Questions tagged [cumulative-distribution-functions]
For questions related to cumulative distribution functions.
615
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Find the behaviour of the cumulative conditional function and match it to a known distribution
The given joint distribution is:
$$
f_{X, Y}(x, y)=\frac{n 2 a^2 x^{n-1}}{y^{n+3}} I_{[0, y]}(x) I_{(a, \infty)}(y), n \in \mathbb{N}
$$
The exercise asks us to find the cumulative distribution of the ...
- 145
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23
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Joint Distribution Function of a Joint Density Function (exponential)
I have a density function $f(x,y)= 2e^{-x-y}$ for $0<x<y<\infty$ and $0$ elsewhere. What is the joint distribution function?
So far I have calculated $F(x,y)= \int_0^x \int_0^yf(x',y') dy'dx' ...
-1
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66
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Inverse of CDF of random variable
$
F(x)=
\begin{cases}
0 & \text{if } x < 0\\
\frac{x}{2} & \text{if } 0 \leq x < 1 \\
\frac{1}{2} & \text{if } 1 \leq x < 2 \\
\frac{x}{4} &...
- 663
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Expectation of normal c.d.f. of a normal random variable
Let $a$ be a constant, X is a standard normal and $\Phi$ is the c.d.f. of standard normal.
what is the $$E[\Phi(aX)]$$
what is the $$E[\Phi(X + a])]$$
my approach would be to derive the density of $...
- 929
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26
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How to get a CDF value from a PDF when the required CDF is not within the defined area?
I have a density function f(x, y) = 1/2 for 0 ≤ x ≤ y ≤ 2 and 0 elsewhere. I am being asked to find the CDF value F(1, 3), but as you can see the three is past the range of the defined triangle, what ...
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47
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How to log a distribution?
My understanding: Equation $(5)$ can be obtained when we let $e^x=\left({e^t}/{\theta}\right)^{\beta}$, which $x=\beta\,(t-\log\theta)$ , and substitute into $(4)$.
My question: Why equation $(7)$ is $...
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Taxi Stand Waiting Time: CDF Calculation
I'm having a problem understanding this solution.
Let's say this solution is correct. If you take the the derivative of the resulting CDF you should get the correspoding PDF. In this case the PDF ...
0
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1
answer
17
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During the process of simulating an exponential random variable
According to the text book, to get a associated CDF $$F_x$$ is given by
$$F_x(a)=\int_{-\infty}^af_x(x)dx=1_{[0, +\infty)}(a)\int_0^a\lambda e^{-\lambda x}dx=1_{[0, +\infty)}(a)(1-e^{-\lambda a})$$
...
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53
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Finding parameters in piecewise CDF [closed]
The CDF of a continuous random variable X is:
$$
F_X(x) =
\begin{cases}
0, & \text{if } x < 4 \\
Ax+B+\frac{C}{x} ,& \text{if } x \geq 4
\end{cases}
$$
(a) Find the parameters A, B,...
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69
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Probability over $\Bbb R$ and density
Let $P$ a probability over $(\Bbb R, \mathcal B)$ where $\mathcal B$ are Borel set, $f:\Bbb R \to \Bbb R$, $f\geq 0$ almost everywhere s.t. $\forall a<b$ rational
$$P([a,b]) = \int_a^bf(x)dx$$
...
- 1,010
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1
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28
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Understanding an expectation formula
Consider a puddle of water which evaporates at random points over time. Let $f(x,t)$ be the fraction of evaporated water at site $x$ on the puddle at a time $t$ after the start of evaporation. I want ...
- 3,093
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44
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Uniform Sampling points on a line using 2 Uniform distributions
I have been struggling with this problem for quite some time but I am not sure how to proceed.
So I am given a sampling algorithm : We would like to uniformly sample points on a line between A and B. ...
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0
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26
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Monotonicity of quantile function divided by its derivative and argument
Let $F$ be a CDF with $f\equiv F'$ having a positive support (edit), i.e. $\text{supp}(f) \subseteq \mathbb{R}_+$. Then $Q\equiv F^{-1}$ is its quantile function and $q\equiv Q'$, where we know $q(p) =...
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Speed of divergence at infinity of inverse cdf
Consider two random variables with cdf F and G and of bounded support. I noted that if $F^{-1}(G(x))/x$ increases, we have that $F^{-1}(G(x))/x\le F^{-1}(G(x))/1=1$. My question is, can the bounded ...
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Set of discontinuity points of a cdf: accumulation points.
A cumulative distribution function (cdf) has a countable set of discontinuity points. They need not be isolated. Let us call non-isolated points 'accumulation points' of this set of discontinuity ...
- 878
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1
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51
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Why isn't $F(X) = 1$ when $F$ is the cdf of $X$? [closed]
This is a really simple question. For a random variable $X$ and $F$ its cumulative distribution function. Since $F$ is defined as $F(t) = \mathbb{P}(X \leq t)$ for all $t$. I was wondering why we don'...
