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Questions tagged [self-learning]

The process of studying mathematics without formal instruction. Don't use this tag just because you were self-studying when you came across the mathematical question you're asking; it is only for when the fact that you're self-studying is what your question is about.

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3answers
38 views

How to evaluate $\sum_{ r=1}^{16}(5r-7)$?

I'm self-studying from Stroud & Booth's "Engineering Mathematics" and in the "Binomials" chapter, one of the last exercises is to evaluate: $$\sum_{ r=1}^{16}(5r-7)$$ This has got me confused, ...
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1answer
18 views

Defining New States to Make it A Markov Property [on hold]

Is there a way to make a process that depends on two time points a markov property, I need help on spotting the new states to be defined I just have problems associating probabilities to them. Please ...
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1answer
27 views

I need to know if.my example for this problem solved

Let $f(x)=\sup(x^3, x^2+1)$, $I=[1,4]$ $\sup(x^3)=64,\ \inf(x^3)=1$ $\sup(x^2+1)=17,\ \inf(x^2+1)=2$ So $\sup(x^3,x^2+1)=\sup(64,17) =64$ Is this correct, can I have good example.
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0answers
28 views

Where can I study polynomials online?

I have one good book on polynomials. It is called (Level of Knowledge of Polynomials) by its author S. L. Tabachnikov. In principle, I could study it, but I am still mastering the Agebra I. Not long ...
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3answers
65 views

I am stumped on this problem

It is from Intro to Analysis by Bartle 3rd ed Let $I:=[a,b]$ let $f:I\rightarrow\mathbb{R}$ be a continuous function with $f(x)>0$ Prove that there exists a number $c>0$ such that $f(x)\geq c ...
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2answers
18 views

Is there a way to write this term different $e^{x+e^x}$?

I am a little confused what this actually means: $e^{x+e^x}$ It is obviously not the same if I for example $$e^{x}:= \lambda \\ e^{x+e^x} \neq \lambda^\lambda $$
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0answers
33 views

Is the zero truncated Poisson Distribution part of the Exponential Family?

This is the density of a truncated Poisson: $$P(X = x \mid X > 0) = \frac{\lambda ^ x e^{- \lambda} }{x ! \left ( 1 - e^{- \lambda} \right )}$$ To show that it's member of the Exponential ...
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2answers
33 views

Convergence of indicator functions series

Say $E$ is a measurable set, and $\{E_k\}$ is a series of measurable sets defined by $$ E_k \subset E, m(E \setminus E_k) < \frac{1}{k}, k = 1, 2, 3, ... $$ Do their corresponding indicator ...
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0answers
32 views

Puzzles and exercises to improve mathematical intelligence and spatial thinking

In your childhood or adolescence, or maybe as an adult, have there been types of exercises or puzzles that you think have improved your mathematical intelligence and in particular the spatial thinking?...
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0answers
29 views

How do we prove that $\sum_{i=1}^{n} X_{i}$ is a sufficient statistic in the present context?

Let $n$ items be drawn in order without replacement from a shipment of $N$ items of which $N\theta$ are bad. Let $X_{i} = 1$ if the $i$-th drawn is bad and $X_{i} = 0$ otherwise. Show that $\sum X_{i}$...
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1answer
36 views

Show that $\min\{X_{1},X_{2},\ldots,X_{n}\}$ is sufficient for $\mu$ when $\sigma$ is fixed

Let $X_{1},X_{2},\ldots,X_{n}$ be a sample from a population with density $p(x,\theta)$ given by \begin{align*} p(x,\theta) = \frac{1}{\sigma}\exp\left\{-\left(\frac{x-\mu}{\sigma}\right)\right\} \...
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1answer
19 views

Projection of point to plane using normal equation

Say the place is parametrized as $x_1 = y_1 + y_2$ , $x_2=y_1-y_2 , x_3 = y_1+y_2$ and a point in $\mathbb{R}^3$ $x_1 = 2 , x_2 = 1 , x_3 = 3$ What would be the most efficient way to find the ...
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1answer
45 views

I am an illiterate at math [closed]

I'm a math illiterate starting from precalculus. I'm 19 and I somehow got into an engineering college in India. I really want to learn math passionately. I tried EdX, MIT opencourse, Khan academy, ...
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0answers
36 views

A coin is tossed until a head turns up and $\$f(n)$ is paid out, find the expected value of the payment

A fair coin is tossed until heads turns up for the first time. If heads turns up for the first time on the nth toss you receive $2^n$ dollars. (a) Show that the expected value of your winnings ...
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0answers
14 views

The relation between the F1 score and the combined work formula?

I am wondering why we should use the harmonic mean(as stated in Wikipedia: "the traditional F-measure or balanced F-score (F1 score) is the harmonic mean of precision and recall") as the F1 measure, ...
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2answers
28 views

Probability question : Normal distribution

An auto parts company, produces cylinder liners for engines of 1.2 inches in average diameter with a standard deviation of 0.1 inches. Every piece has a diameter less than an inch or more than 1.4 ...
7
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1answer
134 views

What is the best way to study graduate level mathematics?

