Probability density Definition and 285 Threads

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. In other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample.
In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. This probability is given by the integral of this variable's PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to 1.
The terms "probability distribution function" and "probability function" have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values or it may refer to the cumulative distribution function, or it may be a probability mass function (PMF) rather than the density. "Density function" itself is also used for the probability mass function, leading to further confusion. In general though, the PMF is used in the context of discrete random variables (random variables that take values on a countable set), while the PDF is used in the context of continuous random variables.

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  1. J

    A Probability density of dirac spinors

    The probability density of the dirac spinor is known to be ∑(Ψ)2 and I know how it is derived. However, I'm just wondering why it should be positive definite. Since the lower two components represent antiparticles, so shouldn't the probability density contribution of those two components be...
  2. T

    I What Does The Probability Density Function Tell You?

    Hello All I was wondering if someone could help explain what the probability density function tells you. I am trying to learn about surveying and the PDF keeps cropping up and I do not fully understand it. For example I have:- measured a single angle 15 times calculated my Standard Deviation...
  3. S

    I Interpretation of probability density in QFT

    Hello! I am a bit confused about the interpretation of probability density in QFT. Let's say we have the Klein-Gordon equation. I understand that this is the field equation for a spin-0 charged particle. So if we find a solution ##\phi(x)## of the Klein-Gordon equation, as far as I understand...
  4. redtree

    A Deriving Probability Amplitude from Markov Density Function

    1. Given a Markov state density function: ## P((\textbf{r}_{n}| \textbf{r}_{n-1})) ## ##P## describes the probability of transitioning from a state at ## \textbf{r}_{n-1}## to a state at ##\textbf{r}_{n} ##. If ## \textbf{r}_{n-1} = \textbf{r}_{n}##, then ##P## describes the probability of...
  5. Saracen Rue

    Probability Density Function problem

    Homework Statement Presume the relation ##\frac{x}{x+y^2}-y=x## is defined over the domain ##[0,1]##. (a) Rearrange this relation for ##y## and define it as a function, ##f(x)##. (b) Function ##f(x)## is dilated by a factor of ##a## from the y-axis, transforming it into a probability density...
  6. G

    Probability Density in an infinite 1D square well

    Homework Statement The wave function of a particle of mass m confined in an infinite one-dimensional square well of width L = 0.23 nm, is: ψ(x) = (2/L)1/2 sin(3πx/L) for 0 < x < L ψ(x) = 0 everywhere else. The energy of the particle in this state is E = 63.974 eV. 1) What is the rest energy...
  7. A

    I Calculating the probability density of a superposition

    This ought to be some simple gap in my knowledge, but it bugs me nonetheless. Let me present the argument as I see it, I'm fairly certain that there is just some tiny part that I didn't learn correctly. Let us assume a wavefunction $$\Psi$$ is defined as a superposition of two wavefunctions...
  8. Saracen Rue

    Probability Density Function problem

    Homework Statement Q6. A function, ##f\left(x\right)=\frac{ax+1}{\left(ax-1\right)^3-\frac{a}{\left(x-1\right)^2-1}}##, can be defined as a PDf over the domain ##(0, 2)##. Express answers to 4 decimal places unless specified otherwise; (a) Find the value of ##a## given that ##f(x)## is a PDf...
  9. T

    I About stochastic differential equation and probability density

    I have two questions about the use of stochastic differential equation and probability density function in physics, especially in statistical mechanics. a) I wonder if stochastic differential equation and PDF is an approximation to the actual random process or is it a law like Newton's second...
  10. T

    B Probability density of a normal distribution

    If the normalized probability density of the normal distribution is ## p(x) = \frac {1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}} ##, then if ##\sigma = 0.0001## and in the special case ## x = \mu##, wouldn't the probability density at this point, ##p(\mu)##, exceed 1 since it is equal...
  11. Saracen Rue

    Difficult Probability Density Function Question

    Homework Statement A function, ##f\left(x\right)=\left|a^{\frac{\sin \left(x\right)}{\ln \left(ax\right)}}-\frac{x}{a}\right|##, intersects with another function, ##g\left(x\right)=\left|\frac{sin(ln(\sqrt{x}-\sqrt{a}))}{x^2-a^2}\right|##, at point ##Q(b,f(b))## and point ##R(c,f(c))##. A...
  12. It's me

    Using Noether's Theorem find a continuity equation for KG

    Homework Statement Consider the Klein-Gordon equation ##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0##. Using Noether's theorem, find a continuity equation of the form ##\partial_\mu j^{\mu}=0##. Homework Equations ##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0## The Attempt at a Solution...
  13. NatFex

    I Sum of Probability Density Function > 1?

