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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How to write down a cumulative distribution function that consists of two distributions

I am being asked to write a CDF of a random variable $X$. I know that there is $0.5$ probability that $X=5$ and $0.5$ probability that $X$ follows the exponential distribution with parameter $7$. I ...
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Problem based on Cumulative distribution function

I have been given the following cumulative distribution function. I need to compute the probability $P(1\leq X<2)$: $$F(x)=\begin{cases} 0 & \text{ if } x<0 \\[1ex] \frac{x^2}{10} & \...
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Is $Z_t = U\sin(2\pi t) + V\cos(2\pi t)$ is strictly stationary?

Let $Z_t = U\sin(2\pi t) + V\cos(2\pi t)$, where $U$ and $V$ are independent random variables each with mean 0 and variance 1. Is $Z_t$ strictly stationary? Answer: I have proven that $Z_t$ is ...
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What information is required to aggregate percentiles

I have several thousand large datasets that are too big to fit into memory at once, so I need to keep them separate. It is easy enough to get the count, mean, std dev, min and max for the whole ...
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Finding the CDF of a a sequence of independent random variables

Let $Z_t$, where $t$ is even, be a sequence of independent random variables defined as, $$Z_t = \left\{\begin{array}{ccc} +1 & , & p = \frac{1}{2} \\ -1 & , & p = \frac{1}{2}\end{array}...
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how to generate data from a distribution whose cdf is not in closed form? [duplicate]

I am working on a distribution whose pdf and cdf is $$f(x,\alpha,\beta)=\frac{(\frac{\beta}{\alpha})(\frac{x}{\alpha})^{\beta}}{(1+(\frac{x}{\alpha})^{\beta})^{2}}\frac{\sin(\frac{\pi}{\beta})}{(\frac{...
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how to generate data from cdf which is not in closed form?

i am working on a distribution whose pdf and cdf is $$f(x,\alpha,\beta)=\frac{(\frac{\beta}{\alpha})(\frac{x}{\alpha})^{\beta}}{(1+(\frac{x}{\alpha})^{\beta})^{2}}\frac{\sin(\frac{\pi}{\beta})}{(\frac{...
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How to Solve For the Inverse Cumulative Distribution Function of a Double-Exponential Probability Density Function

I'm stuck on figuring out how to sample data from a fake/known double-exponential PDF for a lab project involving C. elegans egg-laying rate data. I need help with figuring out if there's an exact ...
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The formula for conditional expectation in terms of joint cdf

We know that covariance can be written as a function of marginals and joint CDFs, namely $$\newcommand{\cov}{\operatorname{cov}}\newcommand{\d}{\mathrm{d}}\cov(X,Y) = \iint (F_{X,Y}(x,y) - F_X(x)F_Y(y)...
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Constructing portability density function by convolution of two PDFs and acquire ICDF

I need Gaussian-convoluted gamma distribution to fit my data, but the program I'm using doesn't allow construction of custom PDF. Let's call it pdf_g2. $$\mathrm{pdf_{g2}}(x,a,b,c,d) = conv[\mathrm{...
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Meaning of uniform percentile rank distribution?

I'm new to statistics and I've been following Think Stats 2. I've just gotten to Cumulative Distribution Functions. I have some questions regarding uniform distribution: What does it mean for the CDF ...
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How to properly implement Student's T CDF efficiently

X-post from SO: The longshot here is to use this function for the sake of calculating accurate P values in pinescript using the student's t distribution. My attempt to replicate this implementation is ...
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Questions related to normal PDF and CDF

A random variable $Y$ has the Gaussian distribution on $\mathbb{R}$ with mean $\mu$ and variance $\sigma^{2}$. Select all the necessarily true statements from the following. a. $\mathbf{P}(Y \geq y) \...
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Probability density function for the max value of independent variables

I need help understanding the last part of this equation. I understand that all the variables are independent so you can multiply all of the probabilities but I'm not sure where the n comes from in ...
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How to assign probabilities to sample data points

I have a list of continuous values that represents a sample of data points observed from my experiment. For instance: myData = [-0.001, 0.0002, -0.015, 0.008] The ...
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How can $X$ be a discrete random variable? [duplicate]

Suppose that the cumulative distribution function of discrete random variable $X$ is given by, $$F(x) = \begin{cases} 0 & \text{$x$ < 0 } \\[1.5ex] \dfrac{x}{4} & \text{$0 \leq x<1$}\\[...
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Finding a distribution with a particular invariance property: F(x/b) - F(x/a) independent of x

Suppose $F$ is a cdf for some random variable on some support, and that $a,b$ are constants with $a<1<b$. I'm hoping to find a distribution such that: $$F \left( \frac{x}{b} \right) - F \left( \...
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How to calculate confidence interval of the CDF of a non-normal distribution?

