Distributions Definition and 337 Threads

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. B

    Fitting distributions that have a singular component

    For example, suppose you have some data where each point takes its value from U(0,1) with probability p and the Cantor distribution with probability (1-p) where p is fixed but unknown. Here the standard MLE approach falls over, so how would you go about estimating p?
  2. M

    What Software Is Best for Fitting Data Distributions?

    Hi. Which is the best software to use working on data fit distributions from your experience? Regards
  3. B

    The sum of two gamma distributions

    let X~gamma(x,λ), Y~gamma(y,λ) then Z = X+Y is gamma (x+y, λ) I'm trying to prove this. Is using the moment generating functions the only way to do it. and in such case, can I assume that MZ(t)= MX(t)*MY(t)
  4. R

    Comparing two multivariate distributions (two matrices)

    I urgently need some help in my problem for my MS thesis. I have two datasets of same variable dimension but different number of observations, ie same # of columns but not same # rows. The variables are indentical for both sets. I want to compare the multivariate distributions of the two data...
  5. ArcanaNoir

    Finding E(x^k) for gamma, beta, and lognormal distributions

    Homework Statement Find the expected value of g(X) = Xk for the a. gamma distribution b. beta distribution c. lognormal distribution Homework Equations gamma distribution: \frac{x^{\alpha -1}e^{ \frac{-x}{\beta}}}{ \Gamma (\alpha ) \beta ^{\alpha }} with paramenter x>0 beta...
  6. J

    Comparing Continuous Probability Distributions: Finding Significance

    Hi, I was searching the forum about comparing continuous probability distributions and came across this post back in 2005. "You could make two variables X(t) = value of the "true" disrtibution (expensive simulation) at point t and Y(t) = value of the alternative dist. (practical simulation)...
  7. T

    Adding two distributions with same moment generating function

    Homework Statement I wanted to know what the result would be if you added two distributions with the same moment generating function. For example, what would the result be of: Mx(t) + My(t) if Mx(t) = (1/3 + 2/3et) and My(t) = (1/3 + 2/3et) Homework Equations The Attempt at a...
  8. S

    Statistics: Proofs and Problems for Random Variables and their Distributions

    Homework Statement Before I get started here I have one really quick basic question: Lets say I want the probability that an survives two hours, and that the probability an engine will fail in any given hour is .02. Then I can get 1 - .02 - .98(.02) = .9604. This is found by a geometric...
  9. E

    Distributions with compact support are tempered

    Homework Statement Let F \in \mathcal E'(\mathbf R). Prove that F \in \mathcal S'(\mathbf R). 2. The attempt at a solution Since F \in \mathcal E'(\mathbf R), there exists a continuous function f \colon \mathbf R \to \mathbf R and a nonnegative integer k such that for every \varphi \in...
  10. R

    Mean and Variance of Lognormal Distributions

    Homework Statement Given that the natural log of the growth consumption rate is conditionally normally distributed. I am trying to convert it to a lognormal distribution, but I keep getting a variance that is different from what is in the solution manual. The problem is #3 in the document below...
  11. A

    Please help on arithmetic mean of continuous distributions.

    PROVE mean (X bar) of a continuous distribution is given by: ∫x.f(x)dx {'a' is the lower limit of integration and 'b' is the upper limit}
  12. L

    Probability Distributions and asymmetry

    Homework Statement In a scattering experiment to measure the polarization of an elementary particle, a total of N = 1000 particles was scattered from a target. of these, 670 were observed to scattered to the right and 330 to the left. Assume there is no uncertainty in NL + NR = 1000 a)...
  13. D

    Probability Theory- Standard Normal Distributions

    Homework Statement Two types of coins are produced at a factory: a fair coin and a biased one that comes up heads 55 percent of the time. We have one of these coins but do not know whether it is a fair coin or a biased one. In order to ascertain which type of coin we have, we shall...
  14. H

    Solving Poisson Distributions for Expected Values of Sales

    Homework Statement A store selling newspapers orders n = 4 of a certain newspaper because the manager does not get many calls for that publication. If the number of requests per day follows a Poisson distribution with mean 3, what is the expected value of the number sold? Homework Equations...
  15. A

    What is the distribution of difference of two Gamma Distributions ?

