Distribution function Definition and 143 Threads

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



X


{\displaystyle X}
, or just distribution function of



X


{\displaystyle X}
, evaluated at



x


{\displaystyle x}
, is the probability that



X


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



x


{\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



F
:

R


[
0
,
1
]


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




lim

x





F
(
x
)
=
0


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




lim

x




F
(
x
)
=
1


{\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



x


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

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  1. T

    Quantum mechanics angular distribution function

    1. Radial and angular distribution functions for an orbital Find the most probable value of theta and r for a 2pz orbital Homework Equations \psi _{2p_{z}} = N \textrm{cos}(\theta) r exp (-r/2) in units of a_0 The Attempt at a Solution Most probable r is when \textrm{d/d}r (P(r))=0...
  2. M

    Integral from distribution function

    Homework Statement hi, so I've got this distribution function: f(z,p,t)=\frac{1}{2\pi\partial z\partial p}exp(-\frac{[z-v(p)t]^2}{2\partial z^2})exp(-\frac{[p-p_0]^2}{2\partial p^2}) where: v(p)=v_0+\alpha(p-p_0) v_0=\frac{p_0}{m\gamma_0} \alpha=\frac{1}{m\gamma_0^3} I have to...
  3. N

    Total mass from mass distribution function

    Homework Statement An obect whose mass is distributed according to function m(r)=m0e-r, for r ranging from 0 to ∞. Calculate total mass of the object. Write down and evaluate appropriate integral. 2. The attempt at a solution well, I wasnt sure really how to start but thought it may be...
  4. S

    Random Variable and Distribution Function Relationship

    I'm in a probability theory class and I feel like I'm missing something fundamental between random variables and their distribution functions. I was given the following questions: 1)Let θ be uniformly dist. on [0,1]. For each dist. function F, define G(y) = sup{x:F(x)≤y}. Prove G(θ) has the...
  5. F

    MHB Mean and variance from a probability distribution function

    f(x)=f(x)={█(2/(√2π) e^(〖-x〗^2/2)@0 otherwise)┤for 0<x<∞ Find the mean and variance of X The hint says, compute E(X) directly and then compute E(X2) by comparing that integral with the integral representing the variance of a variable that is N (0, 1)
  6. kai_sikorski

    Radial distribution function question

    I've been playing around with some MD simulations, a field not really familiar to me. I put together a code in LAMMPS to simulate a Lennard-Jones fluid and compute the RDF. I get the oscillations one would expect, which is good, but what is surprising is that the minimum of g(r) after the first...
  7. P

    Minimum of Normal Distribution Function

    Homework Statement Show that there is no minimum for the normal distribution function e^(-(x-μ)^2/(2 σ^2))/(sqrt(2 π) σ) Homework Equations The Attempt at a Solution I figured I'd take the derivative and set it equal to 0, but then what?
  8. M

    MHB Calculating Probabilities using Distribution Function F

    Hello...!I need some help...! Let the distribution function F of a random variable X given in the following attachment. Calculate the following: P(X=-1), P(X<0), P(X<=0), P(X=1), P(X>5), P(X>=5), P(3<=X<=4). I think that these are the answers:P(X<0)=F(0-)=0.1, P(X<=0)=F(0)=0.2...
  9. E

    Probability distribution and cumulative distribution function q's

    Homework Statement $$P_x(x)=A(1- \frac{|x|}{2}) \ \ \ |x|≤2$$ $$P_x(x)=0 \ \ |x|>2$$ Find A Homework Equations The Attempt at a Solution This one shouldn't be too bad but I wanted to verify that I am on the right track. I basically have ##P_x(x)=A(1- \frac{x}{2})## when x≤2. 0 otherwise...
  10. P

    Stats/prob: finding cumulative distribution function

    Homework Statement given pdf: f(x) = 2/3x for 0<=x<=1 f(x) = 2/3 for 1<x<=2 f(x) = 0 elsewhere Find the CDF. Homework Equations The Attempt at a Solution I've found: F(x) = 0 for x<= 0 F(x) = 1 for x>=2 F(x) = (1/3)x2 for 0<=x<=1 and I found: F(x) = (2/3)x...
  11. C