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If X and Y are independent exponential random variables with parameters λ1 and λ2,How do I express the density function of Z = XY? [duplicate]
My solution is:
1.calculating the cdf(Cumulative Distribution Function) of Z using the F(Z<z)=F(XY<z), written as F(z).
2.the derivation of F(Z) is just f(z), namely the pdf of Z.
But I failed ...
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22
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Permanence properties of the joint CDF under a linear transformation
Let a selection of random variable $X=(X_1,X_2,\dots, X_n)$ defined on a common background space $(\Omega, \mathcal{F},P)$ be given, and let $F_X:\mathbb{R}^n\to \mathbb{R}$ denote their (joint) CDF.
...
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42
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When is the CDF locally Lipschitz continuous?
Given a finite selection of random variables $X_1,X_2,\dots, X_n$ defined on a common background space $(\Omega, \mathcal{F}, P)$, when is the joint cdf $F_X:\mathbb{R}^n \to \mathbb{R}$ locally ...
- 539
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1
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119
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Quantile of Empirical CDF and Tail Bound
Let $F$ be the CDF of $X$ and $p \in (0, 1)$, and $F_n$ be the empirical CDF of $X_1, ..., X_n$; $F_n(x) = \frac{1}{n}\sum_{i = 1}^nI(X_i \le x)$. The $p$ th quantile of $F$ and $F_n$ are defined as ...
- 595
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26
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Find Marginal CDF given joint CDF
This is my approach so far, but I am unsure what to do with part 2. I found the joint PDF and Marginal CDF's but don't fully grasp the question.
- 21
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3
answers
54
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Unique Measure from Right Continuous and Left Continuous CDF
Consider $\mathbb R$ equipped with the usual Borel $\sigma$-algebra.
For each probability measure $P$ on $\mathcal B$ there exists a unique monotone, right-continuous function $F$ on $\mathbb R$ ...
- 517
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0
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41
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Limit involving tail probability of multivariate normal
I am interested in computing $$\lim_{x \to \infty} x \Phi_n(\mu_x; 0,\Sigma_x),$$
where $\Phi_n(\cdot)$ is the cumulative distribution function of a multivariate normal of dimension $n$ evaluated in $\...
- 11
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47
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How to find $E[X \Phi(X)]$ with $X$ normally distributed?
Let $X$ be distributed as a standard normal variable, with CDF $\Phi(x)$.
How do I prove that $E[X \Phi(X)] = \frac{1}{2\sqrt{\pi}}$?
That is, how to show that
$$\int^\infty_{-\infty} X \frac{1}{\sqrt{...
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34
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Existence of PDF given a CDF with a countable number of jumps
This is in the context of the cumulative distribution function (CDF) of a random variable and its corresponding probability density function (PDF).
Here, the important thing to note is that the CDF ...
- 53
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50
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Relationship between Gumbel and Frechet distributions
I have a question regarding the Gumbel distribution. Based on the figure below, it appears that a form of the Fréchet distribution can be derived from it, using a log.
Variable $u_{i}$ represents an ...
0
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0
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6
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Sequence of sets of p-quantiles converging to a given set of p-quantiles
One of my homework problems asks us to show that a sequence of sets of p-quantiles converges to a given set of p-quantiles. I can start the question but don't know how to continue. Here's the problem ...
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$\Pr(X_n\leq x) \to \Pr(X\leq x)$ for $x\geq 0$ and $\Pr(X_n< x) \to \Pr(X\leq x)$ for $x<0$ imply convergence in distribution?
Let $X_n$ be random variables and let $X\sim \mathcal{N}(0,1)$ such that $$\forall x\geq 0, \quad \Pr(X_n\leq x) \to \Pr(X\leq x)$$
and
$$\forall x< 0, \quad \Pr(X_n< x) \to \Pr(X\leq x).$$
Does ...
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Why do the increasing sequences $x_n, y_n$, decrease to $x,y$ to show a bivariate (or univariate) cdf is right continuous?
I have a problem understanding why the concept of right continuity of a cdf has to decrease a sequence $x_n$ or $y_n$ to a limiting value $x$ or $y$ respectively.
I do not understand why in this ...
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Derivation of the cumulative distribution function for the beta-binomial distribution
Let $X$ be a random variable following a beta-binomial distribution:
$$
X \sim \mathrm{BetBin}(n, \alpha, \beta) \; .
$$
According to Wikipedia, the cumulative distribution function of $X$ is
$$
F_X(x)...
- 96
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1
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Expected value of max function of a random variable: understanding the proof
Consider a random variable $\phi$. Assume $\mathbb{E}[\phi]<\infty$. Let $F_\phi(t)$ be its cumulative distribution function. Let $(a)^+=\max\{0,a\}$.
I am trying to understand the proof of the ...
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42
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Joint CDF of two random variables - rectangular region equation
I just started learning about joint CDFs and I think I understand the concept, but I don't understand this equation in my textbook regarding a rectangular region. Why are the top right and bottom left ...
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1
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30
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Convex order stochastic dominance for nonnegative integer valued random variables
Let $X$ and $Y$ be nonnegative integer valued random variables (with same mean). It is customary to define convex order stochastic dominance for such variables (denoted $X<Y$) if $\sum_{j\geq n}\...
