Probability density Definition and 273 Threads

Homework Statement Calculate the probability current density vector \vec{j}for the wave function \psi = Ae^{-(wt-kx)}. Homework Equations From my very poor and beginner's understanding of probability density current it is : \frac{d(\psi...

Hi! I am currently studying Probability Density Functions and I am having a hard time wrapping my head around something. So, from what I have read, P(X=c), i.e. probability that the random variable X takes on any specific outcome, is equal to 0. Yet, the probability X takes on any outcome...

Homework Statement I need to find the probability density function given the eigenfunction Homework Equations \psi=C\exp^({\frac{ipx}{\hbar}-\frac{x^2}{2a^2}}) The Attempt at a Solution I tried to square the function but that gave me a nasty integral that I could not solve. I...

Hi guys. I'm trying to get the idea of probability density for 1s hydrogen atom. I just don't understand that probability density reaches maximum at nucleus (r → 0) if the most probable radius where electron can be found is at Bohr radius according to radial probability (Which also states...

A needle on a broken car speedometer is free to swing and bounces perfectly off the pins at either end, so if you give it a flick it is equally likely to come at rest at any angle between 0 and pi. If the needle has a length r, what's the probability density ρ(x) of the x-coordinate of the...

Homework Statement Let the probability density function##f(x) = (3/4) \cdot (1-x^2)## if x is between -1 and 1, and let ##f(x)=0## otherwise. What is the probability of ##P(X \leq 0.8 | X>0.5)##? Homework Equations The Attempt at a Solution I assume I have to rewrite the p.d.f. into a joint...

Homework Statement Given that, in free space the probability density for a wave function (free particle) is \mid \Psi(x,t)\mid^2=P(x,t)=\frac{\sigma_0}{\mid \alpha \mid^2\sqrt{\pi}}exp(-(\frac{\sigma_0}{\mid \alpha \mid})^4\frac{(x-x_0-p_0t/m)^2}{\sigma_0^2}) What is need to be done is to...

Dear all, Greetings! I was given a problem from Reichl's Statistical Physics book. Thank you very much for taking time to read my post. Homework Statement The stochastic variables X and Y are independent and Gaussian distributed with first moment <x> = <y> = 0 and standard deviation...

Homework Statement The stochastic variables X and Y are independent and Gaussian distributed with first moment <x> = <y> = 0 and standard deviation σx = σy = 1. Find the characteristic function for the random variable Z = X2+Y2, and compute the moments <z>, <z2> and <z3>. Find the first 3...

In class we went over the probability density for an object on a pendulum, and how at the lowest energy states, you would have strange distributions such as the object being more likely to be found at the bottom. But as you increase the energy level, the wave equation becomes more and more like...

Homework Statement Consider a particle whose wave function is: \Psi(x)=\left\{\begin{array}{ccc} 2\alpha^{3/2}xe^{-\alpha x} & \text{if} & x> 0\\ 0 & \text{if} & x\leq 0 \end{array}\right. Calculate <p> using the \hat{p} operator on probability density in x space. Homework...

http://gyazo.com/02812d5d8f1d07c72153c9f66740e147 I've dealt with integrals with infinity before. When considering the part x >= 1 , do I take the limit as if it's a very large number? i.e. ## \int_0^{\infty} x^{-2.5} \ dx = 2/3 ## ?

I am trying to create a simple implementation of the Bayes decision rule with minimum error criterion and I am running into a problem. Specifically, if I have a data set consisting of a number of feature vectors stored in rows, how can I generate a probability density function from this data...

Hello PF! It's been a while. How are things? In my research I'm faced with determining a probability distribution from a function built as follows: Perform three measurements X, Y, Z that have normally distributed errors. Impose a constraint and variable change that allows me to...

Dear, I assume that a signal S is expressed as S = a*S1 + b*S2, where a, b are weight constant, and S1, S2 are the different signals. In addition, S1, S2 have similar distribution such as Gaussian or Laplacian distribution, and theirs pdf is p_S1 and p_S2. In the above...

Dear, I assume that a signal S is expressed as S = a*S1 + b*S2, where a, b are weight constant, and S1, S2 are the different signals. In addition, S1, S2 have similar distribution such as Gaussian or Laplacian distribution, and theirs pdf is p_S1 and p_S2. In the above assumption...

If g and f are two normalized probability density functions is it then true in general that the convolution of f and g is normalized too?

