Normal Definition and 1000 Threads

Let X and Y be independent normal random variables each having parameters $\mu$ and $\sigma^2$. I want to show that X+Y is independet of X-Y without using Jacobian transformation. Hint given by author:- Find their joint moment generating functions. Answer: Now Joint MGf of...

When light, traveling through a rarer medium, gets reflected off at an interface with a medium of higher refractive index, it suffers a phase shift of half-wavelength. Now if it was normal incidence, the phase-shifted light would retrace its path. Does that mean the wave would cancel out?

I can't prove this proposition. I have however managed to prove that the linear combinations of the independent normal rv's are also normal by looking at it's mgf $$E(e^{X_1+X_2+...+X_n})=E(e^{X_1})E(e^{X_2})...E(e^{X_n})$$ The mgf of a normal distribution is $$e^{\mu t}e^{\frac{t^2...

Homework Statement I'm trying to figure out the reaction at D,that's where the system can rotate. Problem: A force P of magnitude 90 N is applied to member ACDE, which is supported by a frictionless pin at D and by the cable ABE. Since the cable passes over a small pulley at B, the tension...

Hello good people of PF, I came across this problem today. Problem Statement Given bivariate normal distribution ##X,Y \sim N(\mu_x=\mu_y=0, \sigma_x=\sigma_y=1, \rho=0.5)##, determine ##P(0 < X+Y < 6)##. My Approach I reason that $$ P(0 < X+Y < 6) = P(-X < Y < 6-X)$$ $$ =...

Hello. I wonder how to see if the distribution is normal or not based on the numbers in the table. I attach the screenshot with the table, and it is written that "the distribution appears to be bimodal", so it appears to have two peaks. I understand the the question sounds silly, but I truly...

Homework Statement A vector n of magnitude 8 units is inclined to x,y and z axis at 45, 60 and 60 degrees resoectively.If the plane passes through (root2, -1, 1) and is normal to n then find its equation. Homework Equations (r-a).n=0 where r is position vector of a point on plane, a is a point...

Hello there! I am Yanna and I am building a racing car. I am now trying to calculate the tire drag of this car when cornering, but I'm stuck on something. In order to find the tire drag, I need the normal load on each wheel. To find this, I made some equations: 1) the sum of the normal load...

I read that 1. dark matter has to be concentrated in galaxies; 2. McGaugh & Co discovered a precise relationship between visible-ordinary matter and the calculated sum of ordinary + dark matter from thorough observation of actual acceleration of more than 150 galaxies. 3. First experiments to...

Homework Statement Part a of the image below. When I try to solve this question, I can't get the solution in the answer key. In the answer key the y component of the normal force is NCos(alpha) in the last line. Does the normal force not point perpendicular to the ramp? Homework Equations...

Homework Statement A car is lifted vertically by a jack placed at the car's rear end 40cm off the central axis, so that the weight of the car is supported by the jack and the two front wheels. The distance between the front wheels is 1.60m, the distance from the axis connecting the two wheels...

Homework Statement (Scroll to bottom for the true question) Suppose we are to find the integral from -∞ to +∞ of (let’s just say) e-x2dx Homework Equations ∫∫f(x)g(y)dxdy = (∫f(x)dx)(∫g(y)dy) The Attempt at a Solution We can square the integral we want to solve for then use my relevant...

Homework Statement "A helicopter drops rocks onto a desert until it hits a marked bullseye. With respect to Cartesian coordinates whose origins is at the bullseye, both the x- and y-coordinates are normally distributed. Denote the r.v.'s taking on values along these axes as ##X,Y##...

Homework Statement Dylan is rushing to catch a flight. As he walks in the airport he pulls his suitcase behind him, which is rolling on its wheels. The mass of the suitcase is 20.0kg. Dylan pulls on the handle so that he exerts a force on the handle of 100N, 30° above the horizontal. Determine...

Write an expression for normal distribution for the data: Measured values are: 4,393; 4,372; 4,381; 4,373 and 4,401 thanks for help

Homework Statement A block of uniform density experiences a force F to the right, applied 5/3 m from the bottom of the block. The block is 2 m high and 1 m wide. Take the pivot point to be the point at the bottom right of the block. Find the value of the force that is just able to tip the box...

Homework Statement Back for more questions. This section has been pretty tricky. The graph of a folium with equation ##2x^3+2y^3-9xy=0## is given. Find the equation for the tangent line at the pont ##(2,1)## Find the equation of the normal line to the tangent line in the last question at the...

