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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52
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10answers
50k views

Why is the sum of two random variables a convolution?

For long time I did not understand why the "sum" of two random variables is their convolution, whereas a mixture density function sum of $f(x)$ and $g(x)$ is $p\,f(x)+(1-p)g(x)$; the arithmetic sum ...
39
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3answers
57k views

Help me understand the quantile (inverse CDF) function

I am reading about the quantile function, but it is not clear to me. Could you provide a more intuitive explanation than the one provided below? Since the cdf $F$ is a monotonically increasing ...
24
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3answers
24k views

Why is the CDF of a sample uniformly distributed

I read here that given a sample $ X_1,X_2,...,X_n $ from a continuous distribution with cdf $ F_X $, the sample corresponding to $ U_i = F_X(X_i) $ follows a standard uniform distribution. I have ...
8
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1answer
8k views

Distribution of a ratio of uniforms: What is wrong?

Suppose that $X$ and $Y$ are two i.i.d. uniform random variables on the interval $[0,1]$ Let $Z=X/Y$, I am finding the cdf of $Z$, i.e. $ \Pr(Z\leq z) $. Now, I came up with two ways of doing this. ...
12
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1answer
12k views

Why the CDF for the Normal Distribution can not be expressed as a closed form function?

I am working my way through Think Stats, where the author states that "there is no closed form expression for the normal cumulative density function" but does not provide any further details as ...
6
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1answer
586 views

How to approximate the student-t CDF at a point without the hypergeometric function?

Is there a way to closely approximate the CDF of a student-t distribution at a point $x$ without involving the hypergeometric function? For example, by using a series expansion, or expressing the CDF ...
15
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1answer
28k views

How are the Error Function and Standard Normal distribution function related?

If the Standard Normal PDF is $$f(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$$ and the CDF is $$F(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-x^2/2}\mathrm{d}x\,,$$ how does this turn into an error ...
6
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1answer
4k views

Inverse function for a non-decreasing CDF

For a CDF that is not strictly increasing, i.e. its inverse is not defined, define the quantile function $$F^{-1} (u) =\inf \{x: F(x) \geq u \},\quad 0<u<1. $$ Where U has a uniform $(0,1)$ ...
3
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1answer
129 views

How to find quantiles and likelihoods of mixture distributions?

My PDF: M was estimated and found to be 5. I need to work out the quartiles for the PDF above. In addition, I need to use different methods of estimation to estimate the parameters. So far I've ...
17
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4answers
6k views

Which to believe: Kolmogorov-Smirnov test or Q-Q plot?

I'm trying to determine if my dataset of continuous data follows a gamma distribution with parameters shape $=$ 1.7 and rate $=$ 0.000063. The problem is when I use R to create a Q-Q plot of my ...
5
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1answer
3k views

PDF/CDF of max-min type random variable

For i.i.d. random variables, we may write the CDF of $t=\max(t_1,\cdots,t_N)$ as $$F_t(t)=F_{t_i}(x)^n$$ and the CDF of $x=\min(x_1,\cdots,x_N)$ as $$F_x(x)=1-(1-F_{x_i}(x))^n$$ When we have $X=\...
3
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1answer
2k views

CDF of Z=XY with X~Uniform(0.5,1.5) and Y~Uniform(0.8,1.5)

I am looking for the CDF of the product of two independent random variables (X and Y) with uniform distributions. Both random variables uniform distributions have interval boundaries (upper and lower ...
1
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1answer
53 views

Sum of indepedent random variables and a constant

Let $X_1$ and $X_2$ be independent Normal random variables with mean $\mu_1$ and $\mu_2$, and variances $\sigma_1$ and $\sigma_2$. Let $Y = X_2-X_1 + c$, where $c$ is a constant. For notational ...
-3
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1answer
5k views

Deriving ordered statistics minimum cdf

Assume ${{X}_{1}}$, ${{X}_{2}}$, ${{X}_{3}}$,...,${{X}_{n}}$ are i.i.d. samples from distribution with density f, and cdf F. Let V=min( ${{X}_{1}}$, ${{X}_{2}}$, ${{X}_{3}}$,...,${{X}_{n}}$) To ...
11
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1answer
4k views

Proving the probability integral transform without assuming that the CDF is strictly increasing

I know that the proof of the probability integral transform has been given multiple times on this site. However, the proofs I found use the hypothesis that the CDF $F_X(x)$ is strictly increasing (...
9
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2answers
9k views

Distribution of maximum of normally distributed random variables

I'm trying to find the closed-form CDF and PDF of $Y = \max(X_1, ..., X_n)$ where $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$. My thought process so far: $$ \begin{align*} F_Y(y) &= \mathbb{P}(\max(...
7
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2answers
2k views