Hi guys, I must solve, I believe, 2 simultaneous PDE's where the unknown function that I must find represent a conditional density of probability. It is a function of 3 variables, namely x, y and t. So it is P(x,y,t). P(x|y,t) means that the density of probability of a certain function...

I'm hoping this will be the last time I call for help, but in any case, here it goes. I thought I had a handle on this before, but in all of my attempts, my code diverges within a few iterations. My problem is creating a spatial distribution of particles given a probability density. I've...

Is there an experimental verification of the radial probability density for the hydrogen ground state given in the introductory texts. See the following link as an example. Thank you in advance. http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydrng.html#c1

Homework Statement The needle on a broken car speedometer is free to swing and bounces off perfectly off the pins at either end, so that if you gave it a flick it's equally likely to come to rest at 0 and ## \pi ## What is the probability density, ## p(x) ##? Homework Equations...

A joint pdf is given as pxy(x,y)=(1/4)^2 exp[-1/2 (|x| + |y|)] for x and y between minus and plus infinity. Find the joint pdf W=XY and Z=Y/X. f(w,z)=∫∫f(x,y)=∫∫(1/4)^2*e^(-(|x|+|y|)/2)dxdy -∞<x,y<∞ Someone told me I can not use Jacobian because of the absolute value. Is that true? So...

A joint pdf is given as pxy(x,y)=(1/4)^2 exp[-1/2 (|x| + |y|)] for x and y between minus and plus infinity. Find the joint pdf W=XY and Z=Y/X. f(w,z)=∫∫f(x,y)=∫∫(1/4)^2*e^(-(|x|+|y|)/2)dxdy -∞<x,y<∞ Someone told me I can not use Jacobian because of the absolute value. Is that true? So far this...

Homework Statement Bearing capacity of soil varies between 6 and 15 kips/sq.ft. If probability density within this range is given as f(x)=1/2.7 * (1- x/15), 6 ≤ u ≤ 15 =0 otherwise Find E(X) and E(X^2) Homework Equations E(x) should be ∫x*f(x) dx...

Hi, I would certainly appreciate it if you could please confirm the result I obtained to the following Statistics problem. Homework Statement A tank is supplied with fuel once a week. If the fuel (in thousands of liters) that the station sells in a week is a random variable which is...

I'm practicing the past year papers to prepare for my coming finals. Please make necessary corrections if you feel something wrong with it, thanks! Also, I'm supposed to do this in less than half an hour, so any suggestions on how to shorten this answer is really much appreciated! Homework...

Homework Statement So a neutron beam is split into two components, one by reflection, the other by transmission. The phase shift undergone by the reflected beam is \pi radians, and the phase shift of the transmitted beam is \Delta. What is the equation of the probability density of the...

Homework Statement The PDF (probability density function) of a Gaussian variable x is given by. $$p_x(x)=\frac{1}{C \sqrt{2 \pi}} e^{\frac{-(x-4)^2}{18}}$$ a) Find C b)find the probability of x≥2 --> ##P(x≥2)## Homework Equations $$ \frac{dF_X(x)}{dx} x=P(x<X≤x+Δx)$$ The...

Homework Statement A random variable x has a probability density function given by fX(x) = e-x , x ≥ 0 and an independent random variable Y has a probability density function of fY(y) = ey , y ≤ 0 using the characterisic functions, find the probability density function of Z = X + Y...

So I understand how for a continuous random variable the probability of an exact value of X is zero, but then what is the value of f(x) if it's not a probability? I thought it was a probability similar to how the pmf for a discrete random variable was a piece-wise function that gave the...

Homework Statement See figure attached. Homework Equations The Attempt at a Solution I'm having trouble getting start with this one, but here's what I've got so far. I assumed R is the signal received by the TDS. P(R=X) = \mu \quad , \quad P(R=N) = 1 - \mu Now in part...

Homework Statement The joint probability density function of X and Y is given by f(x, y) = c( x3 + xy/4 )  0 < x < 1 0 < y < 2 (a) For what value of c is this a joint density function? (b) Using this value of c, compute the density function of Y . (c) Using this value of c, nd PfX...

1. For the \pi-network of \beta-carotene modeled using the particle in the box, the position-dependent probability density of finding 1 of the 22 electrons is given by Pn(x) = |\Psi_{}n(x)|^2 = (2/a)Sin^2 (n Pi x / a) The quantum number n in this equation is determined by the energy level of...

