What is Distributions: Definition and 337 Discussions

In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).For instance, if X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 (1 in 2 or 1/2) for X = heads, and 0.5 for X = tails (assuming that the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc.

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  1. S

    Estimate p from sample of two Binomially Distributions

    1. Suppose X~B(5,p) and Y~(7,p) independent of X. Sampling once from each population gives x=3,y=5. What is the best (minimum-variance unbiased) estimate of p? Homework Equations P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}The Attempt at a Solution My idea is that Maximum Likelihood estimators are...
  2. A

    Minimum of two iid exponential distributions

    Let X_1, X_2 \sim Exp(\mu) and Y = min(X_1, X_2) then find E[Y]. My attempt is as follows: $$E[Y] = E[Y/X_1<X_2]P(X_1<X_2) + E[Y/X_1>X_2]P(X_1>X_2) \\ = \frac{1}{2} (E[X_1/X_1<X_2] + E[X_2/X_1>X_2] ) \\ = \frac{1}{2} (1/\mu + 1/\mu) \\ = 1/\mu$$ But, we know that minimum of two exponentially...
  3. A

    A question about uniform distributions

    If we have a uniform distribution on [0,x]...then the pdf is 1/x right? But what if we have [0,x)? Do we still have the same pdf? Thanks in advance.
  4. D

    Measure of disjointless of distributions

    What is a measure of "jointness" of a joint distribution? Correlation?
  5. C

    Point estimate from multiple sampling distributions

    Dear all, I hope someone can help me. I have two experimental groups, A (n=5) and B (n=8) containing biological samples. The samples are used to estimate my parameter of interest, θ. I do this with Markov-chain Monte-Carlo, which gives me a posterior distribution of θ for each of my samples...
  6. W

    More on Diff. Forms and Distributions as Kernels

    Hi, Again: I'm trying to show that, given a 3-manifold M, and a plane field ρ (i.e., a distribution on TM) on M, there exists an open set U in M, so that ρ can be represented as the kernel of a differential form w , for W defined on U. The idea is that the kernel of a linear map...
  7. J

    Convolution of two probability distributions using FFT

    I've been trying to code an algorithm to compute the convolution of two probability distributions. using the FFT. This relies on the "convolution theorem": (p*q)[z] = FFT^{-1}(FFT(p) \cdot FFT(q)) However, when I test it using the distributions p={0.1, 0.2, 0.3, 0.4} q={0.4, 0.3, 0.2, 0.1}...
  8. B

    Parametric versus no parametric distributions

    Hi there, I'm working on a simulation of the travel patterns of cars. There are many variables and conditional probabilities in the model. My question is, is there anything wrong with fitting all non parametric distributions to variables (both continuous and discrete)? The software I'm...
  9. H

    Binomial and Hypergeometric Distributions

    Homework Statement We have an urn with 5 red and 18 blues balls and we pick 4 balls with replacement. We denote the number of red balls in the sample by Y. What is the probability that Y >=3? (Use Binomial Distribution) Homework Equations The Attempt at a Solution Okay, so we...
  10. A

    Poisson and Gamma Distributions

    Let Y|X be a Poisson(X), and X be Gamma(\alpha, \beta). Find E(X|Y)... Since Y|X is Poisson(X), we have f(Y|X)= \frac{m^x e^{-m}}{x!}... Since X is Gamma(\alpha, \beta), we have f(x)= \frac{x^{\alpha-1} e^{-x/B}}{\Gamma(\alpha) \beta^{\alpha}}... Since f(Y|X) = \frac{f(x,y)}{f(x)} ====>...
  11. B

    Bayesian probability question about Dirichlet prior distributions

    Hi there, I have a question about Bayesian probability. I have a list of the starting times of journeys. I binned the data into 15 minute bins so I have 96 bins in total (4*24=96). So for example a journey start time of 08:05 am would be in bin number 29. As an example here is the data for...
  12. ShayanJ

