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
{\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.
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.
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
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...
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...
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...
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...
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...
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)...
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:
$\int_{-1}^{1} \ \frac{2}{\pi(1+x^{2})} dx = {{\frac{2}{\pi} arctan(x)]}^{1}}_{-1}=1$...
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)##...
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...
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...
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=...
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]
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...
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)...
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...
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...
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?
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...
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...
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...
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...
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...
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!
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...
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=...
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
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...
With X having the exponential $(\lambda)$ distribution and $Y = X^3$, how do I compute the joint cumulative distribution function?
Here is how far I've come:
$F(x,y) = P(X ≤ x, Y ≤ y) = P(X ≤ x, x^3 ≤ y) = P(X ≤ x, X ≤ y^{1/3}) = P(X ≤ min(x, y^{1/3})$,
$f_x(x) = \lambda e^{-\lambda x}$$ for...
Homework Statement
the distribution function: f(x)=
x + 1 when -1 < x ≤ 0
-x + 1 when 0 ≤ x < 1
0 otherwiseHomework Equations
The Attempt at a Solution
on the first interval i found (1/2)x2 +x + c
on the second interval -(1/2)x2 + x + c
and when integrating the c's will cancel each...
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...
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?
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...
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##...
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...
Hello everyone:
Here is the problem:
Under thermal equilibrium, photon's number can be described as the
photonic density of state (PDOS) * occupation number(ON).
Also, the photon's flux can be described as
PDOS * ON * effective particle velocity into certain direction ( V)
The occupation...
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...
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...
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...
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 $Y$.
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...
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...
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...
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...
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