Suppose ##X## ~ N(##\mu##,##\sigma^2##). Then ##\bar{X}## ~ N(##\mu##,##\frac{\sigma^2}{n}##), where ##\bar{X}## is the random variable for sample mean for samples of size ##n##. But when the population variance ##\sigma^2## is unknown and the sample size ##n## is small, ##\bar{X}## no longer...

Hi guys, For my (master) project I am trying to find the probability that a random variable, which is normally distributed, exceeds a quantile that is estimated by a limited number of observations. See attached for my attempt. - Is it correct? - How to incorporate the fact that the mean and...

Homework Statement Let ##G## be a group and let ##A \subseteq G## be a set. The normal subgroup of ##G## generated by ##A##, denoted ##\langle A \rangle ^N##, is the set of all products of conjugates of elements of ##A## and inverses of elements of ##A##. In symbols, $$\langle A \rangle ^N= \{...

Homework Statement A group is called Hamiltonian if every subgroup of the group is a normal subgroup. Prove that being Hamiltonian is an isomorphism invariant. Homework EquationsThe Attempt at a Solution Let ##f## be an isomorphism from ##G## to ##H## and let ##N \le H##. First we prove two...

In this PDF, http://www.math.unl.edu/~bharbourne1/M417Spr04/M417Exam2Solns.pdf, in answering why a subgroup of index 2 is normal, the author says that the only two cosets must be ##A## and ##gA##. Why so? Why there can't be another element ##g'## such that ##G = g' A + g A##?

Homework Statement For each natural number ##n##, let ##V_n## be the subset of the symmetric group ##S_n## defined by $$V_n = \{(i j)(k l) | i,j,k,l \in \{1,\ldots, n\}, i \neq j, \text{ and } k \neq l\},$$ that is, ##V_n## is the set of all products of two 2-cycles. Let ##A_n## be the...

What is the preferred method of measuring how accurate the normal approximation to the binomial distribution is? I know that the rule of thumb is that the expected number of successes and failures should both be >5 for the approximation to be adequate. But what is a useful definition of...

Hi. I'm reading a paper "Transmission of light through a single rectangular hole in a real metal" and the author refers to the incident light shown below as "p-polarized" without further specification. Note that ax > ay. Is there any convention in regarding a certain polarization as...

Hi. I was wondering what is the relationship between Gaussian normal coordinates and Riemann normal coordinates. Thanks.

Homework Statement A 180lb pilot flies a verticle loop with radius 2000 ft at 350 mi/h. With what force does the seat fress upward against him at the bottom of the loop? Problem 6.21 Homework Equations ##F=ma## ##F_c=\frac {mv^2} {r} ## The Attempt at a Solution There seems to be a...

How do i suppose to determine the uncertainty for the slope of my Static friction against normal reaction graph? My data for static friction and normal force has the uncertainty of +/- 0.0001 The uncertainty is too small for me to draw airbox/bar in the graph to draw the max and min slope...

Hello, Let us imagine a solid immersed in a liquid in a container such that the density of the liquid is less than the density of the solid. This means that the solid must sink. Let us study the solid when it reaches the bottom surface and is now at rest. The forces acting on the solid when it...

Homework Statement My question is not a homework problem. I am having difficulty in understanding the concept oof contact forces. Homework Equations What I don't understand is that when two bodies A and B are in contact and left on a frictionless surface, they apply normal reactions on each...

If I am given a function f( x , y , z , ...) = C then the normal direction to it is simply the (unit vector of the) divergence of the function. How has this been proven?

hi guys i recently had built a HV power supply out of an old Flyback transformer i had laying around , i think it produces about 10KV ~ 15KV not very sure but it could arc at about ( 6mm in air ) , and as you can see i tried to connect it to a 15W smal light bulb ( the positive terminal...

Hi all, I am trying to understand two things from a paper The ratio of a standard normal by the square root of a a chi squared divided by its df ( degrees of freedom) is a t distribution. So 1) What is the dist of square root of Chi squared? I know a normal squared is a chi squared, but a chi...

Let $$\phi(x^1,x^2...,x^n) =c$$ be a surface. What is unit Normal to the surface? I know how to find equation of normal to a surface. It is given by: $$\hat{e_{n}}=\frac{\nabla\phi}{|\nabla\phi|}$$However the answer is given using metric tensor which I am not able to derive. Here is the answer...