    Differential form of gauss's law and surface charge distributions

    Consider the integral form of gauss's law \oint \vec{E}.\vec{d \sigma}=\frac{q}{\epsilon_0} Then let's write q in terms of the volume and surface charge densities \rho and \delta and let's assume that the surface charge is distributed over the gaussian surface of the above integral \oint...
  13. Q

    Statistics: Comparing z scores across distributions

    Homework Statement This problem is taken from the 2011 AP Statistics Exam, which we reviewed in class. The exact problem can be found here, and solution after: http://apcentral.collegeboard.com/apc/public/repository/ap11_frq_statistics.pdf...
  14. S

    Un-estimatable properties of distributions

    Are there any properties of commonly encountered probability distributions that cannot be effectively estimated by sampling them? Searching for "inestimable" lead to irrelevant links. Those links discussed not being able to estimate some parameters of a model when certain types of data are...
  15. C

    Finding the product of multiple normal distributions

    Hi all, I've been working on a little side project, but I've hit a road block on the maths for this one. Basically if you imagine a sealed pack of 10 cards, there is a 20% chance that the pack contains one foil (more valuable) card. The mass distribution of the foil cards are (heavier and)...
  16. B

    Travel patterns distributions of a sample representative of a population?

    Travel patterns distributions of a sample representative of a population?? Hi, I was hoping that somebody might be in a position to advise me on a problem that I have. I am recording the travel patterns of a car for 80 different families for 2 years. Specifically I am recording the...
  17. M

    MHB Sample Data from Distributions

    I have a question that is confusing me. If traffic in a network is being generated from a Pareto distribution, And I am estimating this traffic rate at each packet arrival, If I was to take the previous x estimated rates as a sample of data, Would the distribution of these estimated rates...
  18. O

    Why is the electric field ever defined inside volume charge distributions?

    Example: A common problem in undergraduate electromagnetic classes is calculating the electric field inside a solid spherical charge distribution with known charge density. The common method of solution in my experience is as follows: 1) Place the origin of a coordinate system at the center...
  19. E

    Estimating joint distributions from marginal

    Suppose I have the marginal probability density functions of two random variables A and B, P(A), and P(B). Suppose I modeled P(A) and P(B) using a mixture model from some dataset D and obtained a closed form pdf for each. I am interested in finding their joint density function P(A and B) and...
  20. R

    How to estimate return period amount at different distributions

    I have yearly rain amounts and want to estimate the rain with 100 year return period assuming different distribution. I know some ways to do with for example normal dist. but it's not general for all pdfs.
  21. G

    Calculating Energy Distribution of Free Particle - 1D & 2D

    Homework Statement 2. Calculate the energy distribution of the free particle (a) in one dimension, (b) in two dimensions, Use (3-12) to calculate the energy distribution of the state, assuming (a) V(r) = + K r2 (b) U f r ) = - Z e 2 / r Homework Equations (3-12); L^3 g(\epsilon)...
  22. W

    Can Two Different Random Variables Have the Same Distribution?

    Hello , frustrated with my lecturer assignments , i need your help with this : if X,Y and are two different random variable , is it possible that X,X+Y have equally distributed. if it can be give an example , if not prove it. thanks,
  23. J

    Constraints on matrix-variate normal distributions

    Hello, all. I'm wondering about matrix-variate normal distributions. I know they normally assume an n x p random matrix, X, and associated row and column covariance matrices Omega and Sigma, but I'm wondering how the probability density function changes if X is comprised of a square...
  24. N

    Changing a velocity distributions

    Hi Say I have a Gaussian velocity distribution for a collection of particles. Then I make a change to the setup in a way that the velocity distribution changes. This change is such that a part (not all) of the distribution gets bunched into one particular velocity class lower than the mean of...
  25. E

    Joint Distribution: U,Y - Find P(0≤X≤2/3)

    Homework Statement Let U,Y be independent random variables. Here U is uniformly distributed on (0,1) Where as Y~0.25\delta_{0} + 0.75\delta_{1}. Let X = UY. Find the Cdf and compute P(0≤X≤2/3) The Attempt at a Solution Normally a question like this is fairly straightforward but I'm having...
  26. E

