Questions tagged [cumulative-distribution-functions]

For questions related to cumulative distribution functions.

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Clarification about inequality in summation

In my work I am facing the following situation, wherein I am trying to compute CDF of random variable $Y$ such that $F_Y(y) = \text{Pr}(\sum_{m = 1}^M |Z_m|^2\leq \frac{y}{A})$ -----(1) where $Z$ is a ...
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Finding the best formula for this case lottery algorithm?

there, sir. I'm a Developer and now working on a project. So the problem is... A program generates a 6-digit number for a winning ticket with each digit between a range of (0-9). Then user buys some ...
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How to prove an inequality involving second moments and inverse distribution functions

I have recently encountered a problem requiring dealing with two inverse distribution functions simultaneously. For a cumulative distribution function $\Psi$, the function $\Psi^{-1}:(0,1)\rightarrow \...
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Monte Carlo estimation of this probability

Let $p=P(X+Y\geq t)$ and $t\in \Bbb R$. Question: Using the classic Monte Carlo method, find an estimator $p_n$ of $p$ using $F^{-1}_X$ and $F^{-1}_Y$ Attempt: I defined $$Z=X+Y$$ then I expressed $$p=...
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Prove that $F$ (the cumulative distribution function) has at most a countable number of discontinuities.

Let $(\mathbb{R},\mathcal{B}(\mathbb{R}),\mathbb{P})$ be a probability space, where $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra and $F:\mathbb{R}\to[0,1]$ be the distribution function ...
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Second order Taylor expansion of $\int_0^\mu\frac{x}{2\pi} - F(x)dx$ when $F(x)$ is a distribution function

I'm currently reading an article in which the authors perform a second order Taylor series approximation $\int_0^\mu\frac{x}{2\pi} - F(x)dx = \frac{\mu^2}{4\pi} - \frac{\mu^2}{2}F'(0) + o(\mu^2)$ when ...
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What is the counterpart for cdf in measure theory?

I am interested in the correspondence between probability and measure theory. Currently, I know that a random variable is a measurable function, a probability function is a measure, etc., But, I am ...
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calculation of $E[\Phi(X)]$

Let, $X\sim N(\mu, \sigma^2)$ and let $\Phi(\cdot):\mathbb{R}\to[0,1]$ be the CDF of a standard normal distribution. Then, what is the pdf of $Y=\Phi(X)$. Also, find $E[\Phi(X)]$. Note:- Here, $Y=\...
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Find density functions of $Z=Y-X$ when the joint density function is known.

Find the density function of $Z=Y-X$ where the joint density function of X and Y is given by $f(x,y)=1/2,x>0,y>0,x+y<2$ and $0$ otherwise. I know how to do it by finding the CDF first with ...
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Probability problem with Setting up Fz(z) with the jointly continuous density [closed]

Suppose that X and Y are are jointly continuous and have density f(x, y) . Let Z = Y / X ; we want the density of Z. a) Write down the expression for F_{Z}(z) , realizing that you have to be careful ...
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Finding Conditional Cumulative Distribution function from joint density function

The joint probability density function of $X$ and $Y$ is given by: $f_{X,Y} (x,y) =c (4x^2-y^2)e^{-x}$ if $0<x<\infty , -2x<y<2x$ and $0$ otherwise I want to find the conditional c.d.f $F_{...
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How can $ P\big(|X_n| \geq \varepsilon\big) \leq P\big(\big|X_n-\frac{1}{n}\big| \geq \varepsilon+\frac{1}{n}\big) $?

I am following a probability course on MIT OCW, and on a tutorial on probability convergence, in an attempt to use the Chebyshev's Inequality to prove it, the TA claimed that $$ P\big(|X_n| \geq \...
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cumulative distribution function of special random variable

When I see example 2.3.8 in book Elements of Large-Sample Theory, I have a problem: That is P($Y_{n}$=Y)=1-$p_{n}$ and P($Y_{n}$=n)=$p_{n}$. Obviously, $Y_{n}$ is a piecewise random variable, Then, ...
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Cumulative Distribution function support change

Consider parameter $u$ has a cumulative distribution function $F$, with continous density $f$, and that has support $[m,n]$. Suppose the lower support point $m$ is changed to a new point $v$, where $v&...
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A mistaken proof that CDFs are not right-continuous

While trying to prove that CDFs are right-continuous, I wrote the following proof which seems to actually prove that CDFs are right-continuous if and only if the measure of the given point is zero. I’...
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Measuring half-lifetime from 1-cumulative or frequency distributions, what's the difference?

I have a question that is blowing my mind. Let's say that I measure a phenomenon that has a duration in seconds. I can graph the data as a frequency distribution (a histogram), showing a nice ...
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two-dimensional version of "$F_X(X)$ is uniformly distributed"?

