Questions tagged [cumulative-distribution-functions]
For questions related to cumulative distribution functions.
664 questions
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Problem related to distribution function of $X\sim \operatorname{Gam}(n,\lambda)$ and the incomplete gamma function
Let $X\sim \operatorname{Gam}(n,\lambda)$. Show that the the distribution function of $X$ is
$$
F(x)=\begin{cases}
1-\sum_{k=0}^{n-1}e^{-\lambda x}\frac{(\lambda x)^k}{k!} & \text{if $x>0$},\\
...
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Understanding the Glivenko–Cantelli theorem (and almost sure convergence)
Let $\{X_i\}_{n\geq 1}$ be an independent sample from a cumulative distribution function (CDF) $F$. The Glivenko-Cantelli theorem relates $F$ to the empirical CDF defined as
\begin{equation}
F_n(x)...
- 195
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3
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152
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Given a distribution function, can the random variable be determined from it? [closed]
We know that a random variable is given, then its distribution function can be determined.
Question: if the distribution function is given, can the random variable be determined from it?
Can someone ...
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Alternative formula for $\mathbb{E}[X^r]$ using $r\int^\infty_{0} x^{r-1} \mathbb{P}(X > x) \,dx$
I am trying to prove an identity listed in Feller's "An Introduction to Probability Theory and its Applications | Volume 2" on page 150
$$ \mathbb{E} [X^r] = r\int^\infty_{0} x^{r-1} \mathbb{...
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Brenier's Level set formulation of a particular 1D scalar conservation law
Consider the scalar conservation law
\begin{equation} \label{eq:scalar_conservation_law} \tag{1}
\begin{cases}
\partial_t F_t(x) + \partial_x Q(F_t(x)) = 0, \qquad t \ge 0, x \in \mathbb ...
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What is the cumulative kernel distribution using an Epanechnikov kernel?
good morning everyone.
I understand that kernel density estimation is a non-parametric technique used to estimate the probability density function of a random variable from a sample of data, with the ...
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Vector of order statistics is sufficient
I was trying to solve the following problem:
Let $X_1, \dots, X_n$ be i.i.d with some continuous distribution $F$. Show that the vector of order statistics, $(X_{(1)}, \dots, X_{(n)})$ is sufficient ...
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What would the value of this PDF be at $x = 10$? [closed]
I have the following question:
PDF of X
I am really confused on what the value would be at $x = 10$?
From what I understand about the unit step function:
$$u(x) = 1, x>=0$$
$$u(x) = 0, x<0$$
...
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49
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Expected value involving CDF
Let $F$ be a cumulative distribution function with corresponding density $f(x)=F'(x)$. I am interested in the following integral:
\begin{equation}
I = \int_a^b x F(x) f(x) dx,
\end{equation}
where ...
- 195
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Can a CDF be non-monotonic?
The bus company from Blissville decides to start service in Blotchville, sensing
a promising business opportunity. Meanwhile, Fred has moved back to Blotchville,
inspired by a close reading of I Had ...
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Proof assistance: show that $y < F(x)$ is equivalent to $F^{-1}(y) < x$
Exercise:
For a distribution function $F : \mathbb{R} \to [0,1]$, let
$$F^{-1} : [0,1] \to \mathbb{R}, \quad u \mapsto \sup \{ x \in \mathbb{R} \mid F(x) \leq u \}$$
be the generalized inverse of $F$.
...
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Probability Involving Gaps of Order Statistics
Let $U_1, \ldots, U_n \mathop{\sim}\limits^{iid} U([0, 1])$ be independent random variables distributed according to a $[0, 1]$ uniform distribution. Now, denote by $U_{(1)}, \ldots, U_{(n)}$ the ...
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Diferentiating piece wise cumulative distribution function
here is the function I'm trying to diferentiate at 1 (it wouldn't fit in the title)
$$F(x)=
\begin{cases}
{\frac{\pi x^2}{4}} &\quad\text{if }1\ge x\ge0, \\ \newline
\sqrt{x^2-1}...