    I have a Stats exam on Wednesday and while I thought I was quite well-versed, I've gone back over to the very basics only to find myself confused at what should be introductory. Suppose I have a continuous random variable modeled by a probability density function: $$f(x)=2x$$ Obviously the...
  14. I

    I Probability density and kinetic

    energy. Consider a particle in a box of the form ##V(x)= \begin{cases}0 \; \; \; -L < x < 0\\ V_0 \; \; \; 0<x<L\\ \infty \; \; \; \text{ elsewhere}.\end{cases}## One can show that the probability density ##P(x) = \Psi^* \Psi## is greater in the region of lower kinetic energy (that is at higher...
  15. Schwarzschild90

    Calculating Standard Deviation for a Probability Density Function?

    Homework Statement Homework Equations See below The Attempt at a Solution \begin{align} \begin{split} p(x) = C \ x \ exp(-x/ \lambda) \end{split} \end{align} If $p(x)$ is a probability density function on the interval $ 0 \textless x \textless + \infty $ , then it follows...
  16. Jan Wo

    Getting electron density from probability density function

    Hello Lastly I was thinking a lot about electron density definition. It is not intuitive for me and I'm looking for any mathematical tool that could explain it to me more. My friend told me about idea to derivate it from propability density function using Dirac delta distribution. I'd like to...
  17. T

    Integrating a theta function/2d probability density function

    Homework Statement A two-dimensional circular region of radius a has a gas of particles with uniform density all traveling at the same speed but with random directions. The wall of the chamber is suddenly taken away and the probability density of the gas cloud subsequently satisfies $$...
  18. S

    Wave equation indicates the probability density

    First, we know for every wave function $$p(x)=\psi(x)^*\psi(x)$$ indicates the probability density of a particle appearing at the point x. So if we calculate $$P=\int _M p \text{d}x$$ this gives the probability of the particle appearing in the range M. On the other side, I was thinking about...
  19. E

    What probability density is used in Brownian motion?

    Homework Statement I have a free Brownian particle and its coordinate is given as a function of time: And its first moment, or mean, is given as But what kind of probability density was used to calculate this first moment? Homework Equations I know that the first moment is calculated...
  20. tomdodd4598

    Plotting the Probability Density of the Coulomb Wave Function

    Hey there - I think I have an issue with my 3D density plots of the probability density of the Coulomb wave function. The reason I think something is going wrong is because my plots of |ψ(n=2, l=1, m=-1)|² and |ψ(2, 1, 1)|² are identical, while I would expect them to have the same shape but be...
  21. B

    How do I plot this probability density over time?

    Here's the question: http://imgur.com/N60qRmw I normalized the wave function and got A = sqrt(315/8L^9) but how would I plot representative snapshots if the exponential factor will cancel when I square it? It's not a mixed state so it shouldn't depend on time as far as I can tell. Should I use...
  22. T

    Mean and variance of a probability density

    Homework Statement find μ and σ^2 for the following probability density f(x) = x for 0<x<1 f(x) = 2-x for 1 <= x < 2 f(x) = 0 elsewhere Homework Equations μ = ∫xf(x)dx σ^2 = ∫x^2 f(x)dx - μ^2 The Attempt at a Solution first we find the mean. we split up the integral into sums of different...
  23. T

    Finding Probabilities for a Probability Density Function | Homework Solution

    Homework Statement If the distribution function of a random variable is given by F(x) = 1- 1/x^2 for x>1 and F(x) = 0 for x <= 1 find the probabilities that this random variable will take on a value a) less than 3 b) between 4 and 5The Attempt at a Solution since they use the capital F i...
  24. A

    Probability current density in E.M. field

    Derive the probability current density for a particle in an electromagnetic field. (I previously posted this on StackExchange. Please pardon, but I have been spending a lot of time on this and if anyone knows exactly what the subtle trick involved is, I would really appreciate it.)...
  25. S

    Probability density function of simple Mass-Spring system.