Hey, I'm trying to determine the probability density and current of the Dirac equation by comparison to the general continuity equation. The form of the Dirac equation I have is i\frac{\partial \psi}{\partial t}=(-i\underline{\alpha}\cdot\underline{\nabla}+\beta m)\psi According to my...

Hi all, For an exam I'm required to be able to plot the PDF of a fluctuating velocity function, say u(t)=sint(wt), using what they call the "graphical technique", but handily I can't find it anywhere in the lecture notes, and I'm struggling to find anything with a standard Google search...

Homework Statement Find k such that the function f(x)=ke^{-\frac{x-\mu}{\theta}} is a probability density function (pdf), for x > \mu, \mu and \theta are constant. Homework Equations The property of a pdf says that the integral of f(x) from -\infty to \infty equals 1, that is...

A friend of mine recently tried to tell me that the square of the wave function for a particle (that is, \Psi^2) gives the probability density of finding a particle in space. I disagree. I always thought that the wave function multiplied by its complex conjugate (that is, \Psi \Psi^*)...

If probability distribution function is flat like a rectangular signal then probability density function which is differentiation of probability distribution function will have positive and negative impulses, but probability density function cannot be negative. . what's wrong in this . . ...

Hi all, I need to calculate the probability density function f\left( Y \right) of a function Y of two variables A and B with known individual probability density functions f\left( A \right) and f\left( B \right). What is the correct way to combine the PDF's? Specifically, I have a...

Hi everyone Homework Statement What's the probability density of an electron at a distance r (from hydrogen) which is in the stae n=2, l=1. Homework Equations - The Attempt at a Solution I think I have to to solve \int |\Psi_{nml}|^2 dV The solution of the Schrödinger...

Homework Statement The mean of a function is as follows: $${1 \over {a - b}}\int_b^a {f(x)\,dx} $$ So why is the mean of the PDF as follows: $$\int_{ - \infty }^\infty {xf(x)\,dx} $$ I thought it would have been this way: $$\lim \,b \to - \infty \,{1 \over { - b}}\int_b^0...

The manager of a fast food restaurant determines that the average time that her customers wait for their food is 2.5 minutes. The manager wants to advertise that anybody who isn't served within a certain number of minutes gets a free hamburger. She doesn't want to give away free hamburgers to...

Homework Statement How can I derive the probability density function by using the Central Limit theorem? For an example, let's say that we have a random variable Xi corresponding to the base at the ith position; to make even simpler, let's say all probabilities are equal. If we have four...

Homework Statement I want to calculate the probability of a random sample falling between 2 z scores using the way real mathematicians do it not the fake way by resorting to tables. Ok, so the book outlines the equation below but says that it requires calculus which is beyond the scope of...

What is the probability density equation as a function of angle for a simple pendulum using the small angle approximation? I got 1/(2 pi θmax) sec(sqrt(g/L)t) but it doesn't seem right.

Homework Statement Let X be a continuous random variable with parameters \langle x \rangle and \sigma. Calculate the probability density of the variable Y=exp(X). Calculate the mean and the variance of Y. Homework Equations Reichl's 2nd edition book page 180: P_Y(y)=\sum...

Homework Statement The probablity density function of the n-state of an electron is proportional to fn=(\frac{rz}{a_{0}})^{2n}e^ \frac{-2Zr}{\large na_{0}} show that the expectation value of the potential energy of the electron in the n-th quantum state of the hydrogen atoms is...

Deriving Probability Density Functions from Partial Differential Equations? Hiyas, I have been told that it is quite normal to get PDFs (Probability Density Functions) from PDEs (Partial Differential Equations). That in PDEs that the function can be a PDF and you can get this by solving the...

Homework Statement Assume that a typical electron in a piece of metallic sodium has energy - E_{0} compared to a free electron, where E_{0} is the 2.7 eV work function of sodium. At what distance beyond the surface of the metal is the electron's probability density 20% of its value at the...

A continuous random variable X has pdf: f_X(x)=\left \{ k(x+3), 0\leq x\leq 1\right \} 0 otherwise. Find k. I solved the integral (from 0..1) and solved for the result equal to 1. Hence I got k=2/7. Is this the right way to proceed as the question continues and I want to check if I'm...