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Discrete Random Variables May Have Uncountable Images
For a probability space $(\Omega, \mathcal F, P)$, I'm trying to construct a discrete random variable $X$ (one for which $P(X\in K) = 1$ for some countable set $K$) for which
$\text{im}(X)$ is ...
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Is the probability of $X$ in the interval $[0, \mathbb{E}{[X]}]$ at least $1/2$, if $X \sim \chi^2$?
I am working on the $\chi^2$ distribution and have the following assumption:
The cumulative distribution function of a $\chi^2$ distributed random variable is greater than $\frac{1}{2}$ at the right ...
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2
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71
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Commutative diagram involving order statistics
Given the random variables i.i.d. $X_1, \ldots, X_n$ and their cdf $F$ I'm trying to explain why the following diagram is conmutative:
$$
X_1, \ldots, X_n \xrightarrow{\text{sort}} X_{(1)}, \ldots, X_{...
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Distribution of a product of random variables
I have two independent distributions $X$ and $Y$. $X$ is defined by the piecewise CDF
$$F_X(x) = \begin{cases}
F_X^1(x) & x \in (-\infty, a_1)\\
F_X^2(x) & x \in [a_1, a_2)\\
F_X^3(x) & x \...
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How to calculate the following integral related to the exp function multiply the Q function?
Recently, I have encountered a very difficult integral problem that is related to both exponential function and the normal distribution function. I have been searching for relevant solutions for a ...
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Related to PDF of the sum and product of exponential random variable
In my research work, I came across the following situation that involves sum and product of exponential random variable as shown:
$P = X_1 + \beta_0^2 X_2X_3$ ---(1)
where $\beta_0^2$ is constant and $...
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36
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How to find CDF of the following?
I am trying to find out the CDF of the following expression that involves multiple random variables but not getting it properly.
$P = Pr (\gamma_{u_1} < \gamma_{th})$ ----(1)
where,
$\gamma_{u_1} = ...
- 181
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1
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44
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Finding CDF of function of random variables
I have two random variables $X$, $Y$ i.i.d. with CDF $F_X(x)=(1-x^{-4})\boldsymbol{1_{(1, +\infty)}}(x)$ where $\boldsymbol{1_A}$ is the indicator function on a set $A$, and thus PDF $f_X(x)=4x^{-5}\...
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83
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Seeking Algorithm to Solve a Convolution Integral or Directly Convolve Two CDFs
Hi everyone,
I'm working on a project where I need to find a way to directly convolve two cumulative distribution functions (CDFs) given in polynomial form, and solve the following convolution ...
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0
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69
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Is the convolution between two CDF always well defined?
Given the integral convolution:
$$(F_X * G_X)(x)=\int_{-\infty}^x F_X(t)G_X(x-t)dt $$
and also that the CDF of random variables are bounded between the interval $(0,1)$ and the flipping of one of them ...
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How Does One Show That A R.V Has CDF F?
I'm self studying probability using Statistics 101 book.
In chapter 3 there's a question(ex. 9):
Let F1 and F2 be CDFs, 0 <p< 1, and F(x) = pF1(x) + (1 − p)F2(x) for
all x.
(a) Show directly ...
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Is there an interpretation of the pointwise maximum of CDF's?
Let $X$ and $Y$ be two random variables, with CDF's given by $F_X(x)$ and $F_Y(x)$, respectively. Is there a known interpretation/significance of the pointwise maximum of $F_X(x)$ and $F_Y(x)$?
There ...
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Using a double integral to find the CDF of the standard Cauchy distribution
This question overlaps with the second question here. Blitzstein and Hwang's "Introduction to probability" says: "Let $X$ and $Y$ be i.i.d. $N(0, 1)$, and let $T = X / Y$."
...