    Homework Statement We know that after long run of simple mass-spring system, there should be a probability of finding the mass at certain points between -A and A.. Obviously in probability of finding the particle near A or -A is higher than finding the particle at 0, because the speed is the...
  26. diracdelta

    Probability density function of uniform distribution

    Homework Statement Random variable X is uniformly distributed on interval [0,1]: f(x)=\begin{cases} 1 & \text{ if } 0\leq x\leq 1\\ 0 & \text{ else} \end{cases} a) Find probability density function ρ(y) of random variable Y=\sqrt{X} +1 I tried like this. Is it good, if no why not...
  27. Kior

    The Probability Density of X^2?

    Here is a question about probability density. I am trying to work it out using a different method from the method on the textbook. But I get a different answer unfortunately. Can anyone help me out? Question: Let X be uniformly distributed random variable in the internal [ 0, 1]. Find the...
  28. Jimster41

    Does expansion a(t) affect QM probability density?

    I'm picturing a comoving particle, meaning at rest with respect to CMB? Does expansion affect the probability density, density current? I can't see how it wouldn't? But then it seems like there would have to be a uniform negative probability density current everywhere.
  29. S

    Stat mech: probability density - 2 interacting particles

    1. The problem statement, all variables and given/known data Show that the probability density as a function of seperation, r, of two atoms interacting via a potential U(r) (e.g. a function of separation only, such as a Coulombic interaction), is given by $$\rho(r) = Cr^2e^{-\beta U(r)}$$...
  30. K

    Calculating log liklihood: Zero value of likelihood function

    Hello, I am analysing hydrology data and curve fitting to check the best probability distribution among 8 candidate distribution. (2 and 3 parameter distributions) The selection is based on the lowest AIC value. While doing my calculation in excel, how is it suggested to treat very low (approx...
  31. B

    Step Potential with incident and reflected waves

    Homework Statement A woman is walking along a road. She has a mass of 52 kg and is walking at 1 m/s. (a) She is not paying careful attention and is walking straight towards the wall of a nearby building. Assume that the wall is infinitely hard and that she can be described as a plane wave (a...
  32. gfd43tg

    Solving for probability density (Griffith's 1.3)

    Homework Statement Problem statement in image Homework EquationsThe Attempt at a Solution From my reading, I know that the probability density \langle x \rangle = \int x |\psi|^{2} \hspace 0.05 in dx Where ##\psi## is the wave function and ##\langle x \rangle## is the probability density...
  33. S

    -- cumulative distribution function, probability density

    Homework Statement For the next probability function: f(x)=x/4 for 0<x<2 Homework Equations a) Get the probability function b) Get the cumulative distribution function The Attempt at a Solution I don´t know if the problem is well written, and for that I'm lost with the first question...
  34. diegzumillo

    Probability density for related variables

    Homework Statement Say I calculated a probability density of a system containing m spins up (N is the total number of particles). The probabilities of being up and down are equal so this is easy to calculate. Let's call it ##\omega_m##. Then we define magnetization as ##M=2m-N## and it asks me...
  35. J

    Quantum physics - probability density,

    Homework Statement consider a particle at an interval ##[-L/2, L/2]##, described by the wave function ## \psi (x,t)= \frac{1}{\sqrt{L}}e^{i(kx-wt)}## a) Calculate the probability density ##\rho (x,t) ## and the current density ## j(x,t)## of the particle b) How can you express ## j(x,t)## as a...
  36. A

    Quantum harmonic oscillator tunneling puzzle

    My problem is described in the animation that I posted on Youtube: For the sake of convenience I am copying here the text that follows the animation: I have made this animation in order to present my little puzzle with the quantum harmonic oscillator. Think about a classical oscillator, a...
  37. throneoo

    Probability density and rifle shooting

    Homework Statement A rifle shooter aims at a target at a distance D, but has an accuracy probability density ρ(φ)=1/(2Φ) φ∈(-Φ,Φ) where φ is the angle achieved and is bounded by the small angle ΦPart A find the probability density for where the bullet strikes the target , ρ(x) . The target...
  38. G