    Curve fitting of summed normal distributions

    Hi, I have a dataset of a random variable whose probability density function can be fitted/modelled as a sum of N probability density functions of normal distributions: F_X(x) = p(\mu_1,\sigma_1^2)+p(\mu_2,\sigma_2^2)+\ldots+p({\mu}_x,\sigma_x^2) I am interested in a fitting method can...
  27. Q

    Convergence in the sense of distributions

    I have the following problem: prove that the sequence e^{inx} tends to 0, in the sense of distributions, when n\to \infty. Here it is how I approached the problem. I have to prove this: \lim \int e^{inx}\phi(x)\,dx=0 , where \phi is a test-function. I changed variable: nx=x' and got...
  28. W

    Combining Conditional Probability Distributions

    Hi all, My question is the following. Let's say I have two probability distributions; f(x|b)\,g(x|c) b and c are discrete events while x is a continuos variable. i.e When the button b is pressed there is some distribution for the amount of rain fall the next day, x. When the button c...
  29. S

    Potential of spherical and non-spherical mass distributions?

    Homework Statement Suppose a planet whose surface is spherical and the gravitational potential exterior to it is exactly -GM/r, like that of a point mass. Is it possible to know if the inner mass distribution is actually shperically symmetric? Can a non-spherical mass distribution produce...
  30. P

    Separating the product of two probability distributions

    In general, how does one separate the product of two probability distributions with one of them known? Basically, I have the distribution of rcosθ, I know that P(cosθ) = 2/(πsinθ), and I want to find P(r). Wolfram Alpha makes me think that a delta function is involved based on what they say...
  31. J

    How to simulate lognormal distributions?

    I am studying statistics and am interested in understanding the log normal distribution. From some discussion I gather that the log normal distributions arises from multiplicative effects while the normal distribution arises from additive effects. I generated the following MATLAB code to...
  32. M

    Fitting 4 parameter distributions in S-Plus (or R)

    Hi, I am trying to fit sample data to a Johnson SU distribution in S-Plus. It seems not many people use S-Plus, so if you are familiar with R then you could help as well. The code that I have is: f.Jsu.fun.takeslist(x,g,l,r,e) which is a function I have made that calculates the PDF of...
  33. A

    Calculate electric field for cylindrical charge distributions

    (2.16) A long coaxial cable carries a uniform volume charge density ρ on the inner cylinder (radius a) and a uniform surface charge density on the outer cylindrical shell (radius b). This surface charge is negative and just the right magnitude so that the cable as a whole is electrically...
  34. B

    On concave functions over spaces of probabilty distributions

    Given two (dependent) random variables X and Y with joint PDF p(x,y) =p(x|y)p(y) =p(y|x)p(x), let H[X] be real-valued concave function of p(x), and H[X|Y] the expectation of H of p(x|y) with respect to p(y). Examples of possible functions H include the entropy of X, or its variance. The...
  35. B

    Proof involving means in continuous distributions

    I recall reading somewhere that the mean value of a continuous variable is situated at a point that acts as a fulcrum about which all other values are considered "weights". In other words, if we define the mean as μ = \int^{∞}_{-∞} x ρ(x) dx (where rho is the probability density) then...
  36. A

    Continous charge or mass distributions

    Often you want to calculate the electric field of a lot of charges or the center of mass for a lot of small masses. When we have a rigid body or something similar in the world of electrodynamics, my book tells me always to calculate the cm or total field as an integral, because "the...
  37. J

    Generalized exponential family of distributions

    Homework Statement A discrete random variable Y has probability distribution given by f(y;β) = (ky2β(y+k))/((β+3)(y+2k)(y+1)1/2)Homework Equations I know that for a pdf to be from generalised exponential family of distribution it can expressed as f(y) = exp{(yθ-bθ)/a∅ +c(y,∅)}The Attempt at...
  38. S

    Normal Probability Distributions :