I am studying a 400/500 level measure theory math book on my own. Right now, when I read it I try to read the proposition then the following proof. And then try to do the exercises on my own. I ...
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2answers
24 views

Need help in finding bounds on inequality

Determine the condition on $|x-2|$ that will assure that $|x^2-4|<1/2$ will work So $|x^2-4|=|x+2||x-2$| Do I assume that $|x-2|<1$ or $|x-2|<0.5$ I would like to see full details. Can ...
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1answer
43 views

Why $C(u_1,v_1) + C(u_2,v_2) - C(u_1,v_2) - C(u_2,v_1) \geq 0$ is equivalent to $\frac{\partial^{2}C(u,v)}{\partial u\partial v} \geq 0$?

A bivariate function $C(u,v)$ that maps $[0,1]^{2}$ to $[0,1]$ is a copula if it satisfies the following two conditions: (i) Boundary conditions: \begin{align*} C(u,0) = 0\\ C(0,v) = 0\\ C(u,1) = u\\ ...
1
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1answer
50 views

Average approximates definite integral.

In my studies, I read a result similar to the following. Let $g:[0,1]\rightarrow \mathbb{R}$ be twice continously differentiable on $[0,1]$ (this is probably stronger than necessary). Then \...
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1answer
47 views

Moment Generating Function of beta ( Hard )

Given $X$ is a random variable ~ $Beta ( a , b)$ distribution and $X$ belongs in (0,1) Does the (MGF ) $E[e^{tx}]$ exist for every value of $a , b$ ? (Mgf must not be equal to infinity in order ...
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2answers
44 views

How do i solve P(A\B∪C)

I just started set theory is math and with lack of understanding I've come here for some much-needed assistance. How would I go about determining what the probability is if its A given B or C? I've ...
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0answers
23 views

Integral Is not finite?

How can we prove with calculations that the below integral is not finite ` for $t\ge$ $\dfrac{1}{2}$ $\int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}}e^{-x^2(1/2-t)}\, dx.$ I would be pleased if you ...
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0answers
38 views

Where do I find exercises very similar to this one?

In my book Operational Research by author Hamdy A. Taha is this following exercise A home supply store can place orders for fridges at the start of each month for immediate delivery. A cost of $\$ ...
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0answers
28 views

Rotations of Lebesgue measurable sets are Lebesgue measurable

If $E \subseteq \mathbb{R}^n$ and $Q:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is an orthogonal map, then I know that $\mu^*(QE)=\mu^*(E)$, i.e., the Lebesgue outer measure is invariant under rotations. ...
0
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0answers
17 views

Deriving the cumulative distribution and probability density for the following random variables

Given U is randomly chosen from [0,1] with uniform distribution, find the CDF and PDF of Y given: a) $Y = \frac{1}{2+U}$ b) $Y = ln(U+1)$ a) $P(\frac{1}{2+U} \leq y) = P(U \geq \frac{1}{y}-...
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0answers
26 views

Algorithm to optimize redistribution of balls amongst urns

Here is the question: Say we have k urns with 1 ball in each urn. At each iteration of the game, I pick one urn and redistribute its contents amongst other urns and each urn can receive at most one ...
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2answers
28 views

When do exponential functions form a basis?

I understand that exponential functions of the form $e^{a_{i}x}; i=1,2,..,n$ and $a_{i}s$ are real and distinct are linearly independent. Do these form the basis of vector space of continuous ...
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1answer
25 views

Need clarification on question in Bartle’s Intro to real analysis 3rd ed

On page 85 he estimates the following for the cubic equation $x^3-7x+2=0$ His estimate:$|x*-x_0|<\frac{3^{n-1}}{7^{n-2}}*20$ I don’t understand how the estimate for the denominator is achieved ...
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0answers
16 views

Find the profit of a store from given instructions using entropy and probability

A restaurant sells N number of different dishes (Bi where i= 1,2,3,...N). Now we assume that ingredients of each dish to be sold is independent and Assuming the estimated amount of each dish or ...
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0answers
51 views

Karlin Rubin Theorem UMP (Uniformly most powerful test ) Is it wrong?

Suppose $X_1, X_2, X_3,\ldots, X_n$ are i.i.d. random variables with a common Poisson$(\lambda)$ distribution. $$X=(X_1, X_2, X_3,\ldots, X_n)$$ and $g(λ)=\lambda(1 - e^{-λ})$ , with $(λ>0)$ ...
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1answer
18 views

Epsilon transition in NFA to DFA conversion

I worked through this conversion and it all makes sense except for one small part. Shouldn't $(q_1q_2)$ go to $q_1$ in the DFA on input $0$, not a self loop? We have state $q_iq_2$ to begin with ...
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6answers
366 views

Solving $3x^2 - 4x -2 = 0$ by completing the square

I can't understand the solution from the textbook (Stroud & Booth's "Engineering Mathematics" on a problem that involves solving a quadratic equation by completing the square. The equation is ...
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0answers
27 views

How to show that $E(\hat{Y}| Y = y) = E(\theta |Y=y )$?