    What is the distribution of the difference of two gamma distributions with same scale parameter, and shape parameter of the first one is k(1+e), e -> 0 and second one is k. What i exactly want to know is the following. X~Gamma(K(1+e),\theta) Y~Gamma(K,\theta) Prob (X>Y) or P(X-Y)>0...
  16. B

    Probability distributions and other variables.

    Homework Statement I'm just checking my thinking on my understanding of probability and probability distributions as it seems a little rusty. So let say we have a set of boxes arranged in a grid. We throw balls, all of identical mass m, onto the grid at random.First, am I right in saying that...
  17. T

    Distribution of weighted Normal Distributions

    You created a random number generator that works as follows: With probability p it selects a number X from the standard normal distribution N(0,1), and with complimentary probability (1-p) it selects a random number X from an off-central normal distribution N(5, 1). Write the distribution...
  18. O

    Discrete power law distributions

    I'm not a mathematician, but I want to understand how a mathematician would view this issue. I'm working primarily with degree distributions for finite graphs, and when I make a log log plot of the frequency distribution the data points form a nice straight line (at least for low degree...
  19. Rasalhague

    Poisson & normal distributions as approximations for the binomial

    These three quotes talk about the use of the Poisson and normal distributions as approximations for the binomial when n is large. The first two quotes here say Poisson is best when p small, and the normal otherwise. The third seems to change the story; it says Poisson is best for large p too. Is...
  20. M

    Model parameter distributions from Gamma distributed data

    I have a set of data points {xi,yi} where each yi is a Gamma distributed variable where both the shape k and scale \theta depend on i. I then fit the data points with a power law model y=a(x)b. I would like to know the probability distributions for the fit parameters a and b. Is...
  21. S

    Are These Random Variables Independent or Identically Distributed?

    Homework Statement Let \Omega = {w1, w2, w3}, P(w1) = 1/3, P(w2) = 1/3, P(w3) = 1/3, and define X, Y, Z as follows: X(w1) = 1, X(w2) = 2, X(w3) = 3 Y(w1) = 2, Y(w2) = 3, Y(w3) = 1 Z(w1) = 3, Z(w2) = 1, Z(w3) = 2 (a) Show that these 3 random variables have the same distribution. (b) Find the...
  22. S

    Physical intuitions for simple statistical distributions

    I'm trying to understand why various statistical distributions are so common. For the most part, all I can find online is how to calculate and manipulate them... I did finally find a couple of refs that helped with Gaussians, this being one: http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf"...
  23. O

    Probability: Nested Uniform Distributions

    Homework Statement Problem A: A random variable T is selected from a uniform distribution over (0,1]. Then a second random variable U is selected from a uniform distribution over (0,T]. Determine the probability Pr(U>1/2). Problem B: Suppose 3 identical parts are chosen for inspection. Each...
  24. S

    Understanding Markov Processes & Stationary Distributions

    Homework Statement Consider a Markov Process X(t) over discrete set omega = {-1,0,1} with transition probabitities W(-1 | 0,t)=W(0 | -1,t)=W(1 | 0,t)=W(0 | 1,t)=d a) What is master equation b) Find the stationary distrubtion Ps(x) for x element of {-1,0,1} The Attempt at a Solution a) the...
  25. L

    Could someone help me find the Covariance of these two distributions

    Homework Statement [PLAIN]http://img695.imageshack.us/img695/7551/unledsi.png Homework Equations The Attempt at a Solution I get E[u]=1/3 and E[V]=1, can't get E[UV] to be correct as I do not get the required answer, any help would be greatly appreciated! thanks!
  26. Q

    Probability of No Events in First Two Hours for Poisson Process with Rate 2

    Homework Statement Consider a Poisson process for which events occur at a rate of 2 per hour. (a) Give the probability that the time until the first event occurs exceeds 2 hours. Use an exponential distribution to find the probability.Homework Equations The Attempt at a Solution \lambda = 1/2...
  27. M

    About the linear combination of multivariate normal distributions.