    Find value of a and b such that F(x) is a valid cumulative distribution function

    Hi I have a question , the question asks find value of a and b such that F(x) is a valid cumulative distribution function? 1-a*exp(-x/b) x≥0 F(x)= a*exp(x/b) x<0 My attempt to solve the problem: I know F(x) when x goes to ∞ in 1 and when x goes to -∞ is...
  12. D

    Getting to the Boltzmann Equation from a Nonequilibrium Distribution Function

    Homework Statement Hi. I have a course where I am supposed to show how to get to the Boltzmann equation from a nonequilibrium distrubution function. At the moment I'm kinda lost to how this is done, so hopefully a hint or two would be of some help :)Homework Equations The nonequilibrium...
  13. J

    Intuitive explanations for Gaussian distribution function and mahalanobis distance

    Hello I was wondering If anyone could give intuitive explanations for the multivariate Gaussian distribution function and mahalanobis distance? My professor didn't explain these in probability class, they were only defined... Where did the formula come from? Why is the Gaussian function the...
  14. N

    MHB Integral of Probability Distribution Function

    The distance of someone from the center (cable station) of a circle is depicted as r, and the regular radius of that circle(the cable station) is depicted as rc. The Circle represents the entire area the cable station provides cable service for. Given: Probability Distribution Function P(r) =...
  15. F

    Properties of a distribution function at infinite

    Homework Statement Let's consider a distribution function f=f(t,x^i,E,p^i). Is it true that \mathop {\lim }\limits_{p \to\infty}p^{\alpha}f=0 \forall\alpha\in R ?Homework EquationsThe Attempt at a Solution I think so, not sure though. Thanks in advance!
  16. D

    Deduction of Gauss' Normal Distribution Function

    Hello, I'm David. I'm a new member here. Could anyone of you help me? Where can i find the formal deduction of Gauss' Normal Distribution Function? I've read a lot of statistics books and never found that. Where that comes from? It's just curiosity, not homework. P.S.: sorry about my...
  17. O

    Can you bring a limit inside of the standard normal distribution function?

    Homework Statement Let N(x) denote the CDF of the standard normal density. So, it's the integral of the standard normal density from -∞ to x. Is it true that \lim_{b\to 0} N(\frac{a}{b}) = N(\frac{a}{\lim_{b\to 0}b}) = N(+\infty) = 1? 2. The attempt at a solution This is more of a...
  18. S

    Shaping probability distribution function

    Homework Statement Incoming signal has normal distribution, xmin is equal to -sigma, xman is equal to +sigma. What is the governing equation of the nonlinearity through which the signal has to be passed in order to make its pdf uniform? Homework Equations...
  19. A

    Velocity Distribution function of molecule at low temperature

    Dear all, In classical molecular dynamics simulation initial velocities are generated using the so called Maxwell distribution. At low temperature it's no longer effective, so I'm wandering whether there is a similar way to generate velocities at low temperature taking into account quantum...
  20. B

    Simple probability distribution function on gnuplot

    Hi, I am trying to plot a graph I have made using excel on gnuplot. I have converted the excel book to a csv file and the data I have looks like this 0,0.166341305 1,0.000000159 2,0.000000159 3,0.000000159 4,0.000000159 5,0.000000159 ... and so on for 2048 different points. The data...
  21. D

    Cumulative distribution function

    Homework Statement The continuous random variable X has cumulative distribution function given by F(x) = \left\{ {\begin{array}{*{20}c} 0 & {x \le 0} \\ {\frac{{x^2 }}{k}} & {0 \le x \le 1} \\ { - \frac{{x^2 }}{6} + x - \frac{1}{2}} & {1 \le x \le 3} \\ 1 & {x \ge 3} \\...
  22. A

    Statistics - Distribution Function Technique

    Homework Statement (From Probability and Statistical Inference, Hogg and Tanis, Eighth Edition, 5.1-5) The p.d.f. of X is f(x) = \theta x^{\theta - 1} for 0<x<1 and 0<\theta<\infty. Let Y = -2\theta \ln X. How is Y distributed? Homework Equations Um... Fundamental Theorem of...
  23. fluidistic

    3D Visualization of Temperature Distribution Function

    I just downloaded scilab because Wolfram Alpha wouldn't want to plot the function I'd like. In a physics problem I've found the temperature distribution of a 2 dimensional system. I'd like to visualize this function in 3d. The function I want to plot is u(x,y)=\frac{1}{\pi} \arctan \left (...
  24. C

    Radial distribution function, concept.