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51
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$\sigma$-algebra generated by $F_X (X)$
Let $X$ be an $\mathbb{R}$-valued random variable and $F_X$ be its cumulative distribution function.
I am interested in the connection between the $\sigma$-algebra generated by $X$ and that generated ...
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How does the derivative of a (multivariate) CDF relate to its Lebesgue density?
Let $X:(\Omega,\mathcal
{F})\to (\mathbb{R}^n,\mathcal{B}(\mathbb{R}^n))$ denote a random variable defined on some probability space $(\Omega, \mathcal{F},P)$.
By Lebesgue's theorem, the distribution ...
- 539
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Constructing particular sequence of finitely supported measures converging to a given measure
Let $(\Omega, \mathcal{B}, \mu)$ be a measure space where $\Omega \subset \mathbb{R}^K$ is the unit sphere in the $K$-dimensional space, and $\mu$ is a finite, positive measure (not necessarily a ...
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If the p.m.f. of $X$ is given by $f(x)=p(1-p)^{x-1}$ where $0<p<1$, $x \in J^{+}$, determine the cumulative distrubution function of $X$.
The answer is $F(x)=1-(1-p)^{x}$
I don't understand how this might be achieved without Integration.
Is it possible for this question to be answered without Integration?
Thanks.
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CDF of a continuous and discrete random variable - attributes
Something that bothers me as I was just doing an exercise.
We know that the CDF of a function must be:
Continuous from right side.
Monotonic up.
The limit at infinity is 1.
The limit at minus ...
- 179
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How can I show that a particular expression is positive?
I have the expression $G(m) = \frac{\mu}{2e[F(\mu)-F(0)]} - \frac{\int_{-\infty}^\mu F(c)^e}{F(\mu)^e}, e\in(0,1],\mu>0,m>\geq 0,$ and $F(c) = \mu^{-1}\int_0^\mu 1/(1+exp(-m(c-x-1))dx$, and I ...
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Cumulative distribution function of $2X-3Y$, where $X,Y$ are uniformly distributed
Let be $X,Y$ two random variables that are independently uniformly distributed on $[1,3]$. Compute the pdf and cdf of the random variable $Z:=2X-3Y$. (Hint: use convolution formula)
My approach:
We ...
- 4,298
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CDF of a convergent positive series
Let $Y_0, Y_1, \ldots$ be an i.i.d. random sequence such that
$$ \mathbb{P}(Y_k = 0) \;=\; 1 - \mathbb{P}(Y_k = 1) \;=\; p \qquad \text{for each $k\ge 0$}. $$
I am interested in the following random ...
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38
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0
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Finding the cumulative distribution function of a maximum function, based on a uniform distribution [duplicate]
Suppose I have a uniform distribution
$X$ ~ U($0, θ$).
I have an estimator which pertains to the random variable of the largest value among my random sample. We can express this as
max$(X_1, X_2...X_n)...
- 33
1
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1
answer
42
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Seeking help to evaluate an integration
Let Say, $G \left(x \right)$ is the Cumulative distribution function of a random variable $X$ which is continuous, now consider below integral
$\int_{0}^{\alpha} 1 \left(G \leq u \right)du$, where $1 \...
- 194
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votes
1
answer
69
views
Is there a non-trivial max invariant family of distributions?
The Problem Traditionally, the value of a position in a chess engine is computed as the maximum of the values of the subsequent positions. These values are typically represented as a single number. ...
- 33
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0
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42
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First order stochastic dominance with binary interaction
I have two variables $X_1$ and $X_2$ $\in [0,1]$, where $X_2$ (strictly) first order stochastically dominate (FOSD) $X_1$, i.e., $F_{X_2}(x) \leq F_{X_1}(x)$ for all $x$.
Then I have, $Y$ and $Z$, ...
- 109
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0
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61
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Stochastic ordering of dependent random variables given their sum and a common copula
I am interested in pointers for the stochastic ordering of continuous dependent variables with common copulas.
I have a vector $X=(X_1,X_2,...,X_N)$ and a vector $Y=(Y_1, Y_2, ..., Y_N)$. X and Y ...
- 11
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1
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30
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Example for Helly's Selection Principle
Helly's selection principle(in the context of Probability Theory and not Functional Analysis) gives that if $F_{n}$ be a sequence of CDF's then there exists a subsequence $F_{n_{k}}$ such that $F_{n_{...
- 1,057
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0
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29
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Equivalence of two definitions of convergence in distribution
My lecture notes gives the following definition of convergence in distribution of $\mathbb{R}^k$-valued random variables ($x \leq y$ for $x,y \in \mathbb{R}^k \Longleftrightarrow x_i \leq y_i$ for all ...
- 1,461
4
votes
2
answers
65
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Existence of a weird cumulative function
Does it exist a non discrete random variable with CDF $F$ with the following hypothesis :
$F$ is constant in a neighborhood of each point of continuity
I had the idea of Cantor function but it does ...
- 446