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Transformation of a CDF [closed]

Suppose I have a continuous CDF $F_{X}(x)$ where $x \in [0,1]$. Suppose there is another continuous CDF $F_Y(y)$, and we want to transform $F_{X}(x)$ into another continuous CDF. Now there is a ...
Shujun Tan's user avatar
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1 answer
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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 ...
LZ981ko's user avatar
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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 $...
Heretolearn's user avatar
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27 views

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} = ...
Heretolearn's user avatar
2 votes
1 answer
33 views

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}\...
selenio34's user avatar
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1 answer
74 views

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 ...
guttf's user avatar
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59 views

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 ...
Daniel Muñoz's user avatar
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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 ...
BigTittyHooka's user avatar
1 vote
0 answers
27 views

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 ...
WQE's user avatar
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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 $T = X / Y$." ...
johnsmith's user avatar
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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, ...
johnsmith's user avatar
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1 answer
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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 ...
Mouh Kramo's user avatar
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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 ...
Sam's user avatar
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1 answer
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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 \...
cgmil's user avatar
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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,...
joel07 _'s user avatar
1 vote
2 answers
103 views

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\...
Martin Modrák's user avatar
1 vote
1 answer
37 views

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 ...
Disciple's user avatar
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1 answer
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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 ...
Iterator516's user avatar
0 votes
1 answer
20 views

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-...
feetwet's user avatar
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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 ...
juice-r's user avatar
  • 11
1 vote
2 answers
25 views

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 ...
reyna's user avatar
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5 votes
1 answer
177 views

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 $n$ ...
Dotman's user avatar
  • 326
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1 answer
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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}{\...
Victor Yerz's user avatar
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1 answer
58 views

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 ...
matto's user avatar
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1 answer
147 views

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 ...
jwd's user avatar
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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}...
Sam's user avatar
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0 votes
1 answer
34 views

$\mathbb E [f(X)] = \mathbb E [f(Y)]$ 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 $f \in \mathcal C_b (\mathbb R)$ (...
Anacardium's user avatar
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0 votes
0 answers
13 views

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, $X,Y$ ...
chaaru's user avatar
  • 71
2 votes
1 answer
129 views

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, ..., ...
Randy's user avatar
  • 53
1 vote
2 answers
50 views

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}}{...
Calvin Huang's user avatar
0 votes
1 answer
45 views

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 ...
Soumen Maity's user avatar
0 votes
1 answer
34 views

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 \...
Michał's user avatar
  • 675
1 vote
1 answer
55 views

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^\...
Liding Yao's user avatar
  • 2,259
2 votes
1 answer
66 views

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 ...
LifeIsMath's user avatar
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0 answers
50 views

Reference for a good multidimensional portmanteau theorem

I need references for a good n-dim portmanteau theorem that includes the following equivalent assertions: The sequence $(X_n)_n$ converges in distribution to a random vector $X$ of $\mathbb R^\nu$; $...
MikeTeX's user avatar
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1 vote
1 answer
49 views

Convergence of maximum of n independent random variables to a random variable that isn't degenerate

I have a question that follows from this question. That discussion shows that if $F(x) = 1 - \frac{1}{2}x^2$ for $x \in [-1, 0)$ and $X_i$ are independent random variables with CDF $F$, then $\max_{1 \...
johnsmith's user avatar
  • 367
1 vote
1 answer
79 views

Is it true that ℙ{ F(X) ≤ u } ≤ u for all u in [0,1]? [closed]

If X is an arbitrary (not necessary continuous) r.v. and F is its cdf, is it true that $\mathbb{P}(\{F(X)\leq u\})\leq u$ for all $u$ in $[0,1]$? This feels like it should be straightforward but it ...
Simd's user avatar
  • 437
0 votes
1 answer
88 views

CDF of Convolution Z = X + Y Formula is $F_Z(z) = \int_{-\infty}^{\infty}F_X(z-\xi)\ \mathrm dF_Y(\xi)$

This formula is presented without explanation as the CDF for the convolution of any two independent random variables in the textbook I'm reading (introduction to stochastic modeling), and doesn't seem ...
Christoper Robin's user avatar
0 votes
1 answer
22 views

