0
votes
3answers
77 views

Using random number generator to draw from population

Suppose X is exponentially distributed, f(x) = e^(x/10)/10. How would you use a random number generator to generate a sample of observations from this population?
0
votes
2answers
46 views

Geometric Series..

You are offer $2$ lucrative investment schemes. The $1$st scheme awards £$4000$ on the 1st day and every subsequent day this amount is increased by £$8000$. The $2$nd scheme awards £$0.01£ on the ...
2
votes
3answers
48 views

Find the derivative of $\frac{(2x−1)e^{−2x}}{(1−x)^2}$

I need to find the derivative of $$\frac{(2x−1)e^{−2x}}{(1−x)^2}$$ I seems very complex to me so I'm wondering if there is a rule or formula I should be using? I attempted it using the chain rule ...
0
votes
4answers
58 views

For all $x$ in the set of reals, $ |x-2| > 2 \implies x^2 > 4x$.

How do i go on proving this statement? The first step i took was to assume the antecedent. so.. assume $|x-2| > 2$ then $x > 4$ or $x < 0$ if we assume the consequence, $x^2 > 4x$ ...
2
votes
1answer
114 views

Draw a circle tangent internally to a point of an ellipse and to both its axis

I would like to know how could I draw a circle that touches an allipse internally and at the same time both of its axis. There are, thus, four possible circles of this kind, all of them congruent, ...
0
votes
3answers
66 views

How to generically solve polynomial expressions given a minimum value of x?

Given something like $\dfrac{8x^2 + 2x + 7}{3x^2 + 2x}$, and $x > \sqrt8$, what strategy would you employ to simplify this expression?
0
votes
2answers
124 views

Use binomial theorem to calculate n power of a sum (MIT Single Variable Calculus)

During the first lecture of the MIT course (Single Variable Calculus - rate of Change, 46:00) professor David Jerison uses the binomial theorem to explain the following function: $(x + \Delta x)^n = ...
0
votes
1answer
1k views

Markov matrices: finding the initial state vector

I am wondering how can I find an initial state vector for this problem. If the air quality is good one day, it has 95% chance it will be good the next day. If the air quality is bad one day it has ...
1
vote
1answer
17 views

Probability Strategy

A charity fund-raiser has found that active soliciting gains \$5 contributions with probability .7 and passive soliciting gains \$10 contributions with probability .4. No other contributions are made. ...
5
votes
0answers
94 views

Why does the group $\mathbb{Z}^d \rtimes \mathbb{Z}^k$ have exponential growth?

I apologize if this is posted on the wrong forum, I was reading this question on MathOverflow which asks about "growth rate of $\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$". The last comment of the ...
0
votes
1answer
46 views

An simple algebra problem

Could someone check if my algebra is right? If $x_n = \dfrac{1}{a^{a^n}}$ where $a > 1$ and $a$ is natural number, what is $$\lim_{n \to \infty} \frac{|b_{n+1}|}{|b_n|^R}$$? Where $b_n = x_{2n}$ ...
2
votes
3answers
80 views

Stochastic in finance

I need of a undergraduate guide level to study stochastic process with finances. Starting from a review of probability theory. Eg books, papers or posts. I'll apreciate some help.
0
votes
3answers
39 views

Derivative Confusion

I am confused about something. In derivation we learnt that; a^x = a^x . lna Now the question that comes to mind is what is the difference when we have: a^3 =
0
votes
2answers
458 views

how to differentiate a function with square root

Trying to solve $y =7t^4-10 \sqrt {t+\frac{10}{t}}$ I know how to differentiate down to $7(4t^3)- . . .$ and I know a sqrt is equal to $x^.5$ but cannot figure out how to apply that to the rest of ...
0
votes
1answer
48 views

Finding a Householder matrix for row elimination

I was wondering how to find a Householder matrix such that I could apply it from the right side of a matrix and eliminate values along a row. For example, I have a matrix of the form B = ...
0
votes
3answers
209 views

For all sets A, B and C, if $A\setminus(B \cup C) = \emptyset$ ; then $A\setminus C\subseteq B$.

For all sets A, B and C, if $A-(B \cup C) = \emptyset$ ; then $A-C\subseteq B$. Is it true? If it is, how to prove it? I think it's true...
1
vote
2answers
56 views

Sampling from a Normal Distribution

If I am sampling randomly from only the -sigma to +sigma interval of a normal distribution and rejecting all other numbers, does it imply that the probability density changes? If so, by what degree? ...
1
vote
1answer
60 views

Solution of “quadratic equation” involving functional coefficients.

