Questions tagged [cumulative-distribution-functions]

For questions related to cumulative distribution functions.

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Cumulative distribution function of log-normal distribution

In a lot of sources, such as the Wikipedia page for log-normal distributions, the cumulative distribution function for log-normal distributions is denoted as $$F_X(x)=\Phi\left(\frac{\ln x-\mu}{\...
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12 views

Cumulative distribution of dependent normal variable

Let $X\sim N(0,\sigma^2_X)$, $Y\sim N(0,\sigma^2_Y)$ be two dependent normal random variables, with $COV(X,Y)=a$. Let $F_X(\alpha):=P(X\leq \alpha)=0.5+0.5\operatorname{erf}(\frac{\alpha}{\sigma_X\...
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1answer
14 views

Comparing two Standard Normal Correlated Variables

Let's say I have two standard normal variables $X, Y$ (both with mean 0 and variance 1) with correlation $\rho > 0$. Can I make any conclusions about their joint distribution? For example, can I ...
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23 views

Joint Probability Distribution Problem (drawing balls from an urn)

Suppose that 3 balls are randomly selected from an urn containing 3 red, 4 white, and 5 blue balls. If we let X and Y denote, respectively, the number of red and white balls chosen, find the joint ...
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11 views

Integrate beta distribution over a range

I am trying to compute the mass of the beta distribution between two points. This involves computing the following integral $$I(x,y;\alpha,\beta) = \frac{1}{B(\alpha,\beta)}\int_{x}^yp^{\alpha-1}(1-p)...
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22 views

Probability of an event not recurring [closed]

If I have a vector consisting of ones and zeros, where the weight of ones is at the beginning and the weight of zeros is at the end of the vector, when can I be confident (e.g., 95%) that there will ...
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4 views

Generalized equation for bivariate normal distribution

I would like to know how to derive the CDF of a bivariate probit model as shown below: $p^{y_{1},y_{2}} = \Phi(2c_{1} - 1*x_{1}\beta_{1},2c_{2} - 1*x_{2}\beta_{2}, (2c_{1} - 1)*(2c_{2} - 1)*\rho)$ ...
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11 views

CDF of the sum of independent r.v.

I was wondering if there's a known relationship between $F_{X+Y}$ and $F_X+F_Y$, when $X, Y$ are independent r.v. (or even comonotonic r.v.) Thanks in advance!
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16 views

Calculating Correlation Given Joint Cumulative Distribution Function

I am interested in computing $Corr(X,Y)$ for $0 \leq x \leq 1$ and $0 \leq y \leq 1$, given two cases: firstly, $F_{X,Y}(x,y)=min(x,y)$ and secondly, $F_{X,Y}(x,y)=xy$. How would I go about doing ...
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26 views

Why does F(X) have uniform distribution in [0,1]?

Let $X$ be a random variable with strictly increasing $F(t)$ cumulative distribution function (with inverse function $F^{-1}(t)$). We can show that $F(X)$ ~ $unif(0,1)$: $$P(F(X)≤t)=P(X≤F^{-1}(t))=F(...
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35 views

Using the Distribution Function Technique and then linking it to the normal distribution

I have $Z$ as a standard random variable (with zero mean and unit variance), for all $t≥0$ and $X_t=\sqrt t Z$ I need to find the distribution function ${F_X}_t = P(X≤ x)$ and then show that $X_t $ ~ ...
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27 views

Distribution of X-Y for identical independent random variables

I have X and Y which are independent and both have an exponential distribution with density function $f(x) = e^{-x}$ if $x\gt0$ I want to find the distribution of X+Y and X-Y. Let U=X+Y, V=X-Y My ...
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9 views

Find the cumulative distribution function of Laplace random variable

1) Let X be a Laplace random variable (μ; σ). The density is of the form f (x) = σ/2 exp (-σ|x - μ|); σ> 0: Find the CDF of X Find E [X] and Var (X), Find the law of |x - μ| 2) Let X ~ U (0; a), a> ...
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42 views

Normal distribution around N

I have not done stats for several years and seem to have forgotten the basics. I am trying to find the standard deviation of a normal distribution given a desired mean and cumulative probability ...
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34 views

Expectation of g(f(X)+V) given X

Given two unknown functions $f,g : \mathbb{R} \to \mathbb{R}$, how would I go about to compute the expression for $h(X) = \mathbb{E}[g(f(X)+V)|X]$ where $V \sim \mathcal{N}(0,\sigma^2)$ and $X$ and $...
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45 views

Numerically find intergral with cdf.