    Okay, this is a really basic question. I'm just learning the basics of QM now. I can't wrap my head around the idea that the radial distribution function goes to zero as r-->0 but that the probability density as at a maximum as r-->zero. How can this be? they are opposite to each other...
  25. J

    Non Linear ODE whose solution is can be viewed as a cumulative distribution function

    Let X be continuous a random variable who's support is the entire real line and who's cumulative distribution function satisfies the initial value problem F'(x)=s\cdotF(x)a\cdot(1-F(x))b F(m)=1/2 note that a>0, b>0, s>0 and m is real. m is the median of the distribution, Is it...
  26. D

    Stats - find the distribution function of an infinite sample space.

    Homework Statement A die is rolled until the first time that a six turns up. We shall see that the probability that this occurs on the nth roll is (5/6)n−1 · (1/6). Using this fact, describe the appropriate infinite sample space and distribution function for the experiment of rolling a die...
  27. T

    PROBABILITY: Distribution function

    Homework Statement A continuous random variable X has density f(x) = ax2(1- x) for 0 < x < 1, and f(x) = 0 otherwise. Here a is a constant, to be determined. Find the distribution func- tion FX, the constant a, the expectation E(X), the variance Var(X), and the conditional probability p(X...
  28. M

    Cumulative distribution function

    A dart is thrown towards a quadrilateral defined by {(x,y): 0 < x < b, 0 < x < b}. Assume the dart is equally likely to land anywhere within this shape. Let Z be denoted by the (x,y) coordinate with the least value. Find the region in the square corresponding to {Z < z} so i know the sample...
  29. A

    Cumulative Distribution Function problem

    Homework Statement Let the independent random variables X1,X2,X3 have the same cdf F(x). Let Y be the middle value of X1,X2,X3. Determine the cdf of Y. Homework Equations F(x) = P(X<=x) = ∫-inf to x f(y)dy The Attempt at a Solution I don't understand the question, what does it...
  30. T

    Difficult Distribution Function

    Homework Statement ∫3/43x2e-(x/4)3 Homework Equations F(x) = 0∫xf(t) dt The Attempt at a Solution The solution is 1 - e-(x/4)3 I have tried integrating but I am not able to arrive at the solution... What is the indefinite integral of e-(x/4)3?
  31. I

    Need help solving CDF (Cumulative distribution function)

    Hello, I am new. I been looking on the net for a guide how to solve the CDF by hand, i know the answer and I am about to crack this baby but I got stuck... Im trying to calculate Cumulative distribution function by hand: \int^{1}_{-1}\frac{1}{2\pi} e^{\frac{-z^{2}}{2}} dz or wolfram alpha...
  32. I_am_learning

    Trivial Question on Gamma Distribution Function

    If a random variable X follows Gamma Distribution Function with parameters K and thita, what does (X+k) follow? if K is a constant. I think, since adding the constant is just like shifting the origin, the nature of the curve remain unchanged. But what about its parameter? Thanks.
  33. C

    Cellular network Probability distribution function of frequency

    Hi, I am trying to model a PDF of frequency for the WCDMA handset. I have found some info on ways of graphing a PDF of transmit power but nothing on frequency. I am hoping there is a way to model this, but think it might be something that requires field tests with multiple phones. Any...
  34. L

    Integration using a cumulative distribution function

    Hi all, I'm really banging my head on this problem: Let f be a real-valued measurable function on the measure space (X,\mathcal{M},\mu). Define \lambda_f(t)=\mu\{x:|f(x)|>t\}, t>0. Show that if \phi is a nonnegative Borel function defined on [0,infinity), then...
  35. T

    Help with probability distribution function question

    Homework Statement Let X and Y be two independent random variables with the same probability density funtion over: f(x) = {1/a if x € [0,a] {0 if x=0 Find the density distribution of a) X + Y and b) X*Y Homework Equations The Attempt at a Solution Ok, my...
  36. A

    Solving the distribution function

    I want to solve following double integration. I am stuck from long time. Any intermediate solution will also be appreciated. \int\int dvb dvl (feven+fodd) where, feven = (r*(vl^{2}+vb^{2}))^(-2*\beta)*(potential - 0.5*(vl^{2}+vb^{2}+vr^{2}))^(-3*\beta+5.5), where \beta,potential and vr...
  37. T

    Radial Distribution Function for HCP and FCC lattices

    I was wondering, does anyone have any images that show the difference between the radial distribution functions for hcp and fcc lattices? I would be useful for reference purposes.
  38. S

    Joint distribution function of z and |z|

    X and Y are 2 independent gaussian random variables with parameter a. Z = XY / (X-Y) W = XY / |X-Y| I am to find the joint distribution function of z and w. I know how to find pdf of Z but how could I use it to find the joint distribution function of z and w?
  39. L

    What should the normalization condition be for an energy distribution function?

