Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions. Use this tag along with the tags (probability), (probability-theory) or (statistics).

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Finding the expected value of two correlated RVs

$\newcommand{\Exp}[1]{\mathbb{E}\left[#1\right]}$ I am interested in understanding wether the following approach holds when calculating the expectation of two correlated random variables. Suppose $X\...
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4 votes
1 answer
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Prove that $\nu$ is absolutely continuous w.r.t. $\mu$ iff $\sum \alpha_j^2<\infty$

This is question 3 in Chapter 4.12 from Barry Simon - Real analysis I tried to define $ f_j : \{0,1\} \to \mathbb{R} $ where $n$ means that it is a function from $j$th $\{0,1\}$ in the product to $\...
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2 answers
42 views

Suppose that the random vector $(X, Y)$ is uniformly distributed over the unit ball in $R^2$. Calculate $Cov(X, Y)$

Suppose that the random vector $(X, Y)$ is uniformly distributed over the unit ball in $\mathbb R^2$. Calculate $Cov(X,Y)$ I'm not sure how to solve this covariance problem. I would appreciate some ...
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-1 votes
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Probability density function of max(0, z)

Question The full question is from Exercise 4.6 in Statistical Inference (Casella, Berger). It is A and B agree to meet at certain place between 1 PM to 2 PM. Suppose they arrive at meeting place ...
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How to find PDF of a random variable when it is squared and multiplied by constant?

I am getting confused in the following situation. Say we have a random variable $X$ whose PDF is known to us. Now how to find the PDF of another random variable say $Y$, which is related to $X$ as $Y =...
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Monte Carlo simulation at football

Would you please help and clarify how to run a Monte Carlo simulation to determine the probability of one of the two teams winning a match given the average number of goals for each team (in matches ...
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-2 votes
0 answers
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Probability that at least $6$ of $10$ randomly drawn light bulbs are white [closed]

Out of $100$ bulbs produced by a manufacturing company, $35$ are white light bulbs and the rest are yellow light bulbs. If $10$ bulbs are randomly drawn without replacement, find the probability that ...
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1 vote
1 answer
31 views

Consistent estimator of function of moments up to given order

Asume we the moments of a distribution exist up to order $n$. Consider the estimator $$t (x)=f(m_1(x),..., m_n(x)) $$ where $m_k(x):=1/r \sum_{i=1}^r x^k$. We assume we have $r$ samples of the ...
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2 votes
3 answers
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Help finding the probability of a pipe being blocked in a system of parallel pipes

There is a system of pipes from one point to another. Pipe A is the start point, and connects left to right to Pipe B, C & D, which are in parallel and connect to the end point. The pipes can flow ...
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2 votes
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Bayesian Network for wordle

Intro In preparation for stuyding AI, I'm currently studying probability and bayesian inference. As a first challenge in the subject, I want to model and train a Bayesian network that is able to solve ...
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Mutual Information between $v_1$ and $v_2$ coming from the same Inverse-Wishart distribution?

Say that $\left(\begin{matrix} v_1 & c\\ c & v_2 \end{matrix}\right)$ is a bivariate covariance matrix that comes from an Inverse-Wishart distribution $W^{-1}(\Psi, \nu)$. Then what is the ...
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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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1 vote
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Understanding absolute value of random variables

In my work I am facing the following situation: $y = |a+bc|^2$ ----(1) where $a,b,c$ are zero mean circularly symmetric complex Gaussian (ZMCSCG) random variables with variance $\sigma^2_a, \sigma^2_b,...
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PDF of following random variable.

I am trying to find the PDF of random variable $X$ but not getting it correctly. $X = \sum_{m=1}^{N}|a_m+\eta b_m c_m|^2$ ----(1) where $a, b, c$ are Zero mean circularly symmetric complex Gaussian (...
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3 answers
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Entropy for random variable and probability distribution

Let $(\Omega, \mathbb{P})$ be a finite probability space. Then one can define the entropy of probability distribution $\mathbb{P}:\Omega\to [0,1]$ as follows: $$H(\mathbb{P})=\sum \limits_{\omega \in ...
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If two functions are close apart can I prove the difference of their empirical loss is also small?

I am trying to understand the proof of Theorem 3 in the paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. Basically there exist atleast one $w_{L,e}$ in $\...
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1 vote
0 answers
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How to get the Lipschitz constant $L$ using the inequality?

Taken from paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. $$ \begin{aligned} &\mathbb{P}\left(\exists f \in \mathcal{F}: \frac{1}{n} \sum_{i=1}^{n}\left(y_{...
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1 vote
1 answer
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Conditional distribution of one of the two exponential random variables, given one is smaller than the other

Let $X$ be a random variable with exponential distribution with parameter $a$, i.e. $X\sim Exp(a)$. See https://en.wikipedia.org/wiki/Exponential_distribution for the definition of exponential ...
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Statistical mechanics partition function from probability distribution

I am curious about the mathematical background of something I came across while working on a problem in statistical mechanics. As an example, I am going to use the classical canonical ensemble, though ...
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Estimates of parameters of interest $θ_1$ & $θ_2$ ($\pm$ standard errors) were $25\pm10$ & $10\pm3$. Find estimate & Standard Error of $δ=θ_1/5−θ_2$

Problem: In two independent studies, the estimates of the parameters of interest $θ_1$ and $θ_2$ ($\pm$ their standard errors) were computed to be $25\pm 10$ and $10\pm 3$, respectively. If one is ...
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Given a person's age x, population's average life span u, predict life span.

