# 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

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1. ### 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. ### 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. ### 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. ### 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. ### 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...

22. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...