    Hello everybody. I have the following problem: I am given a Speed distribution function, f(v). This distribution is normalized to the total number of particles there are, that's equivalent to: \int_0^\infty f(v) dv = n_0 (particles traveling away from my target are not considered.)Now. I know...
  40. J

    Hard disk radial distribution function

    Hi, i am running a hard disks molecular dynamics simulation. I would like to compare the radial distribution function obtained from my simulation with the theoretical radial distribution function. May i know what is the theoretical radial distribution function? Or what data do people normally...
  41. Shackleford

    Cov(X,Y) Distribution Function

    For (4), when I calculated the Distribution Function, I had to break it up into two cases whether or not the point was above or below the line. For (6.c), I got the right answer for the Cov(X,Y), but I didn't break up the double integral into two cases. Is this because since the density...
  42. P

    Probability Distribution Function

    Homework Statement Given the probability distribution function: See attachment. Determine the: 1. Probability density function. 2. The mean. 3. The median. Homework Equations Hello, I am really struggling with this subject area, this is an example I have found, would...
  43. M

    Java Lognormal cumulative distribution function for JavaScript?

    Hey everyone! I'm building a "Deal or No Deal" calculator that estimates the odds of getting a better deal than the one presented. As you can see in the title, I'm looking for a function that will calculate the CDF. For ease's sake, please only post JavaScript code (no C,Perl,BASIC,etc.)
  44. Shackleford

    What is distribution function?

    http://i111.photobucket.com/albums/n149/camarolt4z28/Untitled.png?t=1298077701 Oops. Nevermind. It is in the family. I think the rest of my work is correct. http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110218_190456.jpg?t=1298077991
  45. T

    Probability Distribution function change of variables.

    Homework Statement I have been given the distribution function F_X of the random variable X and I am asked to find the distribution function F_Y of Y, another random variable which is defined from X in the following way. Y={\stackrel{X^{2} if X<2;}{4 if 2\leq X < 3;}\stackrel{4(4-X) if...
  46. U

    Cumulative distribution function

    i have two random variables x e y independent and they're uniform on the interval [0, 1] find cumulative distribution function of Z= (x+y)/(x-y) i just try to solve... [PLAIN]http://img202.imageshack.us/img202/5647/97250438.jpg is it right?
  47. A

    Probability distribution function question

    Homework Statement A PC generates "random" numbers from [0,1], programmed such that the distribution function F(x) of a continuous random variable X, which is satisfies the formula: F(x) = 0 , x<0 x , 0<=x<0.25 0.25 , 0.25<=x<0.5 x2, 0.5<=x<1 1 , 1<=x THe problem then asks the values of...
  48. M

    Finding the Probability distribution function given Moment Generating Function

    Hi everyone, So I am taking a statistics course and finding this concept kinda challenging. wondering if someone can help me with the following problem! Suppose X is a discrete random variable with moment generating function M(t) = 2/10 + 1/10e^t + 2/10e^(2t) + 3/10e^(3t) + 2/10e^(4t)...
  49. J

    How Do You Find Marginal Distribution Functions?

    Given: P(X=x, Y=y)=\frac{a^ye^{-2a}}{x!(y-x)!} where x=0,1,2,...y and y=0,1,2...\infty, and a>0 Find P(X=x) and P(Y=y) An example is provided in a book on books.google.com Page 96...
  50. S

    Cumulative distribution function question

    consider rolling a die. S= {1,2,3,4,5,6} P(s)=1/6 for all s in S X= number on die so that X(s)=s for all s in S Y= X^2 compute the cumulative distribution function Fy(y) = P(Y<=y), for all y in the set of real numbers. My guess for Y=1 i get...
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