For a non-normal distribution, how to calculate the confidence interval of the Cumulative Distribution Function (CDF) of such distribution? Are there any approximations to calculate confidence limits ...
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Appropriate notation for denoting a multivariate student's t distribution?

Suppose, we have a vector variable $\mathbf{Y}=[y_1,y_2]$, such that $\mathbf{Y}\sim N(0,I)$, where $I$ is the identity matrix. Then the bivariate distribution $P[y_1\leq Y_1,y_2\leq Y_2]$ can be ...
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cdf of autocorrelation statistic, null hypothesis is uniform distribution

Suppose I carry out $N$ observations $X_1,...,X_N$. The null hypothesis will be that these are sampled iid from $U(0,1)$. I would like to check whether observations a distance $k$ apart are correlated....
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If $X$ is a continuous random variable with support $A$, does this imply that the cdf of $X$ is strictly increasing on $A$?

If $X$ is a continuous random variable with support $A$, does this imply that the cdf of $X$ is strictly increasing on $A$? My guess is yes. But just in case, let me know if you can think of any ...
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Is Weibull distribution memoryless?

I googled and it seems not. Only exponential distribution is memoryless. Does anyone have an intuitive explanation why it is not? Thanks
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Fitting a function to ECDF [closed]

I have some ECDFs. I would like to summaries the ECDFs with functional approximations. I was thinking that a polynomial, spline, or other line fitting procedure would generate a nice parsimonious ...
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Finding an expression for the CDF of sum of two random variables $X, Y$ conditioned on the value of one variable $Y$: find $P(X + Y < c | Y = b)$ [closed]

This is related to my other question on renewal processes https://math.stackexchange.com/questions/3947852/renewal-theory-probability-of-residual-lifetime-gamma-t-x-conditioned-on-c $X, Y$ are ...
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Why $\frac{e^x}{e^x+e^{-y}}$ can't be a joint CDF?

I was trying to come up with an example of a joint CDF, and this function looked to me like a proper one: $$\frac{e^x}{e^x+e^{-y}} = \frac{e^{x+y}}{e^{x+y}+1}$$ It goes to 1 if you take x and y to ...
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What should the integral of a CDF be called?

This is strictly a nomenclature question. I have no particular problem finding double integrals of the type $\int\int\text{pdf}(y) \, d y \,d x$, and I find them quite useful. Whereas we have a good ...
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Small sample size fails to approach inverse CDF

When sample size $n$ gets large, we know that a sorted set of the $n$ samples approaches the inverse cumulative distribution function (CDF) sampled at $\frac{1}{n}, \frac{2}{n}, \dots, \frac{n}{n}$. ...
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Bivariate Random Transformation finding CDF

Problem Assume $Y_i, i=1,2$ are independent with pdf-s $$f(y_i;\theta)=\frac{1}{\theta}e^{\frac{-y_i}{\theta}} \forall y_i>0, \, \theta >0$$ Let $Y = Y_1 + Y_2$, and show that $$F(Y) = P(Y \le y)...
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Is skewness visible in the cumulative distribution function (cdf)?

The following two figures are the pdf's of four parametric distributions and their corresponding cdf's. The most left-ward blue line is clearly not skewed, while the most right-ward orange line is ...
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Characterstic function vs Cumulative distribution function

They say that the characteristic function $\varphi_X(t)$ of a random variable $X$ completely determines the behavior and properties of the probability distribution of $X$, just like the cumulative ...
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Integral of difference of density functions of two Continuous Random Variables goes to 0

The problem says : Let $(X_n)_{n=1}^\infty$ be a sequence of continuous random variables with probability density functions $(f_n)_{n=1}^\infty$ , and let $X$ be another continuous random variable ...
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Random variable without pdf but with a cdf?