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The distribution of X + Y - floor(X + Y) where X and Y are independent and uniformly distributed over (0, 1)
I'm having trouble understanding the following discussion on the distribution of $X + Y - \lfloor X + Y\rfloor$ where $X$ and $Y$ are independent and uniformly distributed over $(0, 1)$:
If $t \in [0, ...
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CDF for the distance of a point chosen uniformly inside a square to its boundary
I have the following problem and I want to derive the cumulative distribution function (CDF) $F_X(x)$ for a random variable $X$.
Problem:
A point is chosen at random and uniformly inside a square of ...
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Let $X$ be a random variable with cdf $F$ and pmf $p$. Show that if either of the images of $X$, $F$, or $p$ is countable, so are the two others.
I am trying to prove the result (?) below. I believe I may have proved that $(1\iff 3)$, but I'm struggling to show that $(2)$ implies (or is implied) by either of the other two conditions:
Let $X$ be ...
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If $X\sim N(\mu, \sigma^2)$ and $\Phi$ is the CDF of a standard Normal random variable, what is the distribution of $\Phi(X)$?
Let $\Phi$ be the cumulative distribution function of a standard Normal random variable, $Z\sim N(0,1)$. Let $X\sim N(\mu, \sigma^2)$ follow any Normal distribution. We know that $\Phi(Z)\sim \...
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multivariate distribution function
Show or disprove
$$F_1(x,y) =
\begin{cases}
0, & \text{if } x < 0~~ or ~~y<0 \\ \newline
\\
\min(x,1) \cdot \min(y,1), & \text{if } x \geq 0 \text{ and } y \geq 0
\end{cases}$$
$$F_2(x,...
user1232800
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An integral equation over two CDFs on the unit interval
I have $F_1,F_2$ two CDFs of random variables over $[0,1]$ and a number $0 < m < 1$. I'd like to somehow characterize the solutions to the constraint:
$$
\forall x, 0 < x < 1: m(1 -x) - m\...
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CDF for min(x, 1-x)
Find the cumulative distribution function of $Y=\min(X,1-X)$
if:
a) $X \sim U[0;1]$
b) $$F_X = \begin{cases}0 & : x < 0 \\ x^2 & : 0\leq x < 1 \\ 1 & : 1\leq x \end{cases}$$.
My ...
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Connect definition of 1st-order Wasserstein distance given CDFs and general definition of Wasserstein distance
In this post, the definition of the 1st-order Wasserstein distance is
$\int_{-\infty}^{\infty} |F_1 (x)-F_2(x)| dx$
In Wikipedia, I see something completely different.
How do I connect the 2 ...
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Computing Rice CDF values in spreadsheets
Is it possible to compute CDF values for the Rice distribution using standard spreadsheet functions?
The following probability distribution functions are available in spreadsheets:
Normal
t
F
Chi-...
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0
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Probability distribution of a random variable defined in terms of other random variables
An unbiased cubical dice has number 1 on 1 face, number 2 on 2 faces and number 3 on 3 faces. The dice is rolled twice and the sum of the 2 rolls is denoted by the random variable X.
Obviously, I can ...
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Continuity of random variable and its CDF relation in $R^d$
Problem. For random variables $X_1,...X_d$ with joint CDF $F$, show that for a fixed $x=(x_1,...x_d)\in R^d$, $F$ is continuous if and only if $$P(\bigvee_{i=1}^d(X_i-x_i)=0)=0.$$
(Note. $\bigvee$ is ...
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Analyzing Cumulative Distribution Functions in Sampling Without Replacement vs. With Replacement
I am studying a population of $N$ bits, comprising $K$ ones and $N-K$ zeros. For sampling $n$ bits without replacement, the situation conforms to a hypergeometric distribution. The sum of these $n$ ...
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Taking the inverse of normal CDF inverse after an additive operation
$\Phi\left(x\right)$ is the CDF of a normal distribution with parameters $m$ and $\sigma$. Is there a way to solve for $\rho$ here?