    Skewed Generalized Gaussian Distribution

    I am looking for more information (e.g., reference, the CDF and descriptive stats) about a four-parameter skewed generalized Gaussian (SGG) distribution. I have come across the PDF for this distribution, but with no reference and not a lot of other information. Here is a snippet... On...
  39. L

    Radial probability density of hydrogen electron

    Homework Statement I know this question has been asked before, but I am looking for a different kind of answer than the other poster. Bear with me here. Problem: For a hydrogen atom in the ground state, what is the probability to find the electron between the Bohr radius a0 and (1.01)a0...
  40. J

    Expectation Value vs Probability Density

    I know the difference between the expectation value and probability density, but how do you calculate the probability density of an observable other than position? For position, the probability of the particle being in a particular spot is given by |\Psi|^2, which is the probability density, and...
  41. Q

    MHB Marginal probability density functions (pdf)

    I have a set of two related queries relating to marginal pdfs: i.How to proceed finding the marginal pdfs of two independent gamma distributions (X1 and X2) with parameters (α1,β) and (α2,β) respectively, given the transformation: Y1=X1/(X1+X2) and Y2=X1+X2. I am using the following gamma...
  42. T

    Answer Root-Mean-Square Fluctuation in Box of Length L

    Homework Statement There is an equal probability density for finding a particle anywhere in the box. Assume that the box is of length L. What would the root-mean-square fluctation in position be? Homework Equations root-mean-square fluctation2 = <x2> - <x>2 The Attempt at a Solution since...
  43. N

    Contact rate between individuals of different probability density functions

    Hi all, I'm interested in obtaining some measure of contact (or encounter) likelihood between two individuals, each is spatially distributed with some probability density function at time ##t## such that, Space use of individual 1 = ##u(\mathbf{x}, t)## Space use of individual 2 =...
  44. A

    Average value and probability density

    Homework Statement Consider a particle oscillating according to x(t) = a\cos(\omega t): Find \rho(x), the probability density to find particle at position x. Compute \langle x\rangle, \langle x^2\rangle. Homework Equations So in general we know that \textrm{Prob} = \int f_X...
  45. B

    Relating the CDF to the probability density

    Homework Statement If ##X## is any random variable defined on ##[0,\infty]## with continuous CDF ##F_X(t)##. Prove that ##E(X)=\int_1^\infty (1-F_X(t)) dt##. . Homework Equations The Attempt at a Solution I am not sure how to go about this. I think double integration can be used to prove it...
  46. fluidistic

    Probability density that "seems" complex instead of real

    Homework Statement Hello guys, I am extremely worried because I do not understand something. My statistical mechanics course somehow follows Reichl's book for some parts. For a system of N particles, Reichl's define ##\rho (\vec X^N, t)## as the density of probability on the phase space...
  47. L

    MHB Joint Probability Density Functions

    Problem: Two birds have landed on a power line that spans the 100' distance between utility poles. a) What is the average distance between the birds? b) The line runs north and south. Another bird lands on the line. What is the expected position of the north-most bird from the south-most...
  48. G

    MHB Joint Probability Density from two Gaussian Distributions

    I've been reading the following paper entitled "An Improved Algorithm to Generate a Wi-Fi Fingerprint Database for Indoor Positioning": An Improved Algorithm to Generate a Wi-Fi Fingerprint Database for Indoor Positioning In Part 3.3 (Step 6), it states: Use the fingerprint database to...
  49. M

    How to get from <P> to probability density P(k)?

    Homework Statement Knowing the momentum operator -iħd/dx , the expectation value of momentum and the Fourier transforms how can I prove that <p> = ∫dk [mod square of ψ(k)] h/λ. From this, mod square of ψ(k) is defined to be equal to P(k) right? Homework Equations Momentum operator, p...
  50. D

    MHB Continuous joint probability density functions

    Consider the following joint probability distribution function of (X , Y): a(x + y^2) {0<=x<=2, 0<=y<=2} 0 otherwise Calculate the value of the constant a that makes this a legitimate probability distribution. (Round your answer to four decimal places as appropriate.) And then, For the...
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