# Questions tagged [cumulative-distribution-function]

Cumulative distribution function. While the PDF gives the probability density of each value of a random variable, the CDF (often denoted $F(x)$) gives the probability that the random variable will be less than or equal to a specified value.

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### calculate joint cdf from joint pdf, joint pdf of (x,y) is 2*e^(-x)*e^(-y) with domain 0<x<y<inf [closed]

I have already calculated the first part which is 2, but i have no idea how to calculate the joint cdf from joint pdf, my professor gives solution like this , but i am still confused about it. in the ...
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### Is the cumulative distribution function of a r.v. X strictly increasing (X -) almost everywhere?

Let $X$ be a random variable and $F_X(x) = P(X \le x)$ its cumulative distribution function (cdf). $P_X$ is the probability measure induced by $X$, which is defined by $P_X((a,b)) = P(X^{-1}((a,b))$ ...
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### Confidence interval for sum of product of scaled binomial random variables

I have discrete, independent, but not necessarily identically distributed random variables $X_1,\dots,X_n$ that take on non-negative integer values. Each random variable has unknown distribution ...
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### Transforming data with a fitted distribution function

I have a bivariate dataset on $[0,1]^2$ in which I am interested in fitting a joint distribution. I fit a Gaussian copula but am unsure how to judge if it's a good fit. I tried transforming my data ...
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### Problem with Cumulative distribution function

I can't understand this cumulative distribution function. I would like to calculate the data distribution function: ...
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### Correspondence between the "density function of a probability measure" and the "probability density function" (PDF)

Question. If there is a one to one correspondence between a "borel probability measure" on the line $\mathbb{R}$ and a "cumulative distribution function" (CDF) (please see on page ...
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### Why P(X=x) is not zero here?

From the CDF,I feel that the random variable here is continuous.So shouldn't the P(X=x) equals 0 here and we only need to find P(X>=1).
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### Copula Invariance Principle

I don't get why equation 7 is true, can someone explain me why? This is part of the proof of the invariance principle in copula theory.
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### Calculating PDF conditioned on event