I need to find a value of this integral. $f(\tau) = \int_{b/\sigma}^\infty \dfrac{1}{\sqrt{2\pi}}*\exp(-\dfrac{y^2}{2})*\Phi(-\dfrac{1}{\sqrt{(1-r^2(\tau)}}(\dfrac{b}{\sigma}(\dfrac{\tau}{\tau_0}-1)+r(...
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38 views

Find the random event set and c.d.f. for a random variable.

So, I need to find random event set for as well as find the c.d.f. for a random variable that describes the chance that there are no black balls left to pull from the bag of total 5 balls, 2 of them ...
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1answer
40 views

The cdf and pdf of the random variable $X(\omega)=1/\omega$

Consider the probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\Omega=(0,1]$, $\mathcal{F}$ is the Borel $\sigma$-field generated by intervals of the form $(0,\frac{b}{2^n}]$ with $b\leq 2^n$,...
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1answer
21 views

Set of solutions to probability quantile equation

Let $X$ be a random variable taking its values over some set $S$, with cumulative distribution function (cdf) $F$. Let $\epsilon\geq0$. We define the set $S^*(\epsilon)$ as follows: $$S^*(\epsilon)\...
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1answer
18 views

Subtract two normal cumulative distribution functions rather than plotting a normal one to compare a binomial with a normal variable?

In order to understand the Central Limit Theorem, I am comparing a $Binomial(n,p)$ variable with a large $n$ and a normal variable with mean $\mu p$ and a standard deviation $\sigma = \sqrt{np(1-p)}$. ...
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28 views

Find CDF of a uniform distribution between 2 values

So, I've been struggling on a problem that goes like this: X is a random variable with this probability density function: ...
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27 views

Can you find out what the cumulative distribution function is if you know the random variable X?

I have a theoretic question on my probability class. Suppose that you know the X random variable. Can you calculate the cumulative distribution function? If not, why not?
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1answer
29 views

Solving Normal Probability Distribution (PDF) and Cumulative Probability Distribution (CDF) for given X and comparing to Excel's NORM.DIST() function

I have been trying to understand and implement the Excel function NORM.DIST(x, mean, standard_deviation, cumulative) in another programming language. There exist ...
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19 views

Numerical evaluation of cumulative gaussian distribution function for a particular correlation matrix

I'm looking for an efficient numerial method to calculate the cumulative multivariate distribution function for normally distributed random variables with correlation matrix \begin{equation} \mathbf{\...
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14 views

how to calculate the cumulative distribution function of sums of n Bernoulli distribution?

I'm a beginner on statistics. And I come up with this question when studying Bernoulli distribution: Let $X_n$ be the sum of $n$ independent trials of a Bernoulli experiment, whose result can be $1$ ...
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1answer
16 views

How are the Probability Measure and Cumulative Distribution Function linked when calculating the Expectation of a RV X?

Given a probability space $(\Omega, \Sigma, P)$ and a measure space $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$, let $X: \Omega\rightarrow\mathbb{R}$ be a RV which is $(\Sigma, \mathcal{B}(\mathbb{R}))$-...
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30 views

Find $\alpha$ and $\beta$ so that $f_X(x)$ can be a density function.

\begin{equation*} f_X(x) = \begin{cases} \frac{4x^2}{5} & \text{ , if } 0 < x \leq 1\\ \alpha(5-2x) & \text{ , if } 1 \leq x < 2\\ \beta x^2 & \text{...
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1answer
20 views

the composition of a random variable and its cdf

Let $X$ be a continuous random variable. Let $F(t)=P(X\le t)$ be the cdf (cumulative distribution function) of $X$. Then the random variable $Y=F(X)$ takes values in the unit interval $[0,1]$. What is ...
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1answer
26 views

Maximum Likelihood of P(x<a) = a and P(x<a) = a^2 number generators given a sample

The question asked: There are two number generators. Given a sample, which of these generators is it from? 1) generates a random number uniformly between 0 and 1 with $P(x<a) = a$ 2) generates a ...
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1answer
40 views

X is a Random Variable taking values {1,2,…} with P(X=k)=$c/[k(k+1)]$

Show $P(X\ge k)=c/k$ I tried getting the integral for $X\ge k$. Didn't work out. $$\int_k^{\infty} c/k(k+1)dk=-c*ln(|(1/\infty)+1|)+c*ln(|(1/k)+1|)$$ Which equals $c*ln(|(1/k)+1|)$, which I don't ...
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25 views

The $L^1$ distance of two CDF is the $L^1$ distance of the quantile function coupling

In a book I found the following exercise: Let $F,G$ be two cumulative distribution functions. Then $$\int_0^1 \vert F^{-1} (t) - G^{-1} (t)\vert \text d t = \int_\Bbb R \vert F(x) - G(x) \vert \text ...