Independent Definition and 745 Threads

the solution for the infinite num case the problem is that i only could reach the condition that the coefficients are zero when i substituted n=0 , i am reaching two independent variables i am not sure what am i doing wrong that's preventing me from getting a similar result to them "that...

For part(b), My solution is, ##(c_1, c_2, c_3, c_4) = (c_1, 5c_1, -c_1, -3c_1)## They have taken the case that c_1 = -1, which gives their expression for a linear dependent function as they have shown. However, I'm confused that the functions are linearly dependent for any value of ##c_1 \in...

Good evening, I'm running into a little confusion on the second part of this problem due to finding two different formulas for calculating the uncertainty in multiplied quantities. The way that I was taught was something like this. If ##z = x \cdot y##, then: $$\dfrac{\delta z}{z} =...

I assume since ##X## and ##Y## are independent and their probability density functions are functions of ##x## $$f_{Z_1}=\frac{f_X(x)+f_Y(x)}{2}$$ and $$f_{Z_2}=\frac{f_X(x)\cdot f_Y(x)}{\int_{\mathbb{R}}(f_X(x)\cdot f_Y(x))dx}.$$ The divisions occur because it must be that...

It's a 4th-dimensional 4th-rank tensor so at first we have ##4^4=256## components. According to the book, Given that ##R_{iklm}=-R_{ikml}## 256 components reduces to 96. But I cannot see how. For one pair of i,k 16 components are dependent. We have 12 pairs of i,k(for ##i≠k## becsuse for i=k...

Suppose X and Y are random variables. Is it true that Cov (Z,K) = Cov(X,K)+Cov(Y,K) where Z=X+Y?

I keep coming across this descriptor, "two (or three) independent, non-interacting parts," in many books on QM (for example, Penrose's Shadows of the Mind). It is usually followed by a mathematical description (for example, state vector |A>|B>). I can wrap my mind around the quantum paradox of...

Hi guys, Can you give me some feedback on whether my calculation is correct? I applied the formula below (Boltzmann Distribution) but I didn‘t know what to use for the variable z. I don‘t even know if I used the correct equation. Can you help me further? The task is: Consider a system of...

Hello, I have been pondering on the following: we have data for blood pressure BP (response variable) and data about age and gender (categorical variable with two levels). We can build two linear regression models: $$BP=b_0+b_1 age+b_2 gender$$ $$BP=b_0+b_1 age$$ The first model does not take...

Reference; https://www.statisticshowto.com/probability-and-statistics/t-test/ My question is, can we as well have 'subtract each ##x## score from each ##y## score?' thanks. ...t-tests after all are easy to comprehend; as long as one knows the types; i.e 1. indepedent sample tests (compares...

There are 4 players numbered 1 to 4. There is a room with an entrance door to one side and exit door to opposite side. Inside the room, there are 4 boxes numbered 1 to 4. Inside each box, there is a chit containing a number (with equal probability) from 1 to 4 (inclusive). No two chits can have...

I'm studying Differential Equations from Tenenbaum's, and currently going through non-homogeneous second order linear differential equations with constant coefficients. Method of Undetermined Coefficients is the concerned topic here. I will put forth my doubt through an example. Let's say we are...

I was reading Mechanics by Landau and Lifshitz and I am confused when it is stated in chapter 2 section 6 that one of the integrals of motion is not independent and it can be considered an additive constant of time. Hence I tried searching it up online...

It's a topic that's been giving be a headache for some time. I'm not sure if/why/whether I can always consider velocities and (independent) coordinates to be independent, whether in case of cartesian coordinates and velocities or generalized coordinates and velocities.

Please bear with me I am trying to get a grip with underlying principles. Starting to try and understand Einstein’s second postulate and distinguish that the speed of light is independent of the speed of the source – v - of objects, other than light with travel initiated independent of the...

Is it because the current applied to a capacitor will never be time independent? Please help me out🙏

Can anyone out there give me a hint as to where to start with this problem? I've been looking at it for a while and can't see a way forward. What exactly is "the curvature itself" here?BTW I think the dynamic initial value equations 21.116 and 21.117 are incorrect. MTW should have inserted to...

We can formulate the spacetime in an observer/coordinate independent way, i.e. a particle becomes a worldline in the 4d space. Then relative to each observer, the worldline can be casted to a function in R^3. However, I haven't found any reference on formulating configuration space in a...