Suppose I have a "quadratic equation" whose coefficients are functions of the variable to be solved for: $$f(x)x^2+g(x)x+h(x)=0,$$ with $f,g,h\neq 0$. Would it make sense to apply the quadratic ...
1
vote
2answers
45 views

Convolution product

How can we prove that if $f$ is compactly supported and $g$ is periodic with period $P$ then $f*g$ exists and is also P-periodic ? thanks.
6
votes
1answer
134 views

How to compute the mean average exponent of the naturals? What is the limit for large numbers?

With a friend I was trying to get an understanding for why the expected gap between primes is logarithmic. With that motivation I tried to express the average exponent of numbers. By average ...
2
votes
0answers
128 views

Clarification in stochastic integration

In the book "Stochastic Processes" by Bass R.F. when he constructs the Stochastic Integral, at some point he defines for $Y$ predictable $$||Y||_2= \left(\mathbb E \int_0^{\infty}Y_t^2\text{d} \langle ...
1
vote
2answers
93 views

What is the primitive of $\frac{\ln(1+x)}{x}$

I'm trying to evaluate this integral on $0 \le x \le 1$, but substition and by parts doesn't work here. $$\int {\frac{\ln(1+x)}{x}dx}$$ By Taylor series, we have: ...
1
vote
1answer
92 views

Combinatorics of the coupon collector's problem

I have a set of N numbers. How many possible M-combinations - with M > N - of the N numbers are there which contain every number at least once? Example: Numbers {1,2} and hence N = 2 and M be 3. ...
2
votes
2answers
73 views

Checking multivariable function's differentiability at $(0,0)$

I have a function: $$f(x,y) = \begin{cases} (x^2+y^2)\sin (x^2+y^2)^{-1} \ \ \ &\text{, if } x^2+y^2 \neq 0 \\ 0 &\text{, if } x^2+y^2=0 \end{cases}$$ I calculated the partial derivatives: ...
0
votes
1answer
76 views

Estimating the number of connected components of a curve contained in a given set

Let $X$ be a metric space and $\Omega\subset X$ an open set. Take $x\in\Omega$ and choose $r>0$ such that the open ball $B(x,r)\subset B(x,2r)\subset \Omega$. Let $\gamma:[0,1]\to X$ be a Lipschitz ...
0
votes
2answers
45 views

$P(X>16|X>10)$ - normal distribution

If $X$ is a normal random variable with parameters mean = 10 and standard deviation= 6, compute $P(X>16|X>10)$ ? Can someone help to explain this $P(X>16|X>10)$ in the normal rv. term? ...
0
votes
1answer
55 views

Volume of pyramid where we know the sides

A triangular pyramid has five sides of length $2$, and another side of length $\sqrt6$. What is the volume of the pyramid?
0
votes
5answers
109 views

How to integrate this integral $\int(1-r)/(1+r)^2$??\

How to evaluate this integral $$\int \frac{1-r}{(1+r)^2} \ \mathrm{d}r$$ I was trying to do by parts however it doesn't seems to work here.
2
votes
1answer
80 views

Combination Problem with mulitiple variables

I am new to this, but getting into math more and have a question regarding combinations and permutations with several variables involved. I work for a sales company and this question is based on ...
1
vote
0answers
112 views

Integration property proof (upper and lower Riemann Integral)

Consider $f: I =[a_1,b_1]\times...\times[a_n,b_n]\subset R^n\to R^n$ I have to prove the following property: $f$ is integrable $\Longleftrightarrow \int_\underline{I} f=\int_\overline{I}f $ (Sorry ...
2
votes
2answers
66 views

Constructing a certain bijection

I'm not sure at all which theorem(s) could be applied in order to get started on the following problem. Any suggestion is very much appreciated. Suppose $k$ is a field and $A$ is a finitely ...
2
votes
1answer
109 views

An equality that holds with $v_t \in L^2(0,T;L^2(\Omega))$ but its proof requires $v_t \in L^2(0,T;H^1(\Omega))$

Let $Q=(0,T)\times \Omega$. For all $\varphi \in C_c^\infty(Q)$ such that $0 \leq \varphi \leq 1$, the following holds $$\int_Q \varphi^2 (\Delta v)v_t = \int_Q |\nabla v|^2 \varphi \varphi_t - ...
0
votes
1answer
29 views

which are the correct answers?

let$\{a_n\}_{n ≥ 1}$ be a sequence of positive numbers such that $$a_1 >a_2>a_3>...$$ then which of the following are always true ? $\lim_{n \to \infty}a_n=0$ $\lim_{n \to ...
-1
votes
3answers
50 views

Linear Algebra, find determinant with x1, x2,…,xn as scalars

I have no clue how to even begin solving for $\det(A)$ since $n$ is unknown, HELP!
1
vote
2answers
136 views

Convolution of finite measures

I am puzzled by the following (maybe very stupid) question I stumble upon in the course of a project: let $p$ be a probability measure on some abelian group $E$ (actually, $E=\mathbb{Z}_n$ with its ...
0
votes
1answer
41 views

$A \in L(V)$ where $V$ is a FDVS. Prove that there is an invertible $Q$ such that $AQ$ is a projection.