    Hi Physics Forum! I hope this is the right section... I couldn't find a section on statistics. This is a rather easy question but I can't seem to get the answer that the answer key says! Average Life expectancy μ=72 standard Deviation ( in years ) δ=5 The question: Recent studies have...
  39. T

    Limit involving dirac delta distributions

    Hey All, I am trying to evaluate the limit: \lim_{x\to 0^{+}} \frac{\delta''(x)}{\delta''(x)} Where \delta'(x) is the first derivative of the dirac distribution and \delta''(x) is the second derivative of the dirac distribution. I thought about the fact that this expression...
  40. N

    Probability distributions and particles

    Hi My question is not related to a specific piece of software, but more of a technique. I have a probablity distribution giving the probability of a classical particle having some velocity v. Now, what I have is a function to calculate the trajectory for a particle with some velocity vi. I need...
  41. N

    Integrate mixture of multivariate normal distributions

    I have a mixture of multivariate normal distributions, and I want to compute the integral with the first element of the input vector varying between specified limits, and the other elements varying from -infinity to +infinity. See attached pdf for equations. I've done it numerically but would...
  42. S

    Time between process problems (Probability distributions)

    Homework Statement The time between process problems in a manufacturing line is exponentially distributed with a mean of 30 days. (a) Let T be the waiting time (in days) for four problems. What is the distribution of T? (b) What is the expected waiting time for four problems? (c)...
  43. M

    Probably (yea I know, hilarious) easy qs about Bin and Normal distributions

    hey Gonna get straight to the point. I need to establish the probability difference between two probabilities p_1 and p_2 at 95%. Its the two probabilites that a cabin hook will hold for a certain force (25kN). two samples, each with sizes "the originals" n_1=107, "cheap pirated ones"...
  44. Sigurdsson

    Generalized functions (distributions) problem - Mathematical physics

    Homework Statement Find a distribution g_n which satisfies g'_n(x) = \delta(x - n) - \delta(x + n) and use it to prove \lim_{n \to \infty} \frac{\sin{nx}}{\pi x} = \delta(x) Homework Equations Nothing relevant comes up at the moment. The Attempt at a Solution Well the first...
  45. K

    Pointwise Convergence For Induced Distributions

    Homework Statement If U \subseteq \mathbb R^n find a sequence of locally integrable functions f_n \in L^1_{\text{loc}}(U) which converge pointwise, but whose induced distributions \langle f_n, \cdot \rangle: C_c^\infty(U) \to \mathbb R, \qquad \langle f_n, \phi \rangle = \int_U f_n \phi...
  46. T

    Differential equations with distributions

    Homework Statement Solve x^2\frac{du}{dx} = 0 in the sense of distributions. Homework Equations <u',f> = -<u,f'> for any test function f. The Attempt at a Solution My thinking is that since we want to see the action of the left hand side on a general test function f, we try...
  47. J

    Extreme value theory and limiting distributions for i.i.d. order statistics

    (This question was previously posted to sci.math.research. I only received one reply; sadly the advice therein conflicted with section 9.1 of H.A. David's "Order Statistics" - and probably with the fact that there was such a field of study as "r-extreme order statistics" - hence my reposting it...
  48. M

    Green function as distributions

    If we have Green function g(x,s)=exp[-\int^x_s p(z)dz] we want to think about that as distribution so we multiply it with Heaviside step function g(x,s)=H(x-s)exp[-\int^x_s p(z)dz] Why we can just multiply it with step function and tell that the function is the same. Tnx for the answer.
  49. S

    Programming languages that have statistical distributions

    hello - Do you know which programming languages have probability distributions available to use such as the binomial distribution (as an example). I guess I would like to limit my questions if it's necessary to languages that are either free or cheap - no specialty math packages for...
  50. S

    Thermodynamics, ideal gas, probability distributions

    I need some help with e), but could someone also check to see if the rest is correct? Homework Statement The velocity component v_x of gas particles in the x-direction is measured and the probability distribution for v_x is found to be P \propto e^{-\frac{-m v_x^2}{2 k_B T}} with m the...
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