What is Cdf: Definition and 113 Discussions

In statistics, cumulative distribution function (CDF)-based nonparametric confidence intervals are a general class of confidence intervals around statistical functionals of a distribution. To calculate these confidence intervals, all that is required is an
independently and identically distributed (iid) sample from the distribution and known bounds on the support of the distribution. The latter requirement simply means that all the nonzero probability mass of the distribution must be contained in some known interval


{\displaystyle [a,b]}

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  1. Hamiltonian

    I Finding the pdf of a transformed univariate random variable

    The above theorem is trying to find the pdf of a transformed random variable, it attempts to do so by "first principles", starting by using the definition of cdf, I don't understand why they have a ##f_X(x)## in the integral wouldn't ##\int_{\{x:r(x)<y\}}r(X) dx## be the correct integral for the...
  2. Isopod

    A CDF measures W mass higher than predicted

    The team has found that the particle, known as a W boson, is more massive than the theories predicted. The result has been described as "shocking" by Prof David Tobak, who is the project co-spokesperson. The discovery could lead to the development of a new, more complete theory of how the...
  3. docnet

    Calculate the joint CDF of two random variables

    $$f_{XY}=1$$ $$dzdy=2xdxdy⇒\frac{1}{2\sqrt{z}}dzdy=dxdy$$ $$f_{ZY}=\frac{1}{2\sqrt{z}}\quad \text{on some region S}$$ $$F_{ZY}=\int^y_{g}\int^x_{h}\frac{1}{2\sqrt{z}}dzdy\quad\text{for some}\quad g(x,y),h(x,y)$$ im learning how to find the region S using a change-of variables technique
  4. M

    MHB Joint CDF of a word problem

    So I have this word problem as seen below: Joy and Ethan have agreed to meet for dinner between 8:00 PM and 9:00 PM. Suppose that Ethan may arrive at any time between the set meeting. Joy on the other hand will arrive at the set meeting under the following conditions: • Joy will always arrive...
  5. M

    MHB Cdf, expectation, and variance of a random continuous variable

    Given the probability density function f(x) = b[1-(4x/10-6/10)^2] for 1.5 < x <4. and f(x) = 0 elsewhere. 1. What is the value of b such that f(x) becomes a valid density function 2. What is the cumulative distribution function F(x) of f(x) 3. What is the Expectation of X, E[X] 4. What is...
  6. B

    I Finding CDF given boundary conditions (simple stats and calc)

    I'm not quite sure if my problem is considered a calculus problem or a statistics problem, but I believe it to be a statistics related problem. Below is a screenshot of what I'm dealing with. For a) I expressed f(t) in terms of parameters p and u, and I got: $$f(t)=\frac{-u \cdot a + u \cdot...
  7. E

    I Order Statistics: CDF Calculation for i.i.d. Random Variables

    Suppose I have the random variables ##Z_k=X_k/Y_k## with a PDF ##f_{Z_k}(z_k)## for ##k=1,\,2\,\ldots, K##, where ##\{X_k, Y_k\}## are i.i.d. random variables. I can find \text{Pr}\left[\sum_{i=1}^3Z_k\leq \eta\right] as...
  8. E

    I The CDF from the characteristic function

    This thread will be a collection of multiple questions I asked before over different forums. I will start from the beginning, and I hope someone will follow the steps with me, because I did it before alone, and I ended with a numerical integration that is not finite, which doesn't make sense...
  9. E

    I The CDF from the Characteristic Function

    Is there a way to find the CDF of a random variable from its characteristic function directly, without first finding the PDF through inverse Fourier transform, and then integrate the PDF to get the CDFÉ
  10. E

    I CDF of summation of random variables

    Hi, I have this random variable ##\beta=\sum_{k=1}^K\alpha_k##, where ##\{\alpha_k\}_{k=1}^{K}## are i.i.d. random variables with CDF ##F_{\alpha}(\alpha)=1-\frac{1}{\alpha+1}## and PDF ##\frac{1}{(1+\alpha)^2}##. I want to find the CDF of the random variable ##\beta##. So, I used the Moment...
  11. RJLiberator

    For the cdf F(x) find the pmf f(x), 25th percentile, 60th percentile

    Homework Statement We have F(x) = ∑ (1/2)^j [Sum goes from j=1 to x] For the cdf F(x), find the pmf f(x), the 25th percentile, and the 60th percentile.Homework EquationsThe Attempt at a Solution I've been doing numerous of these for continuous distributions, however this one is tricking up...
  12. aquaelmo

    Find the cdf given a pdf with absolute value

    Homework Statement Consider a continuous random variable X with the probability density function fX(x) = |x|/5 , – 1 ≤ x ≤ 3, zero elsewhere. I need to find the cumulative distribution function of X, FX (x). 2. Homework Equations The equation to find the cdf. The Attempt at a Solution FX(x)...
  13. barryj