$\Phi\left(x\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\frac{x-m}{\...
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1
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CDF of geometric distribution
In Blitzstein & Hwang, there's a problem about getting the CDF of the geometric distribution (support = {1,2,3,...}). I'm trying to use the same approach to get the CDF of the shifted geometric ...
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1
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Probability of collision vs. Mean Free Path
Background
In physics, there is the concept of "mean free path," which is the expected value for the distance a molecule (for example) can travel before it hits another one.
If they're all ...
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Let $X$ be a random variable with cdf $F_X$. If $F_X$ is continuous, is $\text{im}(X)$ uncountable?
Let $X$ be a random variable. I suspect that if the CDF $F_X$ is continuous, then $\text{im}(X)$ is uncountable. I reason as follows:
Attempt: as $\lim_{x\to-\infty}F_X(x) = 0$ and $\lim_{x\to+\infty}...
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$\mathbb E [f(X)] = \mathbb E [f(Y)]$ for all $f \in \mathcal C_b (\mathbb R) \implies \mathbb P (X = Y) = 1.$
Let $X$ and $Y$ be two random variables on a probability measure space $(\Omega, \mathcal F, \mathbb P)$ such that $\mathbb E [f(X)] = \mathbb E [f(Y)]$ for all $f \in \mathcal C_b (\mathbb R)$ (...
- 2,542
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0
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How to find CDF under presence of multiple random variables
I am trying to derive CDF of the random variable $Y$ from the below expression.
$P_Y = \text{Pr}\bigl(Y\cdot k_1 < \frac{u_1}{X}+u_1k_2\bigr)$ ---(1)
where $\text{Pr}$ denotes probability, $X,Y$ ...
- 71
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1
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147
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The distribution of the minimum value among $mn$ non-independent random variables and Expected average distance in greedy matching on a circle
Now we have several independent and identically distributed random variables following the uniform distribution on the interval [0, 1].They are denoted as $x_1, x_2, x_3, ..., x_m$ and $y_1, y_2, ..., ...
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2
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54
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How does this integral based on the $\Phi$ function equal $x$?
So a stats problem involving normal random variables has a solution involving a step where the following simplification occurs (based on inspection):
$$\int_{-\infty}^{ \infty } y\frac{e^{-(y-x)^2/2}}{...
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1
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47
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Expectation of a function of the CDF of a Normal variable
Let X $\sim$ $\mathcal{N(\mu, \sigma^2})$. Find the Expectation of $\left(-log_{e} \left(\Phi\left(\frac{X - \mu}{\sigma}\right)\right)\right)^3$ , where $\Phi \left(. \right)$ denotes the cumulative ...
0
votes
1
answer
54
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CDF of $Y$ given the CDF of $X$ and $Y=X^2$
Let $X$ be a continuous random variable with CDF $F_X$, and $Y=X^2$. Find the CDF of $Y$.
My solution:
$$F_Y(t)=P(Y \leq t) = P(X^2 \leq t) =
P( -\sqrt{t} \leq X \leq \sqrt{t}) = $$
$$P(X \leq \...
- 675
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1
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Determine all $(p,q)$ where Lorentz quasi-norm $\|\cdot\|_{L^{p,q}}$ is norm
On a measure space $(X,\mu)$, for $0<p,q<\infty$ the Lorentz space $L^{p,q}(\mu)$ is defined by $$\|f\|_{L^{p,q}(\mu)}:=p^\frac1q\|t\cdot\mu(|f|>t)^\frac1p\|_{L^q(\mathbb R_+,\frac{dt}t)}=p^\...
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1
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Finding $x$ and $y$ with the information that "On exactly half of the days,No more than one student was absent". [closed]
Henry recorded the number of students present everyday for a total of $40$ days. Also there are a total of $29$ students. Here is the frequency table:
Also it is given that,on exactly half of the ...
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