On P. 94 of Halmos' FDVS, you will find the following theorem: Corresponding to any linear transformation $A$ on a finite-dimensional vector space $V$, there is an invertible linear transformation ...
5
votes
1answer
339 views

groups with same number of elements of each order

When two groups which have the same number of elements of each order are isomorphic? Can we characterize them? I already know the abelian $Z_{p^2}\times Z_p$ and non-abelian $Z_{p^2}\rtimes Z_p$ have ...
1
vote
0answers
47 views

Integral Domain and PID Proof

Prove that, in a domain, $(a)=(b)$ iff $a = bu$ for some unit $u$. By $(a)=(b)$, it also means that $a\mid b$ and $b\mid a$ so we can write them as $a=bu$ and $b=av$ for some $u, v \in R$ where $u$ ...
0
votes
1answer
64 views

How would I solve this question

I'm currently in year 10 and revising for a mock higher paper, which will take place tomorrow. I'm stuck, how would I go about answering this question?
2
votes
1answer
53 views

Help solving a challenge - relational algebra or second order logic

I am a self-taught man and I'm posting my first question here. I'm facing a challenge I'd like to solve. Based on what I know it fits propositional calculus (hope it is). Suppose 3 people: a ...
0
votes
2answers
71 views

Very elementary number theory

Prove that gcd(m+1,n+1)|mn-1 ,where m and n are integers. I have already done: (m+1)(n+1)=mn+m+n+1 Clearly gcd(m+1, n+1)|(m+1)(n+1) And obviously gcd(m+1),(n+1)|(m+1)(n+1) Therefore ...
2
votes
1answer
60 views

Divisibility and Principal Ideal Domain Proof

Let R be a ring. Show that a|b iff $b \in (a)$ iff $(b) \subseteq $ (a). I first just want to write out what I know about this statement: a|b means that a divides b or a is divisible by b and there ...
1
vote
1answer
58 views

Evaluating integral with trigonometric function. [closed]

Evaluate the following integral. $$ \int 2^x~\tan^9(2^x)~\sec(2^x)~\mathrm{d}x $$ The homework hints say that i should make $u = 2^x$. I did but I couldn't continue.
0
votes
1answer
46 views

Nice proof for $\lim_{h\to 0}\frac{f(x+nh)-f(x)}{h}=nf'(x)$ besides LHR

Why is $$\lim_{h\to 0}\frac{f(x+nh)-f(x)}{h}=nf'(x)?$$ A cheap answer would be L'Hospital's rule, but I think there should be a more direct way to prove it, appealing to the first principles of the ...
0
votes
2answers
23 views

which one is correct of these options?

for the following set of simultaneous equations: $ 1.5x-0.5y=2 $ ; $4x+2y+9z=9$ ; $ 7x+y+5z=10 $ the solution is unique infinitely many solutions exist the equations are not compatible finite ...
0
votes
1answer
26 views

How do you use induction on a recursive sequence using different variables?

I've been working on some recursive sequences for my Discrete class. I've understood most of them, but I've come to a new question which I'm confused about. A sequence $C_{0}$, $C_{1}$, $C_{2}$ is ...
0
votes
1answer
64 views

More transcendental numbers than natural numbers [duplicate]

Are there any simple proofs that obtain this result? I haven't been able to find one online.
0
votes
1answer
140 views

keeping c1 continuity in joining several bezier curve

I have some complex curves, I separate the long curves to smallest one to be able to fit them with Bezier curve. However, my Bezier curve has no C1 continuously, if I force C1 continuously, my curves ...
1
vote
0answers
658 views

How to numerically find Floquet multipliers (e.g., characteristic multipliers or Lyapunov exponents for periodic orbits from chaotic systems)?

Anyone have any suggestions for the following situation/question? (help wanted, please!) I understand the theory (c.f., Perko or Nayfeh and Balachandran, Ch.3), but I do not understand how this is ...
0
votes
1answer
122 views

How would I go about solving this probability question?

I am currently in year 10 studying GCSE higher maths. My once issue with maths is probability. I have an exam tomorrow and am currently revising by going through past papers, I saw this question and ...

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