    How to display a normal CDF graph on the TI-84 calculator

    I am not sure where to ask this question but I must try somewhere. 1. Homework Statement I am trying to graph a simple cdf on my ti-84 and I cannot get it to work. I need an example to follow Homework Equations See below. The Attempt at a Solution 2nd, dist, 2 enter -1E99,0 0,.5 [/B]I also...
  14. Z

    Verifying Solution for PDF to CDF and Inverse CDF Calculations

    Homework Statement I was hoping someone could just verify this solution is accurate. p(x) = 0 , x < 0 4x, x < .5 -4x + 4 , .5 <= x < 1 Find CDF and Inverse of the CDF. Homework EquationsThe Attempt at a Solution CDF = 0 , x < 0 2x^2 ...
  15. A

    CDF of minimum of N random variables.

    There's this problem that I've been trying to solve. I know the solution for it now but my initial attempt at a solution was wrong and I can't seem to figure out the mistake with my reasoning. I'd appreciate some help with figuring this one out. 1. Homework Statement I have a set of random...
  16. E

    B Evaluating CDF of a Random Variable with Exponential Components

    Hello all, I have the following random variable ##X=\frac{a_1}{a_2+1}##, where ##a_i=b_i/c_i##, where ##b_i## and ##c_i## are exponential random variables with mean 1. I need to evaluate the CDF of ##X## as F_X(x)=Pr\left[X\leq x\right]=Pr\left[\frac{a_1}{a_2+1}\leq...
  17. E

    B Finding CDF of Gamma_m: Solve Using Functions

    Hello all, I have the following random variable ##\Gamma_m=\frac{a_m}{\sum\limits_{\substack{n=1\\n\neq m}}^Ka_n+1}## where the random variables ##\{a_n\}## are independent and identically distributed random variables. The CDF of random variable ##a_n## if given by F_{a_n}(x)=1-\frac{1}{1+x}...
  18. E

    B The CDF of the Sum of Independent Random Variables

    Hello all, Suppose I have the following summation ##X=\sum_{k=1}^KX_k## where the ##\{X_k\}## are independent and identically distributed random variables with CDF and PDF of ##F_{X_k}(x)## and ##f_{X_k}(x)##, respectively. How can I find the CDF of ##X##? Thanks in advance
  19. E

    B Finding the mean using the CDF

    Hello all, I have the random variable ##X## with CDF and PDF of ##F_X(x)## and ##f_X(x)##, respectively. Now I have a function in terms of the random variable ##X##, which is ##e^{-X}##, and I want to find the mean of this function. Basically this can be found as...
  20. R

    A Using Metropolis-Hastings for Sampling from a CDF

    Hi If I have a CDF = integral wrt t from 0 to r^2/2 of 2*sqrt(pi*t)*e^(-t)*dt but I want to generate samples from the PDF, would Metropolis-Hastings algorithm be the best alternative? Rgds rabbed
  21. J

    Random process involving CDF and PDF of standard normal

    Homework Statement Let $$ \Phi(x)=\int_{-\infty}^{x} \frac{1} { \sqrt{2\pi} } e^{-y^2 /2} dy $$ and $$ \phi(x)=\Phi^\prime(x)=\frac{1} { \sqrt{2\pi} } e^{-x^2 /2} $$ be the standard normal (zero - mean and unit variance) cummulative probability distribution function and the standard normal...
  22. R

    Bivariate transformation using CDF method

    If I have the following relations: X = sqrt(1-V^2)*cos(U) Y = sqrt(1-V^2)*sin(U) Z = V where (-pi < U < pi) and (-1 < V < 1) are independent random variables, both with uniform distributions. How do I use the CDF method to find X_pdf(x)? X_pdf(x) = X_cdf'(x) = ( P( X < x ) )' = ( P(...
  23. W

    Finding the PDF and CDF of a given function Z = X/Y

    Homework Statement Given a Uniform Distribution (0,1) and Z = X/Y Find F(z) and f(z) Homework EquationsThe Attempt at a Solution So I'm just trying to make sure i have the range correct on this one... I'm honestly lost from beginning to end with it. R(z) = {0,∞} because as y is very small, Z...
  24. J

    Solving Gaussian Random Variable Expected Value: CDF & Expectation

    Hi, I have trouble with the following problem: Gaussian random variable is defined as follows \phi(t) = P(G \leq t)= 1/\sqrt{2\pi} \int^{t}_{-\infty} exp(-x^2/2)dx. Calculate the expected value E(exp(G^2\lambda/2)). Hint: Because \phi is a cumulative distribution function, \phi(+\infty) =...
  25. J

    How can I find the CDF and PDF of Y?

