# Questions tagged [cumulative-distribution-functions]

For questions related to cumulative distribution functions.

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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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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 $... • 367 7 votes 2 answers 341 views ### Expectation of a monotone function of CDF:$\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 ... • 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 ... • 118 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 ... • 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})$.... • 63 0 votes 1 answer 68 views ### CDF of$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)... • 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 ... • 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 Givena,b,\lambda\in \mathbb{R}$,$\Phi$the cumulative distribution function of a standard normal random variable, and$f_X$the probability density function ... • 13 0 votes 1 answer 32 views ### Proof of nomalization condition for a Derived Density Function and its associated CDF expression I encountered a knowledge gap while attempting to solve a problem I posed to myself. I have presented the problem along with my reasoning below. Let$G$,$Q$and$\Theta$be 3 random variables such ... • 21 1 vote 1 answer 40 views ### Why this relationship between the prior and posterior of a CDF is true? We use a location/scale transformation of a base CDF which we show by$A$, to have its posterior which we show by$B$.$\alpha$is the location parameter and$\beta$is the scale parameter. Let$X$be ... 0 votes 1 answer 44 views ### Finding the covariance and correlation of two random variables Let$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}{...
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 ...