In this video, the professor says that some random variables have no pdf but do have a cdf. Also, in my course material, I studied that converging in mean was stronger than converging in cdf which ...
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Can't compute CDF for Inverse Gaussian distribution

I am trying to implement in Python the CDF of the Inverse Gaussian distribution: Inverse Gaussian pdf: $$ f(x) = \sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda(x-\mu)^2}{2\mu^2x}} $$ Inverse ...
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How to estimate mean and variance for rate of change when I only have state data at different ages

I'll give you the intuition behind my problem first. I have data on whether children ($n \approx 200$) can read and their age in integers from 0 to 14. For each age, it is straightforward to ...
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Putting into words variable transformation

If Y is a random variable which comes from a transformation of X : $ Y = \phi(X) $, the formula for the tranformation of this random variable can be written as follows : $$ F_Y(y) = P(Y \leq y) = P(X \...
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Intermediate proof for Glivenko-Cantelli Theorem

Show that for any cdf, the following holds: $sup_{x\in \mathbb{R}}|F_n(x)-F(x)|\leq sup_{u\in [0,1]}|\overline{F}_n(u)-F(u)|$ Where $F_n:=\frac{1}{n}\sum_{i=1}^n1_{(-\infty,x]}(X_i)$ is the empirical ...
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Can we sample from both pdf and cdf?

my question is quite generic. I am currently studying the algorithms calculating random numbers from distributions: In inverse transform method we get the cumulative distribution function in the end ...
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Help me understand how the following likelihood function is derived

A week ago, I asked a question concerning the Taylor expansion of an arbitrary distribution function. As noted by a member of the forum, the question was vague and perhaps incorrect. I had asked this ...
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How is an 'ogival function' defined?

Reading on a paper on factor analysis and measurement invariance I find the description of some functions as 'ogival' functions. In Google I find it referenced mostly in papers from the '70s and '80s....
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Prove that $P(X \le a) + P\{Y \le \frac{1}{a}\} = 1$

Prove that if $X$ has the F-distribution with $(m, n)$ d.f. and $Y$ has the F-distribution with $(n, m)$ d.f., then for every $a > 0$, $$ P(X \le a) + P\left\{Y \le \frac{1}{a}\right\} = 1 $$ I ...
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CDF*[1-CDF]/PDF — name? integrable?

Suppose I have a random variable $X\in\mathbb R$ distributed according to a smooth nonzero probability density function (PDF) $f(x)$, with cumulative distribution function (CDF) $F(x):=\int_{\infty}^x ...
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About unique determination of symmetric point (or center) of a distribution based on pdf or cdf

Suppose we have a distribution that is known to be continuous and symmetric, and is otherwise unknown. We want to decide whether it is actually centered at zero using an equation involving pdf or cdf. ...
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How to prove that a function is 2-increasing (copula)

There are three conditions to prove that a function is a copula: $C(u,0)=0=C(0,v)$ grounded. $C(u,1)= u, C(1,v)= v$. $C(u,v)$ 2-increasing function. Here I am concerning in the last condition how to ...
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Is the definition of symmetric distribution using cdf correct?

Based on wikipedia (https://en.wikipedia.org/wiki/Symmetric_probability_distribution), a distribution is symmetric about $x_0$ if and only if it is a distribution whose pdf(or pmf) $f(\cdot)$ ...
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point such that area to the right of that point under one gaussian is 5% of area under a second gaussian

Say I have two gaussian random variables $Z_1 \sim f_1 = f(\cdot|\mu_1, \sigma_1)$ and $Z_2 \sim f(\cdot|\mu_2, \sigma_2) = f_2$, where $f$ is the gaussian density. How can I calculate the value of $x$...
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How to compute the median of a continuous distribution?

I don't have a solid background in statistics so the concept of probability density functions in the statistics course I'm taking is new to me. I need to derive the median of a continuous distribution ...
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Creating a Probability Plot of a Custom Distribution

Let's say we have some icdf function, which I will paste below: ...
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Which $\mu$ hold so that integral of CDF (from $\mu$ to $\infty$) equals to integral of 1-CDF (from $-\infty$ to $\mu$)?

What is the $\mu$ s.t. $$\int_{\mu}^{\infty}1-F(x)dx = \int_{-\infty}^{\mu}F(x)dx?$$ Here $F(x) = P(X\leq x).$ Should $\mu$ be the median of X, i.e. $0.5=F(\mu)$? I think $\mu$ should be the point so ...
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Copula between a distribution and its univariate transformation

I'm trying to compute the copula (or joint distribution) between x and a univariate transformation, like say sin(x). That is compute $C_{XY}$ (or $F_{XY}$) given that $x \sim U(0,1)$ and $y = sin(x)$ ...
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What is the ppf of the truncated normal distribution?

What is the percent point function (ppf), or inverse cdf, of the truncated normal distribution? The distribution and cdf is defined here: https://en.wikipedia.org/wiki/Truncated_normal_distribution $$...

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