Hey! :giggle: We have the table of distribution of $X$, $Y$ and their joint random variable : with $$(c,d)\in \left \{(c,d)\in \mathbb{R}^2\mid 0\leq c\leq \frac{1}{4}, \ 0\leq d\leq \frac{1}{2}, \ \frac{1}{4}\leq c+d\leq \frac{1}{2}\right \}$$ I want to calculate the values of $c$ and $d$...

(This is not about independence of ##q##, ##\dot q##) A system has some holonomic constraints. Using them we can have a set of coordinates ##{q_i}##. Since any values for these coordinates is possible we say that these are independent coordinates. However the system will trace a path in the...

I'm as mainstream as a physicist can be, so you will not see my trying to publish things like "relativity is wrong" or "there is no proof of quantum mechanics". But I live and work in a 3rd world country with an awful policy for teaching and research, and the universities I work for only pay me...

I'm reading "Differential Equations with Applications and Historical Notes" by George F. Simmons and I am confused about something on pages 431-432 He has the second order ordinary differential equation $$\frac {d^2v} {d\phi^2} + \frac {\cos(\phi)} {\sin(\phi)} \frac {dv} {d\phi} + n(n+1)v = 0...

I attached a screenshot of the book (sorry no pdf available for this book). Right above the somewhat central line they give the theorem that if there are m currents and n nodes, then there will be n - 1 independent equations from the current law and m - n - 1 from the voltage law. I count 4...

Summary:: problem with independent events in possibilities A forest has a fire per year with probability 30% and it has 2 fires per year with possibility 5%.The probability of 3 or more fires is 0%.The forest might have disasters from the wind which blows with probability of 100% or has...

(I know how to prove it). Prove that a finite sum of of independent normal random variables is normal. I suspect that independence may not be necessary.

Hello, I am review some basics. In the case of two variables like ##t## and ##C##, which represent the time in year and the cost in $ of a product respectively, we can graphically represent this pair of variables on a Cartesian plane. The horizonal axis is generally assigned the variable ##t##...

Let A be invertible. Show that, if $\textbf{$v_i,v_2,v_j$}$ are linearly independent vectors, so are \textbf{$Av_1,Av_2,Av_3$} https://drive.google.com/file/d/1OuHxfUdACbpK4E5aca2oBzdaxGR0IYKv/view?usp=sharing ok I think this is the the definition we need for this practice exam question...

$\tiny{115.C51}$ find a linearly independent set T so that $\langle T\rangle =\langle S\rangle$ $S=\left\{ \left[\begin{array}{r}2\\-1\\2\end{array}\right], \left[\begin{array}{r}3\\0\\1\end{array}\right], \left[\begin{array}{r}1\\1\\-1\end{array}\right]...

Dear PF, In the figure down below is Q7.47 which asks to determine the voltage v(t) across the capacitor for t > 0. Since it is given that V(0) = 0 there are two scenario's which is between time interval 0 < t < 1 and t > 1 according to the independent sources. For the scenario 0 < t < 1 the...

Matrix multiplication is defined by \sum_{k}a_{ik}b_{kj} where ##a_{ik}## and ##b_{kj}## are entries of the matrices ##A## and ##B##. In definition of orthogonal matrix I saw \sum_{k=1}^n a_{ki}a_{kj}=\delta_{ij} This is because ##A^TA=I##. How to know how many independent parameters we have in...

I think there are 4 essential node. So we can write 3 KCL independent equations.

Hi everyone , I have done Bachelor's in Physics from Delhi University , India . I discontinued Master's from Indian Institute of Technology more-than-1 year before completion , it was 2 year course. I want to do research in Physics independently and publish ground-breaking...

Let ##X_1 X_2 X_3 ## be three independent random variables having Normal(Gaussian ) distribution, all with mean ##\mu##=20 and variance ##\sigma^2##=9. Also let ##S=X_1+ X_2 +X_3## and let ##N## be the number of the ##X_i## assuming values greater than 25. ##E\left[N\right]##=? I did not...

For a star.. Apparent Magnitude = -2.5log10 I K And I = I0/d^2 So in terms of I0... Apparent Magnitude = -2.5log10 (I0/d^2) And the Stefan-Boltzmann law says: Energy Flux = Sigma(T^4) In my reading it says that Intensity is the energy emitted per unit of area per unit of time. It says the same...

I am afraid I had no credible attempt at solving the problem. My poor attempt was writing the matrix ##\mathbb R## as a ##3 \times 3## square matrix with elements ##a_{ij}## and use the matrix form of the orthogonality relation ##\mathbb R^T \mathbb R = \mathbb I##, where ##\mathbb I## is the...