    Problem Let X be a uniform(0,1) random variable, and let Y=e^−X. Find the CDF of Y. Find the PDF of Y. Find EY. Relevant Equations http://puu.sh/kAVJ8/0f2b1e7b22.png My attempt at a solution If I solve for the range of y I get (1, 1/e), but because Y is not an increasing function, my...
  26. W

    Expectation of a function of a continuous random variable

    Homework Statement X ~ Uniform (0,1) Y = e-X Find FY (y) - or the CDF Find fY(y) - or the PDF Find E[Y] 2. Homework Equations E[Y] = E[e-X] = ∫0 , 1 e-xfx(x)dx FY(y) = P(Y < y) fY(y) = F'Y(y) The Attempt at a Solution FX(x) = { 0 for x<0 x for 0<x<1 1 for 1<x } fX(x) = { 1 for...
  27. W

    Conditional PDF question -- I think anyway....

    Homework Statement Suppose you take a pass-fail test repeatedly. Let Sk be the event that you are successful in your kth try, and Fk be the event that you fail the test in your kth try. On your first try, you have a 50% chance of passing the test. P(S1)=1−P(F1)=1/2. Assume that as you take the...
  28. _N3WTON_

    CDF and PDF Calculations

    Homework Statement The current in a certain circuit as measured by an ammeter is a continuous random variable X with the following probability density function: f(x) = 0.057x + 0.272 if 3 <= x <= 5. It is 0 otherwise. a) Verify that the total area under the density curve is 1. b) Obtain the...
  29. D

    Interpretation of the maximum value of a CDF

    I was watching a youtube video from MIT's open courseware series on probability. A scenario was proposed: Al is waiting for a bus. The probability that the bus arrives in x minutes is described by the random variable X, which is uniformly distributed on the interval [0,10] (in minutes). I...
  30. A

    CDF to PDF problem (probability)

    Homework Statement Homework Equations integral from -inf to inf of fx(x)dx = 1, fx(x)=PMF The Attempt at a Solution I get values for probability function PMF of: 0.3, x=0 0.3, x=2 A-0.6, x=3 I guess I try to find area under curve of PMF which will be equal to 1. (0.3)(2) +...
  31. B

    Finding the pdf and cdf of this function

    Homework Statement Let ##X## have the pdf ##f_X(x)=\dfrac{1}{\sqrt{2\pi\sigma^2}}e^{\dfrac{-(x-\mu)^2}{2\sigma^2}}## where ##-\infty<x<\infty,-\infty<\mu<\infty,\sigma>0##. Let ##Z=g(X)=\frac{X-\mu}{\sigma}##. Find the pdf and cdf of ##Z##[/B] Homework EquationsThe Attempt at a Solution...
  32. L

    CDF of Distance and Angle from Origin of N(0, 1) RVs Y and Z

    Y and Z are independent N(0, 1) random variables. Let X = |Z|. Consider the random point (X, Y). (a) Derive the CDF FD(d) = P(D ≤ d) of the distance from the origin D = √(X2 + Y2). Sketch this CDF as a function of all real d. (b) The ratio T = Y/X has Student’s t-distribution with 1 degree...
  33. B

    Relating the CDF to the probability density

    Homework Statement If ##X## is any random variable defined on ##[0,\infty]## with continuous CDF ##F_X(t)##. Prove that ##E(X)=\int_1^\infty (1-F_X(t)) dt##. . Homework Equations The Attempt at a Solution I am not sure how to go about this. I think double integration can be used to prove it...
  34. Mogarrr

    Statistics, inverse of cdf

    Homework Statement Show that the given function is a cdf (cumulative distribution function) and find F_X^{-1}(y) (c) F_X(x) = \frac {e^{x}}4 , if x<0, and 1-(\frac {e^{-x}}4) , if x \geq 0 Homework Equations for a strictly increasing cdf, F_X^{-1}(y) = x \iff F_X(x) = y and for a...
  35. R

    What is Nc in the R Ratio formula? re CDF detector measurements

    Ive attached the R ratio formula to this post, please could someone who is familiar with it tell me what Nc is? Does it relate at all to centre of mass energy? Thanks in advance for any help. source...
  36. N

    CDF Query: Conditional CDF of S

    Hello. I was wondering if the following is correct. Let S= a*X/(b*X+c), where a,b,c are positive constants and X is a positive random variable. Also let H= h, where h is also a positive random variable (S and H are mutually independent). Then, let F_{Z}(.) and f_{Z}(.) denote the CDF...
  37. R

    MHB Transform Random Var CDF to Standard Normal: F(x)=1-exp(-sqrt x)

    How to transform a random variable CDF to a standard normal Given F(x) = 1- exp (-sqrt x), for x greater that 0 Thanks.
  38. N