Inverse transformation sampling for mixture distribution of two normal distributions

I am confused by the special way required to use inverse method in the following problem, Here is the problem: Consider a mixture distribution of two normal distributions, where the desired PDF $...
9
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1answer
1k views

Variance of a Cumulative Distribution Function of Normal Distribution

Suppose, $X\sim N(\mu,\sigma^2)$. Can anyone help in finding the following : $\text{Var } \bigg(\Phi\big(\frac{X + c}{d}\big) \bigg)$ ? Here, c and d are positive. Here, $\Phi(x)$ is the "...
10
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1answer
2k views

Simulate from a truncated mixture normal distribution

I want to simulate a sample from a mixture normal distribution such that $$p\times\mathcal{N}(\mu_1,\sigma_1^2) + (1-p)\times\mathcal{N}(\mu_2,\sigma_2^2) $$ is restricted to the interval $[0,1]$ ...
4
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1answer
1k views

Obtain marginal CDF from joint CDF by simulation

How can I evaluate the marginal cumulative distribution function of a set of random variables for which I do not have the CDF in closed form. I can, however, simulate from a joint distribution ...
4
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1answer
1k views

How do I convert simulated values from a copula to “real” observations? R

I have managed to fit different kinds of copulas to my data in R (mostly Archimedean copulas) using the copula package. I have no problem simulating pseudo observations (u and v), my questions are: ...
4
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1answer
2k views

CDF and PDF of radius of a unit disk

Let X and Y be uniformly distributed on a unit disk such that $x^2 + y^2 \leq 1$ Let $R = \sqrt{X^2 + Y^2}$. What are the CDF and PDF of $R$? I know that the area of the unit disk is $A = \pi r^...
6
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1answer
11k views

How to calculate the probability of a data point belonging to a multivariate normal distribution?

Okay, so I have a $n$-dimensional normal distribution with the means and co-variance matrix defined (for now we can assume that these are the true distribution parameters, not estimates). For a given ...
4
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1answer
391 views

Multi-dimensional CDF on a discrete support

Suppose I have two discrete-support random variables, $X$ and $Y$. They have joint CDF $F(X,Y)$. If I want to find $\Pr(a \leq X \leq b , c \leq Y \leq d)$. It is obviously not: $F(b ,d)-F(a-1 ,...
3
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2answers
121 views

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 ...
2
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1answer
2k views

How do I estimate a smooth cdf from a set of observations?

I have a set of observation, let's call it $X$ and would like to fit a cdf to it. $X$ has a distribution which is roughly approximable with the normal distribution. This CDF should correspond to a ...
23
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4answers
190k views

How to calculate cumulative distribution in R?

I need to calculate the cumulative distribution function of a data sample. Is there something similar to hist() in R that measure the cumulative density function? I have tries ecdf() but i can't ...
28
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3answers
2k views

Gaussian Ratio Distribution: Derivatives wrt underlying $\mu$'s and $\sigma^2$s

I'm working with two independent normal distributions $X$ and $Y$, with means $\mu_x$ and $\mu_y$ and variances $\sigma^2_x$ and $\sigma^2_y$. I'm interested in the distribution of their ratio $Z=X/...
17
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3answers
24k views

Do the pdf and the pmf and the cdf contain the same information?

Do the pdf and the pmf and the cdf contain the same information? For me the pdf gives the whole probability to a certain point(basically the area under the probability). The pmf give the probability ...
6
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1answer
1k views

What is the limiting distribution of exponential variates modulo 1?

I have tried to find the limiting distribution of $X_n\sim\text{Exponential}(\lambda/n)$ by finding the cdf and taking the limit. I got: \begin{align*} F_{X_n}(X)) = \int_{0}^{X} \frac{\lambda}{n} e^{...
2
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2answers
25k views

Distribution function terminology (PDF, CDF, PMF, etc.) [duplicate]

I am confused about the following terminologies: Distribution Function Cumulative Distribution Function (CDF) Probability Distribution Function Probability Density Function Probability Mass Function (...
7
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3answers
7k views

Minimum CDF of random variables

I know that the joint cumulative function of two random variables X and Y is defined as: $F_{X,Y}(x,y)=P(X≤x,Y≤y)$. How can I find the CDF for $F_{X,Y}=\{x,x\}$. In other words is what will be $Pr\{...
15
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3answers
1k views

CDF raised to a power?

If $F_Z$ is a CDF, it looks like $F_Z(z)^\alpha$ ($\alpha \gt 0$) is a CDF as well. Q: Is this a standard result? Q: Is there a good way to find a function $g$ with $X \equiv g(Z)$ s.t. $F_X(x) = ...
9
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4answers
3k views

What is the CDF of the sum of weighted Bernoulli random variables?

