Homework questions are welcome as long as they are asked honestly, explain the problem, and show sufficient effort. Please do not use this as the only tag for a question. For the answers on homework questions, helpful hints or instructions are preferred to a complete solution. Please do not add ...

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0
votes
1answer
25 views

Applying Jensen's formula to polynomials?

Prove that $$\frac{1}{2\pi}\int_{0}^{2\pi}|f(e^{i\theta})|^2d\theta=\sum_{k=0}^n|c_k|^2$$ for each polynomial $f(z)=\sum_{k=0}^nc_kz^k$. The hint given by the homework is: show first that for integer ...
0
votes
0answers
30 views

Show that $Y_i$ is independent of $Y_j$ for any $i$ not equal to $j$

Let $\{X_1,X_2,\ldots\}$ be independent, identically distributed, absolutely continuous random variables. Let $Y_n=I\{X_n>\max(1< i < n)\}$ for $n=2,3,\ldots$ a) Show that $Y_i$ is ...
3
votes
3answers
72 views

Evaluate $\int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x$

My try, using $x = \sec(u)$ substitution: $$ \begin{eqnarray} \int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x &=& \int \frac{\sqrt{\sec^2(u) - 1}}{\sec(u)}\tan(u)\sec(u) \mathrm{d}u \\ &=& ...
0
votes
1answer
37 views

M/M/3 queue - reducing wait time by adding servers

Full question below: You are the manager of the customer support division in your company. Your division uses 3 telephone lines operated by 3 separate customer service representatives. A customer is ...
-1
votes
1answer
50 views

Complexification the real inner product space

Let $V$ be a real inner product space. If $W=V\times V$ with the operations $(u_1,v_1)+(u_2,v_2)=(u_1+u_2,v_1+v_2)$ and $(\alpha +i\beta)(u,v)=(\alpha u-\beta v,\alpha v+\beta u)$, where $u, ...
-6
votes
0answers
71 views

Please give feedback to my answers [on hold]

Problem A Question 1 Are the boolean functions(p\wedge\thicksim q)\vee(\sim r\wedge q) and (p\vee\thicksim q)\wedge(r\vee\sim q) equal?Explain your answer. Solution The boolean function ...
0
votes
2answers
30 views

Discrete Mathematics: Prove that f(x) is in O(x)

Prove that $$\frac{2x^{2}+x}{x+1}$$ is in $O(x)$
0
votes
0answers
4 views

Non-monotonic function but Homothetic function

Is it possible for a function to be non-monotonic, but still homothetic? Thank you for your explanations.
8
votes
5answers
273 views

Why does $n^0 = 1$?

Why is it that $n^0 = 1$? I understand how $n^2 = n*n$ and how $n^1 = n$ but I can't understand why $n^0 = 1$.
-3
votes
1answer
25 views

what is the Number? [on hold]

A car has a number of 4 digit.last digit is the multiple of first digit.second and third digits are same.and last 2 digits are the multiple of first 2 digits.what is the number of car?
1
vote
1answer
69 views

Grade 11 functions [on hold]

I need help with three questions for my homework. Any answers would be appreciated. Please try to explain steps or just show them, as I would like to know how to do them. Thanks. Determine if the ...
0
votes
0answers
22 views

Duality gap problem

I have checked that the objective function is concave and the constraint functions are convex. Now to find the duality gap, one need to find the optimum of the primal and dual problem, and find the ...
0
votes
0answers
22 views

About a practical problem [on hold]

One day a young man came to the store to buy king boss a gift, this gift cost is $ 18$, the price is $ 21$. The result is this young man took out $100$ yuan to buy this gift. Wong did not change, with ...
-2
votes
1answer
32 views

Calculate Summation of series

One of my homework: Compute $\sum_{i=0}^{k-1}\alpha_i - (k-1) $ given $\alpha_i$ is $\frac{k-1}{k}$ Answer given is zero. So i suspect that $\sum_{i=0}^{k-1}\frac{k-1}{k}=k-1$? But how can i get ...
-2
votes
0answers
23 views

Feedback please ! Question : state the claim of the theorem and its logical notation [on hold]

Claim: A theorem of number theory that if every even number is greater than $6$ Logical notation: $\forall n\ge 6\implies(\exists a,b\,| 2n=a+b)$
0
votes
0answers
32 views

Proving equality with finite and pairwise disjoints

I'm having some problems proving this. Let $A_1,A_2,.....A_n$ be finite and pairwise disjoints. So any two sets are disjoint. How do we prove that $$|A_1 ∪ A_2 ∪ ....A_n| = |A_1|+|A_2|+....|A_n|$$
1
vote
1answer
23 views

$X$ and $Y$ have a joint distribution density function. Working out a marginal density function for $X$ and $Y$