It is a well-known fact in probability theory that if $X$ is a continuous random variable and $F_X$ is a cdf of $X$, then $F_X(X)$ is uniformly distributed over $[0, 1]$. Is there a two-dimensional ...
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Integral on triangle Bivariate

I can't see why the integral $$ J=P(Z_1>0,Z_2>0,Z_1+Z_2<1) $$ can be computed as $$ J = \left(F(\frac{1}{\sqrt{2}})-\frac{1}{2}\right)^2 \qquad (*) $$ where $F$ is the cumulative probability ...
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Conditional distribution of $Z$ given $Z=X$

Given that $X$ and $Y$ are independent exponential random variables with parameters $\lambda_1$ and $\lambda_2$. If $Z=min(X,Y)$, find the conditional distribution of $Z$ given $Z=X$ My try: I found ...
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How to find the CDF of $Y=\frac{1}{2}X$?

Let $X$ have an exponential distribution with rate parameter $\lambda=\frac{1}{2}$ I believe that the probability density of $X$ is $$f_X(x) = e^{-x/2}$$ So the CDF of $X$ is then $$F_X(x)=-2e^{-x/2}$$...
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CDF and cumsum of a prbabolity density function

This is a really really stupid question, but why when I plot a CDF and cumsum of a PDF (e.g. exponential): $$ f(t) ~ = \lambda e^{-\lambda t}, ~~~ t \ge 0 $$ I get ...
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Equality in distribution for quantities involving empirical and regular quantile functions

Let $U_1,...U_n$ be i.i.d. random variables of uniform distribution on $[0,1]$ and $X_1,...X_n$ i.i.d. real random variables with common cumulative distribution function (cdf) $F$ and empirical cdf $...
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Inequality between two cumulative distributions

I am having difficulty in solving the following practice problem: Assume that $X_n \leq Y_n$ for every $n$, $X_n \overset{D}\longrightarrow X$, and $Y_n \overset{D}\longrightarrow Y$. Let $F$ and $G$ ...
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Distribution of $Z=\min \left\{U_{1}, \ldots, U_{X}\right\}$

Let $U_{i}, i=1,2, \ldots$ be independent uniform random variables in $(0,1)$. Also $X$ is a discrete random variable whose pdf is given by: $$P(X=x)=\frac{1}{(e-1)x !}, x=1,2,3, \ldots$$ Find the CDF ...
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Singular Value Decomposition (SVD) with Monotonic Constraint

I am trying to compress some cumulative distribution functions (CDFs) which are stored in an $N \times M$ matrix $A$. Each of the $M$ columns contains $N$ monotonically-increasing values which might ...
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Calculating the joint cumulative distribution function from a junction tree

Assume I have the following Junction Tree between random variables $X_1,\dots,X_7$ that exactly describes the sets variables with non-zero Mutual Information (Alternatively it's the last tree in a ...
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Proof that if $g$ is of bounded variation and $|f|\le M$ then $\left|\int_{[a,b]} f \ dg \right|\le MT$.

Suppose that $g:[a,b]\to \Bbb R$ is a function of bounded variation, and $f$ is Borel measurable with $|f|\le M$ on $[a,b]$. Define $g^+,g^-$ to be the Jordan decomposition so that these are ...
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Differentiating a standard normal cumulative distribution function [duplicate]

I am trying to find a way to differentiate the standard normal cumulative distribution function. I think it has something to do with the Leibniz rule but I can't wrap my head around it. I would like ...
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Gradient of the Inverse Normal CDF

I have a stochastic variable modelled as $x \sim \mathbb{N}(\mu,\sigma^2)$, and I am considering thresholds in $x$ based on the probability of reaching these values, e.g. find $x$ such that $p = P(X \...
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Conditional survival function in landmark analysis

In H.Putter & H.C. van Houwelingen's paper "Understanding Landmarking and Its Relation with Time-Dependent Cox Regression" the authors state that the conditional survival function, given ...
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Partial derivatives of a Black-Scholes solution of a "cash or nothing" call option

Currently working on a problem sheet that asks us to find the partial derivative $\frac{\partial C}{\partial r}$ of the following formula for the price of a "cash or nothing" call option: \...
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How to find the tail of cdf for the distribution?

I don't know how to derive the tail of cdf for this situation: We do a Bernoulli experiment every $\frac{1}{n}$ seconds, the probability of success is $\frac{\lambda}{n}$. $Y_n$ is the waiting time ...
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Right continuity of the cumulative distribution function and an almost sure constant $X$

The dozens of proofs I've seen for showing that a tail sigma-algebra measurable random variable $X$ is almost surely a constant use the infimum $T := \inf \{t \in \mathbb{R}\mid \mathbb{P}[\{X \leq t\}...
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Safe to assume Normal Distribution?

