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MAT9004 Mathematics Foundation For Data Science

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MAT9004 Mathematics Foundation For Data Science

0 Download3 Pages / 750 Words

Course Code: MAT9004

University: Monash University

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Country: Australia

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.

Answer

is the derivative of with respect to

power rule, subtraction and additional rule we obtain is the derivative of with respect to , addition and the subtraction rule we obtain Since then

The stationery points are at point

Hence the stationery points will be at the roots of Using the quadratic equation We obtain the roots of the function at the points The value of at the stationery points will be

Finding local minimum and local maximum

The stationery points are at

Now we use the sign test to determine if the points are maximum or minimum

-2

-1

0

0

-12

+ve

-ve

Using -2 and 0 to do the sign test we can see that the sign changes from positive to negative. When the sign changes from positive to negative, then this indicates that we have a local maximum. Hence point (-1,7) is a local maximum point.

Now using the test sign, we test point (-2,6) using the values -2 and 0

-3

-2

-1

-2

0

-ve

+ve

The sign is changing from negative to positive hence the point (-2,6) is a local minimum.

Graphing the function f gives

From this graph we can observe two points which can be classified as global maximum and minimum. That is point; (-1,7) and (-2,6)

Eigenvalues of M

Using the characteristic polynomial

eigenvalues as

And

Finding eigen vector

For every we find its own vectors so, we have a homogeneous system of linear equations we solve it by Gaussian Elimination

Find the variables from the equation of the system (1) so, we have a homogeneous system of linear equations, we solve it by Gaussian Elimination to obtain

Diagonalize M

The diagonal matrix (the diagonal entries are the eigenvalues

The matrix with the eigenvectors ( as its columns

Calculate probability

The profit function is

The optimal points are at the derivative of the profit function equals to zero.

The derivative with respect to b gives

The derivative with respect to a we obtain b

Since the value of b is minimum we take -3

This gives the value of a

Not divisible by 2 and not divisible by 5.

This means the last digit cannot be

Now we must pick 4 digits from a bucket of 10

This can be picked in the following way

The fist number can be picked in 9 ways

2nd in 10 ways

3rd in 10 ways

4th in 4 ways

Hence the total prime like numbers between 1000 and 9999 is

Has at least one even digit

Here we select 4 numbers from a bucket of 10 in the following way

1st number 9 ways

2nd number 10 ways

3rd number 10ways

4th number 5 ways

Now let’s assume that the even digit is in the 1st number then the numbers will be

Leta now assume the even number is in the second digit

Let’s assume the even number is in the 3rd digit

The even digit cannot be in the fourth number as this will make the number prime like anymore

In total the numbers are

Prime like numbers not divisible by 3

Here we must choose the four numbers such that the sum of the digit is not divisible by 3

1st digit 9 ways

2nd digit 10 ways

3rd digit 10 ways

4th digit 3 ways

The numbers are

Y uniformly distributed

Y

-1

0

1

2

2

0

0

2

Since Y is uniformly distributed all the values occur with equal probability the expected value of expected value of

Y

-1

0

1

2

0

0.5

0

0.5

Since all the values occur with uniform probability the expected value will be

Variance of

The variance is 0

Variance of

The variance is 0

Testing independence of and

No, the equations are dependent as they share similar variance

A fair 6-sided dice is rolled 3 times

Probability that sum is 8

Possible outcomes are

Each roll must have a value between 1 and 6

The only way we can have a sum of 8 is

The roll is 3 times so there are

Thus, the probability is

Probability of 1 given sum of 8

Drawing graph

The adjacency matrix is given by the graph does not have a spanning tree.

No, there is no connection that can allow all eight students to seat in a way that any of them knows both neighbours.

References

McQuarrie, D., 2003. Mathematical Methods for Scientists and Engineers, s.l.: University Science Books.

Salas, S. L., Hille, E. & Etgen, G. J., 2007. Calculus: One and Several Variables. 10th ed. s.l.:Wiley.

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