What is Distribution function: Definition and 143 Discussions

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable


{\displaystyle X}
, or just distribution function of


{\displaystyle X}
, evaluated at


{\displaystyle x}
, is the probability that


{\displaystyle X}
will take a value less than or equal to


{\displaystyle x}
.Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by an upwards continuous monotonic increasing cumulative distribution function




{\displaystyle F:\mathbb {R} \rightarrow [0,1]}




{\displaystyle \lim _{x\rightarrow -\infty }F(x)=0}




{\displaystyle \lim _{x\rightarrow \infty }F(x)=1}
In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to


{\displaystyle x}
. Cumulative distribution functions are also used to specify the distribution of multivariate random variables.

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  1. WMDhamnekar

    MHB Math Help: Understand How to Compute $F_{X_1}(x)$

    Now, I don't understand how did author compute $F_{X_1}(x) = \displaystyle\sum_{j=1}^n \binom{n}{1} F^1(x) (1-F(x))^{n-1} = 1-(1-F(x))^n ?$ (I know L.H.S = R.H.S) Would any member of Math help board explain me that? Any math help will be accepted.
  2. WMDhamnekar

    A Derivation of P.D.F. from distribution function

    Author computed ##f_{X_k}(x)## as follows but I don't understand it. Would any member explain me the following computations?
  3. PainterGuy

    I Distribution function and random variable

    Hi, I cannot figure out how they got Table 2.1. For example, how come when x=1, F_X(x)=1/2? Could you please help me with it? Hi-resolution copy of the image: https://imagizer.imageshack.com/img923/2951/w9yTCQ.jpg
  4. M

    Monte-Carlo integration does not work properly after implementing pdfs

    Does anyone have experience with such strange behavior in Monte-Carlo methods? I think it is a conceptional problem and I am just missing a key point in how to set up the integration instead of a error in my code itself. I use data files from LHAPDF and also checked that my variables give the...
  5. L

    MHB Integral limits when using distribution function technique

    I am not sure about finding the limit of the integral when it comes to finding the CDF using the distribution function technique. I know that support of y is 0 ≤y<4, and it is not a one-to-one transformation. Now, I am confused with part b), finding the limits when calculating the cdf of Y...
  6. TheBigDig

    I Visualising an alternative formulation of Planck's Radiation Law

    I've come across this alternative formulation of Planck's Law which links the number density to energy gap n(E) = \frac{2\pi}{c^2 h^3} \frac{E^2}{exp\big(\frac{E-\mu}{k_BT})-1} I've tried visualising this relation and I imagine it will look similar to the spectral density relation but I'm just...
  7. dRic2

    I Question about an "exact" distribution function

    Suppose I have an exact microscopic distribution function in phase-space defined as a sum of delta-functions, i.e $$F( \mathbf x, \mathbf v, t) = \sum_{i} \delta( \mathbf x - \mathbf x_i ) \delta (\mathbf v - \mathbf v_i )$$ Can I conclude that, in absence of creation/destruction of particles...
  8. F

    I What do "marginalized" or "marginalized error" mean? (contours - posterior)

    I am curently working on Forecast in cosmology and I didn't grasp very well different details. Forecast allows, wiht Fisher's formalism, to compute constraints on cosmological parameters. I have 2 issues of understanding : 1) Here below a table containing all errors estimated on these...
  9. H

    Conditional Probability of a continuous joint distribution function

    For 1) I found two ways but I get difference results. The first way is I use P(A|B) = P(A and B)/P(B). I get P(X<1|Y<1)=(∫_0^1▒∫_0^1▒〖3/4 (2-x-y)dydx〗)/(∫_0^1▒∫_0^1▒〖3/4 (2-x-y)dydx〗+∫_1^2▒∫_0^(2-x)▒〖3/4 (2-x-y)dydx〗)=6/7 The 2nd method is I use is f(x│y)=f(x,y)/(f_X (x)...
  10. S

    MHB Given probability density function find its cumulative distribution function

    Hi :) Here's my problem along with what I've done. Here is the problem: That is the p.d.f. of a random variable X. I have to find the cdf. I don't know which I should do so I tried it two ways. First: $\int_{-1}^{1} \ \frac{2}{\pi(1+x^{2})} dx = {{\frac{2}{\pi} arctan(x)]}^{1}}_{-1}=1$...
  11. Eclair_de_XII

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    Homework Statement "Show that if ##P(X=c)>0##, for some ##c\in \mathbb{R}##, then the distribution function ##F_X(x)=P(X\leq x)## is discontinuous at ##c##. Is the converse true?" Homework Equations Continuity of a distribution function: ##\lim_{\epsilon \rightarrow 0}F_X(x+\epsilon)=F_X(x)##...
  12. T

    MHB Cumulative Distribution function in terms of Error function

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  13. D

    A Normalization of Radial Distribution Function

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  14. S

    Radial Distribution Function (RDF) versus Radial component

    Homework Statement For the RDF, we take the square of the radial component multiplied by 4pi r2 (the surface area of a sphere) and this gives us the probability density of finding an electron r distance away. Whats the point in multiplying it by r2? Homework Equations RDF= r2[R(r)]2 The...
  15. mehdi6

    I Computational particle physics

    Hi to all, I have a random number generator in FORTRAN, which gives a random numbers to my particles initial velocity in three dimension (vx,vy,vz). If, I want to make my particles to embark to move with a specific weight (eg. Maxwellian), what should I do? absv = sqrt(vx*vx+vy*vy+vz*vz) wt=...
  16. José Ricardo

