What is Probability density: Definition and 285 Discussions

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

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  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

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  8. Saracen Rue

    Probability Density Function problem

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  9. T

    I About stochastic differential equation and probability density

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  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

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  12. It's me

    Using Noether's Theorem find a continuity equation for KG

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  13. NatFex

    I Sum of Probability Density Function > 1?

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  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

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    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

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  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

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    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

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  24. A

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  25. S

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  26. diracdelta

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    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

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  28. Jimster41

    Does expansion a(t) affect QM probability density?

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  29. S

    Stat mech: probability density - 2 interacting particles

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  30. K

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    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

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  32. gfd43tg

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  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,

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  36. A

    Quantum harmonic oscillator tunneling puzzle

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  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

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  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)

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  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

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    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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