# What is Distribution function: Definition and 143 Discussions

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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### What does the continuity of a distribution function mean?

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. ### MHB Cumulative Distribution function in terms of Error function

Hey guys, I've got this problem I've been trying to solve, but it makes little sense to me. I've tried a few things, but I feel like with each method, I've made no progress, and I haven't been able to make the problem make any more sense to me by trying those things. Here's the question: The...
13. ### A Normalization of Radial Distribution Function

Hello all, I have a Radial Distribution Function in which the y-axis ie., g(r) value goes up to 40. But the other atoms values for g(r) are, say within 5. So when i plot these two it is difficult to see the smaller graph. So how do i normalize these value..?? I have attached an image. Any...

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. ### 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. ### Radial Distribution Function graph

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. ### A Correlation functions - radial distribution function

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. ### 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. ### I How to plot a scaled lognormal function

Hi, I am trying to plot a lognormal function. I have the value of μ=3.5, the value of σ=1.5 and the value of the Area = 1965. I have as well the value of the maximum height (Amp.=4724). I am tryiing to plot these with Excel or with R but I do not know how. I know how to plot a distribution of...

I would like some help to find some additional info on generalized functions, generalized limits. My aim is to understand the strict definition of delta dirac δ(τ).If you could provide a concise tutorial focusing on δ(τ) not the entire theory...it would be of great help. I am not a math...
21. ### 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 How to calculate RDF (Radial Distribution Function)

Dear friends How can i calculate RDF(radial distribution function)? Thanks
23. ### 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. ### Finding C from a speed distribution function.

Homework Statement Gas particles of a particular gas have a speed distribution function of fv = Cv/(v2 +vo2)2 a. Find the value of C b. Calculate the most probable speed c. What fractions of the particles are moving faster than the most probable speed Homework EquationsThe Attempt at a...
25. ### Finding μ and σ from a normal cumulative distribution function

Homework Statement The relationship between the expected value and the variance for a particular normal CDF is known to follow the rule ##E(X)=arcsin(ln(Var(X)))##. Given that ##Pr(0.32<Z<2.32)=Pr(12.9<X<74.275)##, determine the possible values of the mean and the standard deviation correct to...
26. ### I Distribution function for specific 1D problem

Hello! Maybe someone will be able to suggest something about the following quite simple problem: 1D problem on axis "X". Particle moves only along "X" axis and starts its motion from X=0. However, when "X<0" particle disappears. Particle is influenced by some kind of force in such way that we...
27. ### Integration of Maxwell speed distribution function

Homework Statement Show the steps needed to obtain the equation for average molecular speed, cavg=√8RT/πM from the integral (from negative infinity to infinity) ∫v*f(v)dv where f(v) is the Maxwell distribution of speeds function f(v)=4π*(M/2πRT)1.5v2e-Mv2/2RT M is the molar mass of the...
28. ### What's the nature of a force acting on this gas? (Thermo)

Homework Statement A gas in equilibrium has distribution function: f(p,r) = C0*(1+y*x)(2*pi*m*k*T)-3/2*exp(-p2/(2*m*k*T)) where x is the distance along an axis with fixed origin, and y is a constant. What's the nature of the force acting on this gas? Homework Equations Maxwell bolztmann...
29. ### 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. ### Parametrizing a Pair Distribution Function

Hi!, If we have a bidimensional system of N particles. How can i parametrize a radial distribution function g(r) in another single variable ? This variable is the "time to collision", which we define as the duration of time for which two particles could continue at their current velocities...
31. ### Joint cumulative distribution function

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

Fermi-Dirac distribution function is given by f(E)=(1)/(Aexp{E/k_{B}T}+1) here A is the normalization constant? How we can get A? E is the energy, k_{B} is the Boltzmann constant and T is the temperature. thank you
33. ### Lorentz Transforms & Distribution Funcs: Physics Intro Help

Hi there, kinda new here so please let me know if this question has been answered. I am hoping to get a link or two to some good sources of information on Lorentz transforms and distribution functions (as used in physics). I understand DF's in class and I understand the math behind them I just...

46. ### Basic Maxwell Speed Distribution Function

Homework Statement Homework Equations The Attempt at a Solution I DO NOT NEED HELP WRITING THE PROGRAM! I'm just trying to figure out the basics behind it. Since this is an integral I will be solving this problem by using riemann sums. Where I'm having the most trouble is in the...
47. ### Probability distribution function

The number of hours, N, of daylight at a certain location can be expressed as N(d)=12+6sin(2πd/365) where d=day of the year starting with March 21. (a) What is the probability distribution function for hours of daylight if you assume the day of the year is a random variable? (b) What is the...
48. ### What Is the Most Likely Radius for a 1s Electron in a Hydrogen Atom?

Homework Statement Hey Guys! Here's my problem: ψ=2(Z/a)^3/2*e^ρ/2 for the 1s orbital of a hydrogen atom. Write down the radial distribution function expression (P) of a 1s electron and determine the most likely radius. ρ=2Zr/a Z nuclear charge r radius a Bohr's radius Homework...
49. ### Radial distribution function for H atom in ground state

Homework Statement The normalized energy eigenfunction of the ground state of the hydrogen atom is ##u_{100}(\underline{r}) = C \exp (-r/a_o)##, ##a_o## the Bohr radius. For this state calculate 1)##C## 2)The radial distribution function, the probability that the electron is within a...
50. ### Getting a random number with a distribution function

I have a mean mu, and an exponential distribution function. How do I use a random number, generated with a PRNG, to get a random number from the distribution? I know this is a really basic question. Please help :) Thanks