Given a person's age x, population's average life span u, predict how long person can live? e.g. Say x is 6 and u = 76 one would expect life span of 74. But when x is 90, for same u = 76, one would ...
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1 answer
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What is the distribution of a quadratic function of normal distributions? [closed]

Suppose $Z_i$ is independent standard normal distributions, i.e. $Z_i\sim N(0,1)$, $i=1,2,\cdots, d$. What is the distribution of $$ \sum_{i=1}^d (a_iZ_i+b_iZ_i^2). $$ I know when $a_i=0$, it is the ...
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1 vote
2 answers
50 views

Expected number of rounds for a product of uniform random variables on $[1/2,3/2]$ to be for the first time below a given threshold

Starting with w=1, each time we multiply w by a number x sampled independently and uniformly from [1/2, 3/2] until it is smaller than a given value c. What's the expected number of rounds for this ...
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3 votes
1 answer
48 views

Law of the square of a martingale divided by its bracket

Let $(M_t)_{t\geq 0}$ be a continuous martingale such that $M_0=0$ almost surely. There exists an increasing process $(\langle M\rangle_t)_{t\geq 0}$ which is called the bracket of $M$ such that $M^2-\...
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What positive distribution has the form $x^{-\ln(x)}$

What is the name for the continuous distribution $f(x)\propto x^{-\ln(x)}$ for $x\geq 0$? More precisely, I'd like the name for the family of distributions of the form $f(x)\propto x^{-c \ln(x)}$, ...
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1 vote
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What positive distribution has the form $e^{-x-\frac{1}{x}}$

What is the name for the continuous distribution $$f(x)\propto e^{-\left(x+\frac{1}{x}\right)}$$, where $x\geq 0$? Does it even have a name? My interest in it is because it is the simplest ...
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2 votes
1 answer
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Expected value of $X/(X+Y)$

I have a task. $X$ and $Y$ are independent random variables with exponential distribution and $\mathbb{E}X=1$, $\mathbb{E}Y=\frac{1}{2}$. Calculate $\mathbb{E}\big(\frac{X}{X+Y}\big)$. I tried to ...
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Determining the pdf of the product of two independent random variables

I have two iid random variables $A_{1}$ and $A_{2}$ with a probability density function as such: \begin{equation} f(x) = \frac{1}{12x} \end{equation} In the domain $x \in [0.5, 0.6]$, and $0$ ...
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Determine the distribution of the number of cars in the highway during a two hour period. [closed]

In a highway vehicles are passing according to a Poisson process having a rate of $300$ per hour. Suppose each vehicle is a car with probability $86 \%$ and at truck with probability $14 \%$. $(a)$ ...
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1 vote
0 answers
33 views

What is the expected value when two dice are thrown?

Two dice are thrown and a random variable X is taken as a sum of the values on both dice plus a product of those values. What's EX? Could you please check if I constructed pmf correctly? Is my answer ...
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1 vote
1 answer
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Likelihood of censored exponential random variables

Consider $X_1, \dots, X_n \stackrel{\text{iid}}{\sim} \text{Exp}(\lambda)$ and define \begin{equation*} Y_i = \begin{cases} X_i & X_i \leq c \\ c & X_i > c \end{cases} \end{equation*} for ...
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0 votes
1 answer
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Let $X$ be a random variable following the standard cauchy distribution. Show that $E[X^{\alpha}]$ exists, $\forall\alpha\in(0,1)$

Let $X$ be a random variable following the standard cauchy distribution. Show that $E[X^{\alpha}]$ exists, $\forall\alpha\in(0,1)$ According to me, $E[X^{0.5}]=\int^{\infty}_{-\infty}\dfrac{\sqrt{x}}{...
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0 votes
1 answer
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How a uniform distribution may induce different solution regarding Bertrand's paradox

I have a neophyte question regarding the formulation of the Bertrand's paradox. This is related to the definition of what we call a uniform distribution. If we consider the 3 point of views of “random ...
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0 answers
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Data required for Dual Dirac PDF

I have a simple question regarding the dual Dirac PDF. If I have a set of deterministic data, e.g., d = [-2ps, 2ps, -2ps, 2ps, -2ps, 2ps] Would the resulting PDF look like a dual Dirac PDF? Where ...
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1 vote
0 answers
63 views

How the second inequality stands?