    How can I prove that the any linear combination of multivariate normal distribution is also normal? I can prove it but I'm not sure that this is right or not. The point of my proof is as follows. --- The X and Y has the same dimensional random vector, and each random vector is...
  28. G

    How Do You Find the Density Function of a Sum of Exponential Random Variables?

    Let X1,...,Xn be independent, identically distributed random variables with exponential distribution of parameter λ. Find the density function of S = X1+...+Xn. (This distribution is called the gamma distribution of parameters n and λ). Hint: Proceed by induction. At first I tried computing...
  29. S

    Multivariate Distributions, Moments, and Correlations

    So if I start with a multivariate distribution f(x,y), I can find the marginal distributions, the conditional probability distributions, all conditional moments, and by the law of iterated expectations, the moments of both X and Y. It seems to me that I should be able to relate the conditional...
  30. D

    How to Measure the Distance Between Two Distributions?

    Hello, I've some two distributions, how can I find the distance between those two distributions? is the difference between the mean values would be the distance ?
  31. Z

    Why distributions can not be multiplied ?

    Why distributions can not be multiplied ?? why in general can not give a meaningful expression for \delta (x) \delta ^{m} (x) or H(x) \delta (x) for example the Fourier transform (with respect to 'x') of the expression (theoretically) \int_{-\infty}^{\infty}dt (x-t)^{m}t^{n} =g(x)
  32. E

    Solving the Trinomial & Binomial Distributions: A Challenge

    can anyone help me please can anyone solve this problem for me please Q) The Binomial distribution allows the calculation of the probability of k successes in n trails where there are only two outcomes: success or fail with probabilities p and q respectively. The Binomial probability is...
  33. S

    Help with joint distributions?

    Homework Statement Suppose X and Y have joint density f(x,y)=2 for 0<y<x<1. Find P(X-Y>z) According to the textbook the answer should be (1-z)^{2}/2 Homework Equations The Attempt at a Solution \int \int 2dxdy for x=[0, z+y] and y=[0,1] =\int 2(z+y) dy =2z+1 since we...
  34. S

    Help with simulating distributions

    Help with simulating distributions... Homework Statement For each of the following c.d.f F, find a formula for X in terms of U, such that if U~Uniform[0,1], then X has c.d.f F. a) F(x) = 0 if 0 x<0 x if 0<=x<=1 1 if x>1 b) F(x) = 0 if 0 x<0 x^2 if 0<=x<=1 1 if x>1 c) F(x) =...
  35. M

    Combining results from multiple distributions

    Suppose I have a result where the outcome is that with 95% confidence interval of the sample is between 27 and 42. With a second method the test result of the same sample gives a 95% confidence interval between 27 and 48. And with a third method 95% confidence interval of the sample is...
  36. K

    Convolution of densities and distributions

    Hello everyone, I have a quick theoretical question regarding probability. If you answer, I would appreciate it if you would be as precise as possible about terminology. Here is the problem: I'm working on some physics problems that do probability in abstract spaces and the author freely...
  37. S

    Help with cumulative distributions

    Homework Statement suppose Fy(y)=y^3 for 0<=y<1/2 and Fy(y)=1-y^3 for 1/2<=y<=1. Compute these. 1. P(1/3<Y<3/4) 2. P(Y=1/3) 3. P(Y=1/2) Homework Equations The Attempt at a Solution Is this right for the 1. ? P(1/3<Y<3/4) P(1/3<Y<1/2) + P(1/2<=Y<3/4) (...
  38. C

    Iterative expectation of continuous and discrete distributions

    Homework Statement Suppose X ~ uniform (0,1) and the conditional distribution of Y given X = x is binomial (n, p=x), i.e. P(Y=y|X=x) = nCy x^{y} (1-x)^{n-y} for y = 0, 1,..., n. Homework Equations FInd E(y) and the distribution of Y.The Attempt at a Solution f(x) = \frac{1}{b-a} = \frac{1}{1-0}...
  39. G