$f_{X,Y}(x,y) = \frac{3}{2}(x^2+y^2)$ if $0 \lt x \lt 1$ and $0 \lt y \lt 1,$ or $0$ otherwise. I want to find the marginal probability density function of $X$ and $Y$ and then find $Pr(0 \lt x \lt ...
0
votes
0answers
16 views

Expected value of joint p.d.f. with unknown constant

X and Y are random variables that have a joint p.d.f. $$p(x,y)=c*(x^9)*(y^6)$$ when $0<=x, y<=1$ and $c>=0$ is a constant that should be found. What is the expected value of $Y$? I am having ...
1
vote
1answer
44 views

Prove or find a counter-example

Please give me feedback for my answer to this question. Question: Prove or find a counterexample: The product of any three consecutive natural numbers is divisible by 6 My answer: True, Suppose n is a ...
2
votes
1answer
12 views

Expected value, variance and probability from a joint distribution function

Lets say I am given the following table that shows the joint probability function of X and Y: $$\begin{array} \\{}&y=1&y=2&y=3 \\x_=1&0.1&0.2&0.1 ...
1
vote
1answer
40 views

Probability of the sum of independent standard normal random variables

Let $X_1, X_2, X_3, X_4$ be independent standard normal random variables and $$Y = X_1^2 + X_2^2 + X_3^2 + X_4^2$$ Find the probability that $Y \leq 3$. For this problem I know that the ...
1
vote
0answers
52 views

Number theory claim

Can someone please give me a feedback on my response to one of my homework. We were asked to look for any claim on the theorem of numbers and to write that claim in logical notation. My claim: For ...
0
votes
2answers
236 views

Maximum volume of parallelepiped

Find the dimensions of the parallelepiped of maximum volume circumscribed by a sphere of radius R. I would normally be familiar with this using lagrange multipliers, but how do I do this? It ...
2
votes
1answer
11 views

Joint distribution probabilities

I have a question that is similar to the following(made up here): The construction of a tower of cards is done is two stages, procrastination and the actual building. The time in minutes needed to ...
2
votes
1answer
40 views

Question about vector fields and Lie group

Notation: $\chi(G)$ is the set of smooth vector fields on Lie group $G$, which in fact forms a vector space. Given a Lie group $G$, show that there exists a smooth vector field $X\in \chi(G)$, ...
0
votes
2answers
73 views

Limits that approach zero. [closed]

I am so confused on this concept. When $$f(x) = \begin{cases}x-1 &, x < 0\\ 2x-1 &, x \geqslant 0, \end{cases}$$ what is $\lim\limits_{x\to 0} f(x)$?
0
votes
1answer
16 views

How to handle a double inequality where all 3 spots have unknowns

Given the problem $4x \lt 2x + 1 \le 3x + 2$ solve for x. I'm not sure how to go about solving this problem. No matter how I subtract or add the x's or multiply/divide the coefficients I cannot ...
1
vote
1answer
20 views

Equivalence Class.

Let R be the relation of congruence mod 4 on Z: a R b if a - b = 4k, for some k in Z. What integers are in the equivalence class of 31? How many distinct equivalence classes are there? What are ...
1
vote
2answers
29 views

Jacobian matrix with two equations

Evaluate the Jacobian for: $$f(x,y)=(x^2+x+y, yx+x^2)$$ at the point $(1,2)$.
1
vote
1answer
33 views

What are the limits for this joint pdf?

I'm given equation that the joint pdf is $(2/3)(x + 2y)$ when $0 < x, y < 1$ and we want to find the probability that $X < 1/3 + Y$. I understand how to do the actual math part, and that I ...
0
votes
1answer
23 views

Let $M$ be a bounded subset of the space $C_{[a,b]}$. Prove that the set of all functions $F(x)=\int^{x}_{a}f(t)dt$ with $f\in{M}$ compact.

Let $M$ be a bounded subset of the space $C_{[a,b]}$. Prove that the set of all functions $F(x)=\int^{x}_{a}f(t)dt$ with $f\in{M}$ compact. Some helpful definitions: bounded - A subset $S$ of a ...
0
votes
1answer
14 views

${\bf E}[Y]$ of a joint distribution

So, I have that a joint pdf is given by the formula: $$ 5e^{-5x} / x, \quad 0 < y < x < \infty $$ and I have to find the $Cov(X,Y)$. I know that $Cov(X,Y) = {\bf E}[XY] - {\bf E}[X]{\bf ...
1
vote
1answer
31 views

Let $S_n:= \frac{b-a}{n}\sum_{i=1}^{n}f(t_{i,n})$. Prove: $\lim_{n\to\infty}S_n = \int_a^bf(x)\ dx$.