I know that normal force is not a reaction force to the gravitational force; - First, because the gravitational force is from Earth on the object (action force) thus the reaction of force is simply form the object on Earth according to Newton's third law. - Also because in certain situations...

Homework Statement Homework EquationsThe Attempt at a Solution I've resolved all the relevant forces (tension, weight, normal force), but upon looking at the mark scheme, there is no normal force to be resolved. Forgive me if I'm missing out on something obvious, but shouldn't there be a...

Homework Statement Given $$\phi = x^{2} +y^{2}-z^{2}-1 $$ Calculate the unit normal to level surface φ = 0 at the point r = (0,1,0) Homework Equations $$ \hat{\mathbf n} = \frac{∇\phi}{|\phi|}$$ $$ z = \sqrt{x^{2}+y^{2} -1} $$ $$ \mathbf n = (1,0,(\frac{\partial z}{\partial x})_{P})...

Thought experiment: single use of "magic" Setup: Let's assume we have a giant ball of water in space. Magic: Let's assume the water does not compress its center under its own gravity. (Constant density of 1 g/cm^3) Basic stuff: The mass of this ball of water, (since it does not compress)...

I'm looking at what should be just a simple spring system where four identical springs are holding up a square, load-bearing pallet plate in a warehouse. Now, someone says the equation of motion for the vertical normal mode of vibration is simply d2z/dt2 = -4(k/m)z. Right away however, I see no...

Homework Statement Homework Equations f=ma v^2/r The Attempt at a Solution ∑FX= FN ∑Fy= -mg We are not given an angle, i would have put FNcos(θ). I have my radius as 0.25 by relation of the triangles at that point of the height being .50m so ∑FX= FN = v^2/r = (1.2)^2/.25 = 6.25N

If we take 1 high gravity object and 1 smaller object at a distance from the first object, the first is stationary in relation to the smaller object, and it is in orbit around the first. Will the orbit be circular the center of mass of the two? Now if we put an other object in a different orbit...

Hello, I don't know how to compute that number, but probably is a kind of reference value... Q: How many ion-pairs are there in normal conditions* in the air? (This is not homework, it is just out of curiosity.) Thank you for your time. Regards, ORF * 1 atm, 300K.

I"m wondering, at the abstract level, why different mathematics is used to calculate the Normal Force in an inclined plane versus a banked turn (which a vehicle is driving around). For an inclined plane, the standard approach is take weight and resolved into parallel and perpendicular vectors...

Homework Statement Homework Equations direction of normal and shear stress The Attempt at a Solution I view it like A and C are the same; B and D are the same... as both shear stress of A and C points outwards and shear stress point to the corner with 135 deg vice versa for B and D what...

AIUI, every normal matrix has a full eigenvector solution, and there is only 1 *normalized* modal matrix as the solution (let's presume unique eigenvalues so as to avoid the degenerate case of shared eigenvalues), and the columns of the modal matrix, which are the (normalized) eigenvectors, are...

Homework Statement A 5.0kg block of wood sits on a table with a string running over a pulley suspending a mass. The largest mass that can be suspended without moving the block of wood is 0.75kg. a) Calculate the coefficient of friction in this case. b) What type of coefficient of friction has...

Homework Statement Mass of block: 19kg Applied force: 225N at 16 degrees to accelerate block from rest Coefficient of static: 0.55 Coefficient of kinetic: 0.30Homework Equations Unknown The Attempt at a Solution I have solved for parallel tension and perpendicular tension Tperp =...

I am a bit confused. We have an inclined plane. So I know that Ffmax=μ*N and μ is the coefficient of friction calculated as tgθ (θ is the angle at which the body is on the verge of sliding) and also Ff<=Ffmax And we know that the force that makes the body slide is mgsinθ. So as the angle...

When analyzing hydrological and climatological timeseries/observations it is common practice to compare statistics made on normal periods. As WMO calls them "WMO Climatological Normals". These periods consist normally of 30 years of data. If you want to compare two normal periods with each...

Homework Statement I have an empty cylinder with an external diameter of (23.0 ± 0.5) mm, an internal diameter of (22.5 ± 0.5) mm and a height of (60.0 ± 0.5) mm. I need to calculate its volume with its uncertainty/error. Homework Equations The Attempt at a Solution I do it like this...