I'm confused about problems where we calculate a PDF conditioned on an event. Consider this simple problem: We have two random variables, X and Y, X is uniformly distributed on [a,b], and Y is uniform ...
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Let $(A,B,C)$ follow a multinomial distribution: $$(A,B,C) \sim \text{MultiNom}\left(n=100,p_1=p_2=p_3=\frac13\right).$$ Define $X$, a discrete random variable as $$X = f(A,C) = 2A + 3B + 4C = 2A + 3(... • 5,364 0 votes 0 answers 32 views ### Approximate distribution of random variable similar to studentized mean R.V? It is well known that the distribution of the studentized mean, i.e., T_0 = \frac{n^{1/2} (\bar{x}- \mu)}{\left(n^{-1} \sum \limits_{i=1}^n (x_i^2 - \bar{x}^2)\right)^{1 / 2}} , can be approximated (... 1 vote 1 answer 31 views ### Problem with deriving the cumulative distribution from the density function Consider the continuous distribution with density function$$ p(x) = \frac{1}{2}\cos(x) \;, \quad -\frac{\pi}{2} < x < \frac{\pi}{2}. $$I want to derive the cumulative distribution function for ... • 13 2 votes 1 answer 85 views ### Fitting a nonlinear model for a CDF I have two questions in general here. Suppose I am recording data in time and the response that I am collecting is a monotonic curve that goes from 0 to 1 (sort of a like a CDF). I was thinking of ... 0 votes 1 answer 75 views ### Compare CCDF of datasets with different sample size I have 3 dataframes with the same structure (each dataframe includes a different type of tweet). Here are the columns of dataframes: id, ... • 103 1 vote 0 answers 39 views ### Is there a package/function in R, Matlab or Python to perform smooth non-parametric fitting of empirical CDF data? [closed] I have numerical process that generates percentiles values of a certain random variable X, meaning that I have x_i values and their corresponding probability values \hat{F}(x_i). I want to fit a ... 0 votes 0 answers 75 views ### Approximation on Inverse Mills ratio for the normal R.V I've come across several approximations for Mills ratio, but I haven't found any good ones for the Inverse Mills ratio. Is there any known closed-form approximation for the Inverse Mills ratio (link) ... 2 votes 1 answer 92 views ### Distribution of sum of a discrete uniform and a uniform on (0,2) Let U be uniformly distributed on the interval (0, 2) and let V be an independent random variable which has a discrete uniform distribution on {0, 1, . . . , n}. i.e. P\{V = i\} =\frac{1}{n + ... • 281 0 votes 0 answers 26 views ### Question about joint cdf we have that P(X \leq x, Y \leq y) = \int \int_{s \leq x, t \leq y} f_{X,Y}(s,t) dsdt But how would for example P(X \leq x, Y \geq y) Be defined? Would it just be: P(X \leq x, Y \geq y) = \int \... • 83 1 vote 1 answer 75 views ### CDF of \max under conditions Let, $$g(\alpha,\beta) = \begin{cases} \frac{\alpha}{\beta}, & \text{if } \alpha > \beta \\ 0, & \text{if } \alpha \leq \beta \end{cases}$$ I want to find ... • 223 0 votes 1 answer 89 views ### Auxiliary randomization and the generalized CDF inverse I'm trying to solve Homework 4 from professor Ryan Tibshirani's class on "Advanced Topics in Statistical Learning" [pdf] at UC Berkeley. It deals with basic facts about CDFs and quantiles. ... 0 votes 0 answers 42 views ### Estimation of Distribution using multiple ECDFs Every day, I keep track of the processing times for each input to my CPU and create empirical cumulative distribution functions (ECDFs) based on this data. Let's assume I have 100 observations per day ... • 43 1 vote 0 answers 23 views ### Fit generalized linear mixed model (with lme4 or other) to cumulative data of a continuous variable I have measurements of resin production of pine, which are taking tapping the tree, that is, making a physical wound and collecting the resin in a pot. When the pot is full we replace it with an empty ... • 111 1 vote 1 answer 54 views ### Parameters of the log-normal from CDF of a composition of n i.i.d Let X_1,\ldots,X_n be i.i.d. log-normal random variables such that$$\log(X_i)\sim N(\mu,\sigma^2)\ \ \forall i=1,\ldots,n$$Now let Y be equal to the \min(X_1,\ldots,X_n). It is quite easy to ... 0 votes 0 answers 36 views ### Exact confidence interval for the binomial distribution I am confused about the way to obtain an exact confidence interval for the binomial distribution. Say, we have n samples, X_1, X_2, ..., X_n \in \{0,1\}. Then the probability of having k ... • 217 0 votes 1 answer 127 views ### Determine CDF and PDF from quantiles I would like to determine CDF and PDF from quantiles that I have determined via quantile regression. I have read here in the forum (Find the PDF from quantiles) that it is possible to interpolate this ... 0 votes 0 answers 40 views ### The probability integral transform when the CDF is non-decreasing I'd like to ask about middle of the discussion in this answer. Writing things out in reverse, \mathrm{Prob}(F_X(X) \leq y) = \mathrm{Prob}(X \leq \mathrm{inf}\{x: F_X(x) \geq y\}). Why is \mathrm{... • 315 2 votes 2 answers 135 views ### Integral of cdf of a symmetric random variable How to compute$$\int_{-k}^{k}F(x)dx$$where F(x) is the cumulative distribution function of continuous random variable X which has symmetric pdf about x=0 and k>0. • 31 0 votes 0 answers 23 views ### Statistical Inference by Casella Exercise 4.51 [duplicate] I am self-studying statistical inference by Casella and Berger and having difficulty solving exercise 4.51: let X, Y, Z \sim U(0,1) and they are independent. Find P(X/Y \leq t) and P(XY \leq t). ... 1 vote 1 answer 53 views ### Cumulative distribution function of mixed variables Given the probability density function:  f_{X, Y}(x, y)=\begin{cases} \frac{xy}{3}, & \text{if } x=1,2,3 \text{ and } 0 < y < 1.\\\\ 0, & \text{otherwise}. \... • 11 0 votes 1 answer 88 views ### How do we define the pdf in the multi-variate case and compute expectations? Apologies if this is a very simple question but trying to work through a result in a paper made me realize I missed something a bit fundamental in my undergrad probability and analysis courses. Lets ... • 75 4 votes 3 answers 233 views ### How to show that the negative binomial CDF converges to the Poisson CDF? (Incomplete beta vs incomplete gamma functions) Question: Is there a straightforward proof of the following relationship between the (lower, non-regularized) incomplete beta function \mathcal{B}(x; a ,b) and the (upper, non-regularized) ... 2 votes 1 answer 84 views ### Cumulative Distribution Function of the skewed generalized error distribution I´m trying to calculate the cumulative distribution function of the skewed generalized error distribution with the probability density function: where u = y - m. From Theodossiou in (https://www.... 0 votes 0 answers 34 views ### Find the mean of distribution given its cdf [duplicate] P[X<y]=integral from 1 to y of (1/x)dx how to find mean of this distribution? 4 votes 1 answer 384 views ### On the proof of right-continuity of the distribution function In today's statistics class, we saw properties of the distribution function, i.e. defined by F(x) = P(X\leq x) for a random variable X. One of these was: F(x) is right continuous. The proof ... • 145 3 votes 1 answer 551 views ### cmprsk cumulative incidence - comparing between two curve with different outomes I have a question regarding the extracting p-values from the cumulative incidence curves that considered competing risks. I used cmprsk packages and ... • 31 2 votes 1 answer 191 views ### In statistics how does one find the mean of a function w.r.t the uniform probability measure? I am unfamiliar in statistics. My knowledge is in pure mathematics. Suppose n\in\mathbb{N}, where X is in the \sigma-algebra of Caratheodory-measurable sets such that X\subseteq\mathbb{R}^{n} ... • 161 1 vote 1 answer 148 views ### What is the probability the expected value is undefined or infinite? What is the probability from a uniform probability measure (pg.37) on sample space \left\{N(\theta,1)|\theta\in[0,1]\right\} that for some random variable X in the sample space, the Expected-Value ... • 161 1 vote 1 answer 57 views ### How to combine two integrals containing the PDFs of a variable and its linear transform? Original Post: Suppose we have two random variables X and Y with cumulative distribution functions F(x) and G(y). We know that Y = aX + b. I want to compute Z(x) = F(x) - G(y). What I have ... • 13 6 votes 1 answer 2k views ### PDF does not integrate to 1 - where is my mistake? I am trying to solve a question which gives me a random variable with the distribution function below$$ F(x) = 1 - \left(\frac{\mu}{x}\right)^{2n} $$where 0 < \mu \le x < \infty I ... • 665 5 votes 2 answers 295 views ### Help with a proof regarding empirical CDF Given X_1,X_2...X_n \sim F and (\hat{F}_n(x)) is an empirical CDF I need to prove that$$Var(\hat{F}_n(x)) = \frac{F(x)(1 - F(x))}{n} So what I did is: The variance of an estimator $\hat{F}_n(x)$...
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I was reading a paper and couldn't understand the following transition. Could someone tell me where the term of $p^k (\frac{1}{2} − c′)$ comes from in the following transition? Def: Cumulative ...