I am reading a textbook called Bayesian Statistics: the Fun Way, and chapter 10 touches on integrating the pdf of a normal distribution for a sample of 6 observations with a mean of 20.6 and standard ...
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2 votes
1 answer
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Area Under The Curve of CDF and Alpha Value - Chi-Sq Test Interpretation

Can you help me understand the meaning of the area under the curve of a PDF in the interpretation of a Chi-Square test? This is what I think I know: If my test statistic (i.e. the chi-square value) ...
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Expected minimum of two independant random variables

This is part of an alternative problem of the newsvendor model. Let $D_1$ and $D_2$ be two independant and positives random variables and let $f_{D_1}$ and $f_{D_2}$ be their density functions. I need ...
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2 answers
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calculating mean of a CDF

I have this complex CDF: F(x)=1-exp(-X^2/c) when c is a constant. How can I calculate mean? In my calculation, I come to calculate this: ...
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Generate random values using an empirical cumulative distribution function

I have a set of data points that I have used to generate my empirical cumulative distribution function (CDF), which looks like this (to simplify things I have reduced the number of points for this ...
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Non-standardised central limit theorem

Let $(X_n)_{n}$ be a sequence of i.i.d. real random variables with mean $\mu \in \mathbb{R}$ and variance $\sigma^2 > 0$. Let $S_n = X_1 + ... + X_n$. The usual central limit theorem ensures that $\...
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Is the limit $x \to - \infty$ of the cumulative distribution function always equal to $0$?

I've the probability density function: $$ f_X(x) = \begin{cases} \frac{1}{2} x^2 \, e^{-x} & \text{if } x \geq 0 \\ 0 & \text{if } x < 0 \end{cases} $$ The CDF is: $$ CDF_X(x) = 1 - \left( ...
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2 votes
1 answer
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Proving that $C(x_1,x_2)=\max\{x_1+x_2-1,0\}$ is a copula

I have the following problem. I need to prove the copula property for the function $C(x_1,x_2)=\max\{x_1+x_2-1,0\}$, better known as the lower Fréchet-Hoeffding-Boundary. A copula is defined as a ...
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1 answer
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Finding CDF from given piecewise PDF

Cheers, I have the following PDF $f_X(x) = \frac{2|x|}{5}, \quad -1 \leq x \leq 2,$ and I am asked to find the distribution function $F_X(x)$. I know that to find it I must solve the following ...
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How to find the probability between two jobs given only standard deviation and mean?

I'm aware, for a standard normal distribution using $Z$ score, we can find the probability $Pr(x<a)$. However, how to solve a problem as follows? There are two jobs, the industrial job with the ...
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Transformation of random variables by method of CDF confusion

My lecturer has given these instructions for transforming a random variable by method of CDFs: The trouble is he doesn't specify much beyond this. Could anyone break these down and give a little more ...
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1 vote
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Criteria for convergence in law proof

We say that a sequence of random variables $X_1,X_2,\cdots$ converges in law to a random variable $X$ if $F_{X_n}(t) \to F_X(t)$ (where F is the cdf) for every $t$ that is a continuity point of $F_X$, ...
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Transformation of random variable using CDF not making sense?

I have two exponential random variables $X,Y$ that are both identical (i.e. they have the same $\lambda$). I am attempting to find their sum using the method of CDFs (please note, I understand how to ...
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Calculate PDF for 2 random variables?

Given $(X,Y)$ uniformly distubuted in $([1,2]\times[1,4]) \cup ([2,3]\times[2,3])$. I found that $f_{X,Y}(x,y) = 1/4$ if $(x,y)$ in $([1,2]\times[1,4]) \cup ([2,3]\times[2,3])$, $0$ otherwise. I'm ...
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How to Find CDF of $T_1-T_2$? [duplicate]

Given that $T_1, T_2$ are iid $\text{exp}(\lambda)$ variates. I want to find the cdf $F_T(t)$ where $T=T_1-T_2$ My Attempt $F_T(t) =_1 P(T<t) = P(T_1-T_2<t) = P(T_1<T_2 + t)$ Where $=_1$ is ...
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2 votes
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How to find $\beta$

so my problem is I have to maximize $\bar{p}^Tx$,subject to $\phi(\frac{\alpha - \bar{p}^Tx}{\sqrt{x^{T}\Sigma x}})\leq \beta$, where $\phi()$ is cumulative distribution function. I have to find $\...
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1 vote
1 answer
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Computing CDF with the total probability rule

I tried to come up with the answer without success. I hope somebody can help me. Given $$X∼Exp(1)$$ $$Z∼Bernoulli(1/4)$$ $$W=(-1)^ZX$$ I have to compute the CDF of $W$. I know that the total ...
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