Show that $$E(\hat{Y}| Y = y) = E(\theta |Y=y ),$$ where $\hat{Y}$ is conditionally independent to Y given $\theta.$ Note that $E(\hat{Y}| Y = y) $ is equal to the mean of the predictive ...
2
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1answer
34 views

Solving simultaneous equations via $u$ and $v$

I'm self-studying from Stroud's amazing "Engineering Mathematics" textbook, and have run into an issue in one of the final end-of-chapter exercises for in the "Linear Equations" chapter. Here's the ...
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3answers
25 views

Simplyfying each side separately and solving linear equation

I'm given an equation that first needs simplifying: $$\frac{x-2}{x-4} - \frac{x-4}{x-6} = \frac{x-1}{x-3} - \frac{x-3}{x-5}$$ My next step is: $$\frac{(x-2)(x-4)(x-6)}{x-4} - \frac{(x-4)(x-4)(x-6)}{...
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1answer
23 views

Why $f(x)=x^2$ is local Lipschitz, general question about local/global Lipschitz.

I'm trying to understand the difference between global lipschitz and local lipschitz. Let $f(x)=x^2$ while $x \in \mathbb{R}$ if we look at global lipschitz, for all $M \subset \mathbb{R} \times \...
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2answers
35 views

When can we Central Limit Theorem approximation with good approximation?

I think we an use it when n(no. of trials) is large. But my textbook used this approx. by stating that since the expectation is large, we use the approx. I'm unable to understand this, would ...
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3answers
491 views

Can't solve system of linear equations (that need simplification first)

I'm self-studying from Stroud & Booth's amazing "Engineering Mathematics", and am stuck on a problem at the end of the "Linear Equations" chapter. I've a system of two linear equations: $$\frac{...
0
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1answer
14 views

Dart board probability using line method with Poisson application

You randomly throw darts at a dartboard, one dart every second. Suppose that every dart independently hits the dartboard at distance X from the center, where X is a Unif[0,30] random variable. Your ...
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1answer
37 views

find P( X$\lt$ Y) given X and Y are two independent exponential RV

The question is as follows: You arrive at the post office and as you enter, each of the two clerks, Jim and Jack, starts serving a client. The amount of time needed by Jim to serve his client is X ...
0
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1answer
21 views

Let $(X,Y)$ be a uniformly chosen point of a region $A$, given $A$ compute $EX$

Let $(X,Y)$ be a uniformly chosen point of a region $A \subset \mathbb{R}^2$ Given we have the following joint pdf: $$ f(x,y) = \begin{cases} \dfrac{a}{ \text{area of}~ A} & (x,y)\in A \\ 0 &...
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1answer
32 views

let z1 and z2 be independent standard normal RV, find the pdf of $e^{3Z_1+2Z_2}$

Let Z1 and Z2 be independent standard normal random variables. Find the following The probability density function of $e^{3Z_1+2Z_2}$ The given solutions is as follows: But what doesn't make sense ...
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1answer
31 views

How to understand this probability equation?

$\mathbf { x } ( t ) = g ( \mathbf { s } ( t ) ; \xi ) + \mathbf { n } ( t )$, where n(t) denotes the noise or modeling error and ξ the parameters of mapping $g$ How to understand the following ...
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1answer
28 views

Boundary of the sum of an Infinite horizon discounted model

The sum of an infinite horizon discounted model is given as follows: $$R_t = r_{t+1} + \gamma r_{t+2} + \gamma^2r_{t+3} + ... = \sum_{k=0}^\infty \gamma^kr_{t+k+1}.$$ As can be seen, the sum is ...
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0answers
48 views

probability that sum of geometric random variables is odd

Let $X$ and $Y$ be independent geometric random variables with the same parameter $p$. What is the probability that $X+Y$ is odd? Based on the provided solutions, I can follow through until this part ...
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2answers
40 views

PMF of function of random variables

Let X be a Geom($\frac{1}{2}$) random variable, and define Y=$X^{-1}$ What is the p.m.f. of Y ? attempt: pmf of a Geom RV in general form is $p(1-p)^{k-1}$ There is this similar question, not ...
4
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1answer
158 views

Ars longa, vita brevis.

There's little use studying mathematics without actually doing mathematics. There is a plethora of exercises in any textbook worth its salt. I suppose those with some business in looking up something ...
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2answers
32 views

Coin game between A and B

Consider the following simple game. In a single round, Player A tosses a fair coin, and then Player B tosses a fair coin. Two rounds are played. The winner is the player with the larger number of ...
2
votes
2answers
42 views

Identifying simple tensors.

Let $S$ be a domain. I want to determine whether or not, every element of $\text{Frac(S)}\otimes_S M$ is a simple tensor, where $M$ is any $S$-module. I couldn't produce a tensor that is not pure in ...