1.

a. Show that for any y ∈ Rⁿ, show that yy^T is positive semidefinite.

b. Let X be a random vector in Rⁿ with covariance matrix Σ = E[(X − E[X])(X − E[X])^T]. Show that Σ is positive semidefinite.

2. Let X and Y be real independent random variables with PDFs given by f and g, respectively. Let h be the PDF of the random variable Z = X + Y .

a. Derive a general expression for h in terms of f and g

b. If X and Y are both independent and uniformly distributed on [0, 1] (i.e. f(x) = g(x) = 1 for x ∈ [0, 1] and 0 otherwise) what is h, the PDF of Z = X + Y ?

Please show your work. Thanks!

Make a list of examples of Markov chains with different properties : irreducible, regular, has a limiting distribution, does not have a limiting distribution, does not have a limiting matrix, has infinitely many stationary distributions.

Please show SPSS steps and graph

Problem Set 2: A product developer wanted to know if the preferred type of graphics differs between video-gamers and non-gamers. He had 7 gamers and 7 non-gamers evaluate both types of graphics. The data is presented below, with higher numbers indicating a higher preference. Did graphic preference type differ between gamers and non-gamers? (Note – this is a small sample size so that you can practice selecting an appropriate statistical test and write-up results.)

1. Paste appropriate SPSS output. (2 pts)

2. Paste appropriate SPSS graph. (If there is no significance, create an appropriate graph to display the interaction of the two variables). (2 pts)

3. Write an APA-style Results section based on your analysis. All homework “Results sections” should follow the examples provided in the presentations and textbooks. They should include the statistical statement within a complete sentence that mentions the type of test conducted, whether the test was significant, and if relevant, effect size and/or post hoc analyses. Don’t forget to include a decision about the null hypothesis. (4 pts)

Interpret what this Truth table tells you. What’s your claim regarding Elmo and Big Bird’s colors using the truth table you constructed? Does the information (T and F) from your Truth Table align with reality? How is this process and constructing a Truth Table useful?

4.(12 pts) A 20,000 liter tank is initially half full with 10,000 liters of 2% nitric acid. A solution of 5% nitric acid is pumped in at a rate of 30 liters per minute and the well mixed mixture is pumped out at the same rate of 30 liters per minute.

(a)(8 pts) Set up (But Don’t Solve) an IVP whose solution gives the amount of x(t) of pure nitric acid (in liters) in the tank at time t

(b)(4 pts) Use a quick qualitative analysis of the IVP you found in part (a) to predict how much pure nitric acid will be in the tank as t→infinity(Show a phase line or direction field to predict what happens as t→infinity

spring-mass system has a spring constant of 3 Nm. A mass of 2 kg is attached to the spring, and the motion takes place in a viscous fluid that offers a resistance numerically equal to the magnitude of the instantaneous velocity. If the system is driven by an external force of 15cos(3t)−10sin(3t) N,determine the steady-state response in the form Rcos(ωt−δ).

R= _________

ω=__________

δ=___________

A dorm at a college houses 1900 students. One​ day, 20 of the students become ill with the ​flu, which spreads quickly. Assume that the total number of students who have been infected after t days is given by:

N(t)=1900/1+25e^-0.65t

​a) After how many days is the flu spreading the​ fastest?

​b) Approximately how many students per day are catching the flu on the day found in part​ (a)?

​c) How many students have been infected on the day found in part​ (a)?

2. (16 pts) Use binary modular exponentiation to find 45 mod 84. (Use either table approach taught in slides or the pseudocode algorithm.)












3. (20 pts) Use the Euclidean Algorithm to find the greatest common divisor of the integers

Question B: The following integral is important in a particular geometry problem

(you will learn later which geometry problem):

integral of sqrt( a^2 * cos(t) + b^2 * sin(t) ) dt from t=0 to t=2pi.

For each of the following methods, write down what happens as you try it, but don't spend very long on it.

i) Try a simple u-substitution.

ii) Try integration by parts.

iii) Try trig substitution.

iv) Any other ideas?

v) optional: what if b=a? Can you simplify it and solve it?

Let p be an odd prime and a be any integer which is not congruent to 0 modulo p.

Prove that the congruence x 2 ≡ −a 2 (mod p) has solutions if and only if p ≡ 1 (mod 4).

Hint: Naturally, you may build your proof on the fact that the statement to be proved is valid for the case a = 1.

. Suppose that the sequence (x_n) satisfies

|x_n –α| ≤ c | x_n-1- α|² for all n.

1. Show by induction that c | x_n- α| ≤ c | x₀ - α|²ⁿ , and give some condition

That is sufficient for the convergence of (x_n) to α.

2. Use part a) to estimate the number of iterations needed to reach accuracy

|x_n –α| < 10^-12 in case c = 10 and |x₀ –α |= 0.09.

Find two power series solutions of the given differential equation about the point x=0

(x^2+2) y″+3xy'-y=0

List the three types of radius of convergence. Give an example of each type

The annual National No Spying Day is celebrated at KAOS headquarters this year. There are 8 Control agents and 18 KAOS agents attending.

How many ways can we choose a team of 7 agents if 2 team members need to be from Control and 5 from KAOS?

How many ways can we choose a team of 7 agents if at least 1 team member needs to be from Control?