Hey! 😊 Let $1\leq n\in \mathbb{N}$, let $\mathbb{K}$ be a field, $V$ a $\mathbb{K}$-vector space with $\dim_{\mathbb{K}}V=n$ and let $\phi:V\rightarrow V$ be linear. - Let $0\neq v\in V$ and $1\leq m\in \mathbb{N}$, such that $\phi^{m-1}(v)\neq 0=\phi^m(v)$. Show that $\phi^0(v)=v, \phi (v)...

Hey! 😊 Let $\mathbb{K}$ a field and let $V$ a $\mathbb{K}$-vector space. Let $1\leq m, n\in \mathbb{N}$ and $n=\dim_{\mathbb{K}}V$. Let $v_1, \ldots , v_m\in V$ be linearly independent. Let $\lambda_1, \ldots , \lambda_m, \mu_1, \ldots , \mu_m\in \mathbb{K}$ such that...

Summary:: Roll a dice twice. Event A: First dice roll yield 2 or 5. Event B: Sum of the two results are atleast 7 Is A and B independent? If they are independent then $$P(A \cap B) = P(A) * P(B)$$ P(A) = 2/6 = 1/3 P(B) = 1 - 9/36 = 27/36 $$P(A \cap B) = 7/36 \neq P(A)*P(B) = 1/4$$Yet book...

In a previous thread it was concluded that both torque and angular momentum are taken about a chosen origin, and these quantities are generally not invariant under translations of the origin (since ##\vec{r}## changes but ##\vec{v}## does not, etc.). The moment of inertia tensor doesn't...

Most undergrad textbook simply say that it is intuitive that boundary conditions should not play a role if the box is very large. Other textbooks suggest that this should be taken for granted since the number of particles at the surface are orders of magnitude smaller that the number of bulk...

This topic is quite confusing. For instance, if I write ##14_6##, do I pronounce this as "ten", even though we'd probably just say "one-four, base six"? That is to say that we'd treat "ten" as a number of things that we could count out (i.e. corresponding to a certain number of sticks). Another...

When we calculate the curl of magnetic field, that is the curl of Biot-Savart equation for magnetic field. Please consider these . The working of last equation $$ \nabla \times \mathbf {B} = \frac...

Summary:: I attach a picture of the given problem below, just before my attempt to solve it. We are required to show that ##\alpha_1 \varphi_1(t) + \alpha_2 \varphi_2(t) = 0## for some ##\alpha_1, \alpha_2 \in \mathbb{R}## is only possible when both ##\alpha_1, \alpha_2 = 0##. I don't know...

I need to use some property of the relalation between the coordinate systems to prove that g_{hk} is independent of the choice of the underlying rectangular coordinate system. I will try to borrow an idea from basic linear algebra. I expect any transformation between the rectangular systems to...

Objects fall on Earth at 9.8 m/sec² independent of mass. If gravity is independent of mass why does Jupiter pull an object more than the Earth does? Is that inconsistency within the law or in my perception?

ok not sure what forum this was supposed to go in,,so... If the independent variable of $W(\theta)=2\theta^2$ is restricted to values in the interval [2,6] What is the interval of all possible values of the dependents variable?

Hey! :o We have that the vectrs $\vec{v},\vec{w}, \vec{u}$ are linearly independent. I want to check if the pairs $\vec{v}, \vec{v}+\vec{w}$ $\vec{v}+\vec{u}$, $\vec{w}+\vec{u}$ $\vec{v}+\vec{w}$, $\vec{v}-\vec{w}$ are linearly indeendent or not. Since $\vec{v}, \vec{w}, \vec{u}$...

The question is a bit confused, but it refers to if the following integration is correct : $$I=\int \frac{1}{1+f'(x)}f'(x)dx$$ $$df=f'(x)dx$$ $$\Rightarrow I=\int\frac{1}{1+f'}df=?\frac{f}{1+f'}+C$$ The last equality would come if I suppose $f,f'$ are independent variables.

How do I find the probabilty density function of a variable y being y=ab, knowing the probabilty density functions of both a and b? I know how to use the method to calculate it for a/b - which gives 1/pi*(a²/b²+1) - using variable substitution and the jacobian matrix and determinant, but which...

Hello everyone. Let us consider 3 events A,B,C such that: $$P((A \cap B )\cup C)=P(A)*P(B)*P(C)$$ Notice that the second term is a union and not an intersection. Are they independent? And what if the assumption was: $$P(A \cap( B \cup C))=P(A)*P(B)*P(C)$$? I know that the independence condition...