I will post the assignment and then my attempt at solving it. Let $a,b \in \mathbb{R}$ with $a<b$ and let $f: [a,b] \rightarrow \mathbb{R}$ be a continous function. We'll now define a sequence ...
2
votes
1answer
50 views

what to do when the multivariable second derivative test is inconclusive?

What do we do when the second derivative test fails? How do we approach it, and is there a general method to further find whether a critical point is a maximum, minimum or a saddle point? For ...
1
vote
1answer
37 views

feedback on my answer regarding set intersections.

Prove or find a counter-example to the claim that for all sets $A,B,C$ if $A\cap B = B \cap C = A \cap C = \emptyset$, then $A \cap B \cap C=\emptyset $. the above statement is not true so i need a ...
1
vote
0answers
38 views

Logical expression

Please can someone give me feedback on my answer to the question below. Question. Surf the internet and find a theorem of number theory. State the claim of the theorem, and then express it in logical ...
0
votes
0answers
19 views

How can I write this in Divergence form

Consider the PDE $u_{xx}-(yu_y)_x-y(u_x)_y+yu_y+(y^2+\frac{1}{H^2(x)})u_{yy}$ I need to write this in divergence form. That is, I need to write it in the form $\sum_{i,j}\frac{\partial}{\partial ...
0
votes
3answers
43 views

proving or providing counter example in disrete mathematics

Prove or find a counterexample: The product of any three consecutive natural numbers is divisible by 6. if we take a few consecutive natural numbers such as 1 ,2 ,3. and multiply i get 6 which is ...
1
vote
1answer
583 views

Solving a recurrence realtion using backward substitution.

So I've been trying my best to do this, and I have made some good progress, I just need to know if what I have done is correct and if not, what the hell am I doing wrong? :P I start off with this ...
0
votes
2answers
34 views

Intermediate value theorem problem

Problem: The equation $x=-5\cos(x)$ has at least $3$ distinction solutions. Use the intermediate value theorem to show that this is true. I drew the function,but I don't know what to do next.
1
vote
1answer
17 views

finding conditional expectation under binomial distribution.

Suppose X and Y independent and are both binomial random variables with parameter N, p Compute E(X|X+Y).
0
votes
1answer
9 views

Solving a recurrence relation using forward substitution. [on hold]

How can I solve this? $$T(n)=3T\left(\frac{n}{4}\right),$$ for $n>1$, $n$ a power of $4$, and $T(1)=3$.
0
votes
1answer
21 views

Simplifying Trig Identity

I have an equation I have been given to solve, I know how to start but I do not know what to do after I use the Trig Identities. Any help? Here is what I was given $$ \frac{\cos(A + B) + \cos(A - ...
1
vote
1answer
80 views

Find the conditional expectation of N given that there were exactly 2 heads in the first 3 tosses.

We have two biased coins. The first one yields heads with probability 0.1 and the second one yields heads with probability 0.9. We choose one of the two coins randomly (with probability 0.5 each; we ...
0
votes
0answers
18 views

linear algebra - fourier coefficients of piecewise

Find fourier coefficients of given function: f(t) = {-1 if t $\leq$ 0; 1 if t > 0} so do I do this? $a_{0} = \int_{a+-\pi}^{a+\pi}1$, $a_{k} = \int_{a+-\pi}^{a+\pi}1*cos(kx)$, $b_{k} = ...
0
votes
0answers
8 views

Finite Difference Scheme for a PDE on non-rectilinear coordinates

Consider poisson's equation on the domain $0 \leq x \leq 1$ $0 \leq y \leq H(x)$. Change the coordinates to $\xi=x$, $\eta=y/H(x)$. Construct a FDS that gives a positive definite symmetric matrix. ...
0
votes
0answers
20 views

For which values of $t$, is $x$ moving to the left or the right?

$x=t^2-2t$ , $y=3t-2$ . I've already found tangent lines for these parametric equations, but how can I determine when $x$ is moving to the left or the right? Is it the second derivative?
-4
votes
2answers
32 views

Find the coordinates of all points that satisfy certain conditions. [on hold]

Find the coordinates of all points whose distance from $(-3,6)$ is $\sqrt{13}$ and whose distance from $(2,7)$ is $\sqrt{13}$.
0
votes
1answer
59 views

Number of surjective functions from $\{1,2,…,n\}$ to $\{a,b,c\}$

Ok so following questions are given in my text book Let $A = \{1, 2, 3,...., n\}$ and $B =\{a, b, c\}$ then the number of functions form $A$ to $B$ that are onto is. I have no idea how to find ...
0
votes
0answers
31 views

max and min values on symmetric polytope

Let $-N\leq t \leq N$. Let $A$ be regular $(N-1)$-dimensional simplex with vertices $(t,0, \ldots, 0)\ldots (0, 0,\ldots, t)$ and $B$ be regular $(N-1)$-dimensional simplex with vertices $(t-N+1,1, ...