2, 3].
(a) What is f
(b) What is f
(c) Find all the stationary points of f. Find the value of f(x) at each stationary point. [2]
(d) For each stationary point of f, find whether it is a local minimum, a local maximum or neither. [2]
(e) Find all the global minima and maxima of f in the interval [−2, 3]. [2]

2. Let M =

(a) Find the eigenvalues of M. [3]
(b) Find one eigenvector of M with respect to each eigenvalue. [3]
(c) Diagonalize M, i.e., write M in the form M = NDN −1
, where D is a diagonal matrix. [2]
3. X is a continuous random variable with density f where f(x) = x + x f(x) = 0 elsewhere. What is the probability of the event {X > 1/2}? [6]
4. Suppose you want to launch your startup in some innovative area. Currently, there exist two enterprises A and B competing for this market. You want to find out their investment strategies (which are given by real numbers a, b) prior to commiting into the competition. A friend who works in A told you that their profit is given  but unfortunately he does not know the values of a, b. He also mentioned that both enterprises are apparently not very good in keeping secrets, so the function f(x) = p(x, b) achieves the maximum value at point x = a, while x = b minimises g(x) = p(a, x). Use this information to find a and b [8]

5. A positive integer is called prime-like if it is not divisible by 2 and not divisible by 5.
(a) How many prime-like numbers with 4 digits (from 1000 to 9999)? [2]
(b) How many prime-like numbers with 4 digits have at least one even digit? [3]
(c) How many prime-like numbers with 4 digits are not divisible by 3? [4]

6. Let Y be a random variable uniformly distributed on the set {−1, 0, 1, 2}. Let U1, U2 be the random variables defined by U1 = Y

(a) Find the expected values of U1 and U2. [2]
(b) Find variances of U1 and U2. [2]
(c) Is U1 independent of U2? Justify your answer.

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