Taken from paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. Theorem 3. Let $\mathcal{F}$ be a class of functions from $\mathbb{R}^{d} \rightarrow \mathbb{R}$ and ...
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1 vote
1 answer
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If a random variable has an atom at zero, does it have a density?

Let $Y$ be a random variable with distribution function $$ F_Y(x) = \begin{cases} 0 &\quad x<0\\ p &\quad x=0\\ p + (1-p)F_X(x) &\quad x>0 \end{cases} $$ where $X$ is a continuous ...
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1 vote
1 answer
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What is the result of XOR between two Bernoulli random variables?

Consider two Bernoulli random variables $X$ and $Y$ with probabilities $p_1$ and $p_2$. Now $Z = X \oplus Y$, $\oplus$ is a logical XOR operator. Is $Z$ a Bernoulli random variable and what is its ...
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0 votes
1 answer
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Probability of multiple distributions

I have a question concerning the calculation of the probability of an event among multiple probability distributions. Let's say we have two continuous probability distributions $D_1$ and $D_2$, given ...
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0 answers
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Probability mass function of bank loan, find expected payment.

We have a random variable X that is the number of months that a certain owner of an estate needs to pay a loan in the Bank if he/she has a contract with insurance company to help him/her to pay a loan....
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0 answers
16 views

Checking a 1-Lipschitz condition for Normal density integral [closed]

Let $\phi$ denote the density function for a $N(0,1)$ random variable. Is the following true? $$\int_{\mathbb{R}} \vert \phi(x)-\phi(x-\theta)\vert dx = O(\theta) $$ as $\boldsymbol{\theta \rightarrow ...
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1 vote
0 answers
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Integrating bi-variate Gaussian density with respect to their correlation coefficient

I'm wondering if there is any literature that deals with the integral of the following type $$ \int \dfrac{1}{2\pi\sqrt{1-\rho^2}} \cdot \exp(-\dfrac{(a^2+b^2+2\rho ab)}{2(1-\rho^2)}) \cdot d\rho $$ ...
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0 votes
1 answer
18 views

Finding Conditional expectance, variance from joint pdf

X and Y have the joint pdf: $f(x,y)=e^{-y}$ for $0<x<y<\infty$ Compute $E(Y|X)$ and $Var(Y|X)$. I computed $f_X(x)= \int_{x}^{\infty} f(x,y)dy=e^{-x}$ $f_{Y|X}= e^{x-y}$ for $ x>0$ Then $...
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0 votes
1 answer
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The projection of n-joint distribution onto 2-joint distribution

Let $P$ be a probability distribution on the product of Polish spaces $X_1\times \cdots \times X_N$. In particular, $X_N=X_j$. Suppose the marginal distribution of $P$ on $(X_i)$ are $(\mu_i)$, where $...
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Distribution of $Z = \frac{\sin (XY)}{\cos (X+Y)}$ where $X$ and $Y$ have a joint bivariate normal distribution?

Let us suppose that $X$ and $Y$ have a joint bivariate normal distribution with mean vector $\vec{\mu}$ and covariance matrix $\Sigma$. What distribution does $Z$ have if $$Z := \frac{\sin (XY)}{\cos (...
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3 votes
1 answer
32 views

how to approach the discrete counterpart of a continuous CDF of uniform random variable

I am following an online example to get the CDF of a function $Y$ of uniform random variable $x \in [0, 1]$ $$ Y = \frac{30}{2-x} $$ Based on the example, CDF is a distribution of probability for the ...
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3 votes
1 answer
29 views

Using Jensen's inequality on $\mathbb{E}[1/x]$ when x can be both positive and negative

We know the function $f(x)=\frac{1}{x}$ is convex when $x$ is positive and concave when $x$ is negative. I want to show if $\mathbb{E}[\frac{1}{x}]$ is bigger than or smaller than $\frac{1}{\mathbb{E}[...
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1 vote
1 answer
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CLT Application / R code Question

A store sells two different coffee makers: a basic model for \$30, and a fancier model for \$50. We assume that different buyers’ choices don’t affect each other, and that different buyers share the ...
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0 votes
2 answers
44 views

Conditional continuous probability on a fixed point

I was asked to show the following statement: Let $X, Y$ both be equally distributed random variables on $[0,1]$, we define $$ P(\{Y \leq y\}|\{X=x\}) = \lim_{h \downarrow 0} P(\{Y \leq y\}|\{x \leq X \...
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-1 votes
0 answers
24 views

Calculating probability with a given pdf [closed]

Please check out the image attached. I have tried integrating the function from 0.3 to 1 but I am unable to remove the delta term.
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0 votes
1 answer
36 views

There exists a non-negative random variable such that $P( \{ \omega \in \Omega: E(X)-2\sigma<X(\omega)<E(X)+2\sigma \})=\frac{3}{5}$. True or False?

Problem State whether given statement is True or False. There exists a non-negative random variable such that: $P( \{ \omega \in \Omega: E(X)-2\sigma<X(\omega)<E(X)+2\sigma \})=\frac{3}{5}$ My ...
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