    A problem in understanding distributions exercise

    I'm reading the first chapters of "A Guide to Distribution Theory and Fourier Transforms". On page 10, Exercises 3,6,7 the distribution is defined in terms of integrals. The first one is always without integrand (there's only the integral sign). What does that mean? Am I missing something? The...
  40. T

    Probability with normal distributions

    Homework Statement The time taken by Smith to travel from A to B is a random variable which follows the normal distribution , with mean 5 minutes and standard deviation 1 minute. The time taken by Jones to travel from A to B follows the normal distribution with mean 15 minutes and SD 2...
  41. N

    Are There Distributions Different from Fermi-Dirac and Bose-Einstein?

    Please teach me whether it is possible there are any distributions different from Fermi-Dirac and Bose-Einstein distributions.Because the Statistic Theorem only demontrates that integer spin particles can't obey Fermi-Dirac distribution and spin-haft particles can't obey Bose-Einstein distribution.
  42. D

    Prove Idntity - Dirac Delta - Distributions

    Homework Statement The Identity to prove: Homework Equations Using Integration by parts The Attempt at a Solution I couldn't produce the denominator.
  43. A

    Ramp function, Dirac delta function and distributions

    r(x) = x if x \geq 0 and r(x) = 0 if x<0 I have to show that: 1-\[ \int_{- \infty}^{+ \infty} r(x) \varphi ''(x) dx = \varphi(0) \] And 2- that the second derivative of r is the Dirac delta. And I managed to do this by integrating by parts. Howver, I don't get why I can't just write: \[...
  44. D

    Relating probability distributions huh?

    Homework Statement Problem description: A variable X has expected value 0.002 in meters. Consider X - 0.002, scale to millimeter, and we get Y. Tasks: a) Express Y as a function of X b) Relate the probability distributions FX and FY c) Relate the probability density functions fX and fY...
  45. R

    Conditional expectations of bivariate normal distributions

    Hey guys, I'm having a bit of a problem with this question... Homework Statement If X and Y have a bivariate normal distribution with m_X=m_y=0 and \sigma_X=\sigma_Y=1, find: a) E(X|Y=1) and Var(X|Y=1) b) Pr(X+Y>0.5) Homework Equations N/A The Attempt at a Solution...
  46. E

    How Do You Calculate Total Angular Distribution in Particle Decay?

    Hi there! I have to find the angular distribution of a decay where I suppose I don't know if the parity is conserved. I made my calculus and I found that I have two possible final states, one with total angular momentum L=0, m=0 and one with L=1, m=1. Now I have to find the total angular...
  47. L

    MATLAB Fitting Bimodal/Unimodal Distributions in MATLAB

    i have some values that seems to have 2 modes and i don't know how to fit a distribution to them in matlab. Does MATLAB have any function for fitting bimodal/unimodal distributions? edit: it seems like the function gmdistribution have something to do with it but this only concerns gaussian...
  48. P

    Distributions and Intergration by Parts

    "Distributions" and Intergration by Parts Homework Statement Has it been proven that it is ok to use Integration by parts on "Distributions" like Dirac Delta functions inside an integration? Homework Equations Need to figure out how to write integral signs and Greek alphabet symbols...
  49. I

    Sampling Distributions and Normal Approximation

    Homework Statement A sample survey interviews an SRS of 267 college women. Suppose (as is roughly true) that 70% of all college women have been on a diet within the past 12 months. Use a Normal approximation to find the probability that 75% or more of the women in the sample have been on a...
  50. Z

    Arithmetical function and distributions

    can any Arithmetical function A(x)= \sum_{n\le x}a(n) be regarded as the train of dirac delta functions (its derivative) dA = \sum_{n=1}^{\infty}a(n)\delta (x-n) from this definition could we regard the explicit formulae for chebyshev function d\Psi(x) =1- \sum_{\rho}x^{\rho...
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