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    Homework Statement Use a (simplified) graph to compare the maximum probability (electron density) of the Radial Distribution Function for the 1s and 2s orbitals. Homework Equations xxx The Attempt at a Solution The rest I don't how to solve.[/B]
  17. SchroedingersLion

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    Greetings, I am about to start my master thesis in computational physics and I need to make myself familiar with correlation functions, in particular with the radial distribution function of a system of N identical particles. At Wiki, there is a short explanation of the definition of the...
  18. P

    Finding Probability Density Functions for Independent Random Variables

    Homework Statement Hello! I'm trying to understand how to solve the following type of problems. 1) Random variables x and y are independent and uniformly distributed on the interval [0; a]. Find probability density function of a random variable z=x-y. 2) Exponentially distributed (p=exp(-x)...
  19. E

    I How to plot a scaled lognormal function

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  20. M

    I Looking for additional material about limits and distributions

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  21. V

    Calculate vmax and voltage from X-ray distribution function

    Homework Statement The X-ray intensity distribution function for an X-ray lamp is given on the figure. What is the maximum velocity of the electrons? What is the potential difference under which the lamp is operating? Figure: http://imgur.com/01kCBc8 Homework Equations Kmax = 0.5m(vmax)^2...
  22. A

    A How to calculate RDF (Radial Distribution Function)

    Dear friends How can i calculate RDF(radial distribution function)? Thanks
  23. T

    Continuous uniform distribution function

    Homework Statement Can someone explain why f(x) = 1/(b-a) for a<x<b ? Homework EquationsThe Attempt at a Solution shouldn't it be 0? since its a continuous random variable and so that interval from a to b has an infinite number of possible values?
  24. C

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  25. Saracen Rue

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  26. G

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  27. T

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  28. R

    What's the nature of a force acting on this gas? (Thermo)

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  29. Korbid

    Normalization of a pair distribution function

    Hi, i'm performing a simulation about this potential http://motion.cs.umn.edu/PowerLaw/ I calculated the radial distribution function succesfully but i don't know how these guys are normalized the other pair distribution function, as a function of time to collision. Could anyone help me? Thanks!
  30. Korbid

    Parametrizing a Pair Distribution Function

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  31. Linder88

    Joint cumulative distribution function

    Homework Statement Compute the joint cumulative distribution function $F_XY(x,y)$? Homework Equations The marginal distribution function $F_X(x)$ \begin{equation} F_X(x)=P(X\leq x)= \begin{cases} 0,x<0\\ 0.6,0\leq x<1\\ 1,x\geq 1 \end{cases} \end{equation} and $F_Y(y)$ \begin{equation} F_Y=...
  32. akk

    A Normalization constant of Fermi Dirac distribution function

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  33. R

    Lorentz Transforms & Distribution Funcs: Physics Intro Help

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  34. V

    MHB Computing joint cumulative distribution function

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  35. C

    Distribution function primitive

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  36. D

    Cumulative distribution function problem question?

    Hi, so this is the example I have: Suppose that A, B, C are independent random variables, each being uniformly distributed over (0,1). (a) What is the joint cumulative distribution function of A, B, C? (b) What is the probability that all of the roots of the equation Ax^2 + Bx +C=0 are real...
  37. P

    Definition Radial distribution function

    In some books the radial distribution function is defined as 4pi*r^2*R(r)^2 and in others as r^2*R(r)^2. Thus, a factor of 4pi is differing. Which expression is correct and why?
  38. S

    Probability distribution function

    Homework Statement The technical specification of a particular electrical product states that the probability of its failure with time is given by the function: f(t) = 1 - ke^(-t/t0) if 0 < t < tmax f(t) = 0 if t > tmax where t is the time of service...
  39. C

    Radial distribution function in Debye-Huckel theory

    Homework Statement For a plasma containing two ionic species with opposite charges but the same density ##n_{\infty}##, calculate the radial distribution functions ##g_{++}(r), g_{-+}(r), ##where ##n_{\infty}g_{ij}(r)## is the conditional probability density for finding a particle of type ##i##...
  40. S

    -- cumulative distribution function, probability density

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  41. J

    Effective occupation number of photon

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  42. P

    Orientation distribution function

    Hi everybody, i have a problem that i wanted to share with you if we consider a polycrystal made of cylindrical fibers following a von mises-fisher distribution equation (17) in http://bit.do/vmisesfisher (called orientation distribution function of fibers) . i must change the probability...
  43. T

    MHB Please help with this question (cumulative distribution function of X)

    The probability density function of the lifetime of a certain type of electronic device (measured in hours), X, is given by  f(x) = 10/x^2, 0, x > 10; elsewhere. (a) Find the cumulative distribution function of X, namely F(x) and hence find P(X > 20). (b) What is the...
  44. C

    Non equilibrium boson distribution function

    In statistical mechanics the boson distribution function has the well known form ##f = \frac{1}{e^{E/T} - 1},## (in the special case of zero chemical potential). As one considers the non-equilibrium variant this generalize to ##f = \frac{1}{e^{\frac{E}{T(1+ \Theta)}} - 1},## for some function...
  45. Julio1

    MHB Cumulative distribution function

    Hi !, my problem is the following: Let $F_X (x)$ an distribution function strictly monotone for the random variable $X$ and it's defined the new random variable $Y=F_X (X).$ Find the cumulative distribution function of $Y$.
  46. M

    Basic Maxwell Speed Distribution Function

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  47. T

    Probability distribution function

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  48. crador

    What Is the Most Likely Radius for a 1s Electron in a Hydrogen Atom?

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  49. C

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  50. O

    Getting a random number with a distribution function

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