    CDF of correlated mixed random variables

    Hello, i m trying to evaluate the following: r*x - r*y ≤ g, where r,x,y are nonnegative random variables of different distribution families and g is a constant nonnegative value. Then, Pr[r*x - r*y ≤ g] = Pr[r*x ≤ g + r*y] = ∫ Fr x(g + r*y)*fr*y(y) dy, where F(.) and f(.) denote CDF and...
  39. T

    Finding the CDF from a PDF with absolute value function

    Homework Statement Find the CDF of f(x) = |\frac{x}{4}| if -2<x<2 \\ 0 otherwise Homework Equations The Attempt at a Solution I have to integrate the pdf and to do so, I have to split it into two parts \int_{-x}^{0}\frac{-t}{4}dt + \int_{0}{x}\frac{t}{4}dt integrating I get \frac{x^2}{8} +...
  40. I

    Moment generating function, CDF and density of a random variable

    Assume X is a random variable under a probability space in which the sample space ?= {a,b,c,d,e}. Then if I am told that: X({a}) = 1 X({b}) = 2 X({c}) = 3 X({d}) = 4 X({e}) = 5 And that: P({a}) = P({c}) = P({e}) = 1/10 P({b}) = P({d}) = 7/20 Find the C.D.F of X, the density of X...
  41. S

    Is there an expression for the integral of an arbitrary CDF?

    Let F be any distribution function. With either the indefinite integral, or taking limits at plus and minus infinity, is there an equivalent expression to ∫ F(x)dx ? Can we derive one? Thanks.
  42. U

    MHB Finding the conditional variance and CDF

    Question: Assume a bivariate GARCH process as follows: \begin{align} r_{mt} &= \sigma_{mt}\epsilon_{mt} \ \ \ \cdots \ \ \ \text{(1)} \\ r_{it}&=\sigma_{it}\rho_{it}\epsilon_{mt}+\sigma_{it}\sqrt{1-\rho_{it}^2}\xi_{it} \ \ \ \cdots \ \ \ \text{(2)} \\ (\epsilon_{mt}, \xi_{it}) & \sim S...
  43. N

    MHB CDF, and setting the integral for this

    Please refer to the attached image. I can't quite get the bounds for question a) right. it's so confusing. Would it be wise to split the double integral into two parts? I guess that's usually favourable with when dealing with absolute values. But the bounds are still confusing me. would it be...
  44. F

    CDF: Prove a>0 for F(x) = 1 - e^(-ax) - axe^(-ax)

    Homework Statement Given the function F(x) = 1 - e^(-ax) - axe^(-ax) for x>=0 and 0 elsewhere, for which values of 'a' does the function constitute a CDF? Homework Equations The Attempt at a Solution I started with saying for that to occur, provided x1>x2 -> F(x1) > F(x2)...
  45. N

    MHB Finding the CDF: Solving for c & Understanding Results

    Please refer to the attached image. For part a) when I want to find the CDF, don't I simply take the indefinite integral of e^-|x|, multiply it by c and solve for that = 1? I am unsure of how to take the integral for this, am i correct in saying it is -e^-x, for all x ? that would leave me...
  46. N

    Survival Analysis: find marginal CDF from marginal hazard rates?

    Homework Statement Homework Equations My professor gives us the following formula: [ tex ] F_j(t) = \int\limits_0^t exp\Big\{-\sum\limits_{j=1}^J \int\limits_0^u \lambda_j^{\#}(v)dv \Big\} \lambda_j^{\#}(u)du[ / tex ] where [ tex ]\lambda_j^{\#}(t) [ / tex ] are the cause-specific...
  47. N

    How derive a CDF from MGF directly ?

    Hi, i am trying to develop a CDF from a given MGF. The standard way of using the inverse Laplace transform etc.. is not feasible due to complexity of MGF. I was woldering if there is another straighforward direction via integration or differentiation method to produce the CDF (or PDF)...
  48. C

    Integrating a normal density to find a CDF

    Homework Statement Let X ~ norm(5,10). Find P(X>10).Homework Equations f(x) = \frac{1}{δ\sqrt{2π}} e^{-\frac{(x-μ)^2}{2δ^2}} F(x) = P(X<x) = \int_{-∞}^x f(u) du The Attempt at a Solution P(X>10) = 1 - P(X<10) P(X<10) = \int_{-∞}^{10} \frac{1}{δ\sqrt{2π}} e^{-\frac{(x-μ)^2}{2δ^2}} dx =...
  49. J

    Cdf (Cumulative Density Function) Confusion

    Hi there, So regular i thought that the procedure was F(s) = ∫s0 f(x) dx However i am doing a problem with a kinked pdf and it is telling me to do something like F(s) = ∫s0 f(s) ds for 0=<s>=1/2 then... F(s) = F(1/2) + ∫s1/2 f(s) ds I am confused at the process of using f(x) or...
  50. F

    CDF & PDF: Statistics Basics for Tomorrow's Test

    how are cdf and pdf related in statistics? please help i have a test tomorrow