Convergence of maximum of n independent random variables with quadratic CDF in a closed interval

If $F(x) = 1 - \frac{1}{2}x^2$ for $x \in [-1, 0)$ is the CDF of independent $X_i$, then what does $\max_{1 \leq i \leq n} X_i$ converge to as $n \rightarrow \infty$? ($F(x) = 0$ for $x < -1$ and $...
johnsmith's user avatar
  • 367
7 votes
2 answers
341 views

Expectation of a monotone function of CDF: $\mathbb E \left [g(F(X)) \right ]$

If $X$ is a random variable with distribution function $F,$ then $\mathbb{E}[(F[X])^{-1/2}]$ can be computed by integration by parts, if $X$ has a continuous density $f$. What happens in the general ...
STrick's user avatar
  • 379
0 votes
0 answers
21 views

CDF of entity of paired random variables?

If $2n$ samples are generated in the following way: Draw $U_i\sim Unif[0,1]$ i.i.d. for $i=1,...,n$ Define $\tilde U_i := 1 - U_i$ for $i=1,...,n$ If I would now draw a random sample $X$ among these ...
Nick Halden's user avatar
0 votes
0 answers
20 views

Càdlàg property not compulsory for CDF in Cox theorem approach of probabilities?

The proof that any Cumulative Distribution Function is Càdlàg within the axioms of Kolmogorov is based on countable additivity. However, countable additivity is (and this needs to be taken with a ...
niobium's user avatar
  • 1,231
0 votes
0 answers
40 views

Quantile function

I have the following problem: I have the function $f(x,y)=\frac{1}{(2\pi)}(1+x^2+y^2)^{(-3/2)} $. I have found the quantile function $Q_{T}$ of $T$, where $T=|Y|$, to be: $\tan(\frac{\pi \cdot y}{2})$....
Amy A's user avatar
  • 63
0 votes
1 answer
68 views

CDF of $Z=\min(X,c)$ when $c$ is a constant

For independent continuous (non-negative) random variables $X$ and $Y$, we may have CDF for $Z=\min(X, Y)$: \begin{align*} F_Z(t) &= \Pr(X \le t \;\text{or}\; Y \le t) = \Pr(X \le t) + \Pr(Y \le t)...
Frey's user avatar
  • 1,103
0 votes
0 answers
19 views

How to use cumulative distribution functions within an interactive simulator

I am building a simple simulator in python that should simulate an event taking place based on its ...
FTM's user avatar
  • 101
1 vote
0 answers
50 views

Integral involving the CDF of a normal and a nonlinear function

I know that the following result holds Given $a,b,\lambda\in \mathbb{R}$, $\Phi$ the cumulative distribution function of a standard normal random variable, and $f_X$ the probability density function ...
FM89's user avatar
  • 13
0 votes
1 answer
32 views

Proof of nomalization condition for a Derived Density Function and its associated CDF expression

I encountered a knowledge gap while attempting to solve a problem I posed to myself. I have presented the problem along with my reasoning below. Let $G$, $Q$ and $\Theta$ be 3 random variables such ...
Kevin B.'s user avatar
1 vote
1 answer
40 views

Why this relationship between the prior and posterior of a CDF is true?

We use a location/scale transformation of a base CDF which we show by $A$, to have its posterior which we show by $B$. $\alpha$ is the location parameter and $\beta$ is the scale parameter. Let $X$ be ...
user avatar
0 votes
1 answer
44 views

Finding the covariance and correlation of two random variables

Let $X$ be a random variable that has a standard uniform distribution $U(0,1)$, let $Y = X^k$, $k > 0$. I have performed the random variable transformation receiving $g(y) = \frac{1}{k}y^{\frac{1}{...
milosz7's user avatar
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0 answers
92 views

Compute standard normal CDF from CDF using Gil-Pelaez inversion formula

I am trying to compute the series representation of the standard normal distribution CDF from its characteristic function. Given a random variable $ X $ following a standard normal distribution with ...
Pierre Cayet's user avatar

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