Let's say we have a random variable $Y$ defined as the sum of $N$ Bernoulli variables $X_i$, each with a different, success probability $p_i$ and a different (fixed) weight $w_i$. The weights are ...
7
votes
4answers
8k views

What is the proper way to estimate the CDF for a distribution from samples taken from that distribution?

Given $n$ samples from a (continuous) distribution X, the obvious thing to do is sort them, and distribute them equally across $[0,1]$ by taking $(x_{(k)}, (k-1/2)/n)$ as estimates of particular ...
5
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2answers
10k views

Integral of a CDF

I'm solving a problem where I've this 'expectation': $$ \int_{0}^y x\cdot f(x) dx $$ where $f(x)$ is a PDF with support on $[0, z]$, with $z>y$. Is there a way to rewrite it without the integral ...
3
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1answer
4k views

Quantiles for non-normal cdf

When I compute the five-number summary on my sample, I obtain quantiles that differ from the quantiles I got from the empirical cdf, since they are not normally distributed data. Can you help me in ...
9
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1answer
540 views

What is the two-sample CDF of of $D^{+}$ and $D^{-}$ from the one-sided Kolmogorov-Smirnov Test?

I am trying to understand how to obtain $p$-values for the one-sided Kolmogorov-Smirnov test, and am struggling to find CDFs for $D^{+}_{n_{1},n_{2}}$ and $D^{-}_{n_{1},n_{2}}$ in the two-sample case. ...
6
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3answers
5k views

CDF and logistic regression

Is the probability calculated by a logistic regression model (the one that is logit transformed) the fit of cumulative distribution function of successes of original data (ordered by the X variable)? ...
6
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4answers
3k views

Why is the empirical cumulative distribution of 1:1000 a straight line?

Why does plot(ecdf(1:1000)) produce a straight line? Since Fn($x_n$) = $x_1$/(total sum) +$x_2$/(total sum) +...+$x_n$/total sum = ($x_1+x_2+x_3+...+x_n$)/total sum. the fact that Fn(200) roughly ...
4
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3answers
406 views

Expected value of a random variable by integrating $1-CDF$ when lower limit $a\neq 0$?

I have found several past answers on stack exchange (Find expected value using CDF) which explains why the expected value of a random variable as such: $$ E(X)=\int_{0}^{\infty}(1−F_X(x))\,\mathrm dx $...
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1answer
2k views

How are percentiles distributed?

I was taking a look at this page, and I can't seem to understand why the frequency plot of the percentiles is uniformly distributed. Distances between percentiles are not equal, so why is the ...
8
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3answers
325 views

Can a CDF from data cross with another CDF

Given two data sets of positive real numbers X and Y, both of the same size, and 0<=Y<=X for each row; can the empirical CDF of X ever cross the empirical CDF of Y?
5
votes
1answer
121 views

Inverse transform sampling and ambiguous Intervals

Let $F_i:\mathbb R\to[0,1]$ be a distribution function$^1$ and $$F_i^{-1}(t):=\inf\left\{x\in\mathbb R:F_i(x)\ge t\right\}\;\;\;\text{for }t\in[0,1].$$ I've got a computer program where only $F_i^{-...
4
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1answer
3k views

Cdf of minimum of two iid random variables

I am struggling with the following sentence: Using the fact that the cumulative distribution of the minimum of two i.i.d. random variables can be expressed as $1 - (1 - F(x))^2$.... Can anyone ...
4
votes
3answers
577 views

Defining continuous random variables via uncountable sets

At several sources I have encountered the following two definitions of a continuous random variable associated with uncountable sets: a) uncountable range: The random variable X is continuous if its ...
3
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1answer
225 views

What is the link (discrepancy?) between these PDF/CDF and p-value distributions?

I have created a mixed distribution model comprising 80% $H_0$ plus 20% $H_1$ to illustrate the link between the expected proportions of true and false positives and negatives in the PDF, CDF and p-...
3
votes
0answers
388 views

Constructing a joint distribution from pairwise bivariate marginal distributions?

It's fairly well-known that given univariate distribution functions $F_X, F_Y, F_Z$, one can construct the joint distribution $F_{(X, Y, Z)}(x, y, z) = C(F_{X}(x), F_{Y}(y), F_{Z}(z))$, where $C$ is ...
3
votes
4answers
135 views

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{...
2
votes
1answer
233 views

How to compute the CDF of this random variable?

I'm working on a game theory model of incomplete information, where players observe certain attributes via noisy signals. Specifically, one player has the opportunity to choose any value $\eta$ from ...