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Questions tagged [random-variables]

Questions about maps from a probability space to a measure space which are measurable.

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Example of subGaussian with var(X)=1 and $||X||_{\psi_2}$ = M$.

I have an exercise that states: Show that for any $M \geq 1$ there is a r.v. $X$ with var$(X)=1$ and $||X||_{\psi_2} = M$. Where $||\cdot||_{\psi_2}$ is define as follow: For a sub Gaussian r.v. $X$ ...
isaac's user avatar
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1 vote
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How can I get the probability of three random variables satisfying equations containing sums and functions of the variables

Consider three independent random variables, X, Y and Z. The cdf and pdf of these are known to be $F_{X}(x)$, $F_{Y}(y)$, $F_{Z}(z)$ and $f_{X}(x)$, $f_{Y}(y)$, $f_{Z}(z)$. X and Y are positivie ...
WilliamO's user avatar
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31 views

Proving that "a distribution has a density function if and only if its cumulative distribution function $F(x)$ is absolutely continuous".

Wikipedia states that A distribution has a density function if and only if its cumulative distribution function $F(x)$ is absolutely continuous. What is the exact statement of the result above? It ...
Sam's user avatar
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40 views

Find a majorant of two random variables [closed]

Let $X,Y \in \mathbb{L}^{\infty}(\Omega,\mathcal{F}, \mathbb{P})$ two random variables such that $\mathbb{P}(X<0)>0$ and $\mathbb{P}(Y<0)>0$. Is it true that there exists a $Z \in \mathbb{...
leobgg's user avatar
  • 193
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0 answers
8 views

k-independent hash functions vs. orthogonal arrays

In some randomised algorithms, such as Alon-Matias-Szegedy (AMS) algorithm, two different strategies are used for the generation of a family of random numbers with some special correlation properties: ...
yarchik's user avatar
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1 answer
32 views

Show that $XY$ is absolutely continuous and determine its density

If $X$ and $Y$ and absolutely continuous and independent random variables, how to show that $XY$ is also absolutely continuous and what is its density? I know that $X$ and $Y$ are absolutely ...
JD Maximo's user avatar
0 votes
1 answer
25 views

Median of Mixed Random Variable [closed]

I have the following CDF $$ F_X(x) = \begin{cases} 0 & x < 0 \\ 1 - p e^{-x} & x \geq 0 \end{cases} $$ I found $\mathcal{X} = \{ 0\} \hspace{0.1cm} \cup (0,\infty)$ with $P(X = 0) = 1-p$ ...
daniel's user avatar
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-1 votes
1 answer
34 views

Probability space of a random sample and almost sure convergence [closed]

When working with random samples in statistics, there are 2 approaches: To have the single-outcome space $(\Omega, F,P)$, on which $n$ i.i.d. random variables $(X_1,...,X_n)$ act. I.e., a single $\...
jose89'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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Convergence to zero in L2 implies probability of being outside a bounded open set also goes to zero [closed]

Suppose I have a sequence of random variables $\{X_n\}_{n=1}^{\infty}$ taking values in $\mathbb{R}^{k\times k}$ and $S \subset \mathbb{R}^{k\times k}$ is a given bounded open set. If I know that $$ \...
sixtyTonneAngel's user avatar
2 votes
1 answer
43 views

Is every collection of discrete random variables a function of independent random variables?

Let $\Omega = \{0,1\}^n$, and let $(\Omega,2^\Omega,\mu)$ be a probability space. Does there exist $\Omega' = \{0,1\}^N$, product probability measure $\mu' = \mu_1 \otimes \dots \otimes \mu_n$ on $(\...
Julius's user avatar
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Dependent random variables whose convolution adds up

I want to find two dependent random variables $X$ and $Y$ with values in $\mathbb{Z}$ such that $P_X\star P_Y=P_{X+Y}$, where $\star$ is the convolution. What I've tried: I was "tickling" ...
Christoph Mark's user avatar
-2 votes
1 answer
37 views

Question based on iid cont. rv [closed]

Let $X_1, X_2, X_3, X_4$ be iid random variables with a continuous probability distribution. How do I compute the following probability? $$P(X_3 < X_2 < \max\{X_1,X_4\})$$
Chandrahas Nhayade's user avatar
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1 answer
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Can't understand expected gain calculation in first price auction

I'm reading this article: https://en.wikipedia.org/wiki/First-price_sealed-bid_auction#Example What it says: Suppose there are two bidders A and B, whose valuations (for some item being bid on) are $a$...
user9343456's user avatar
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0 answers
44 views

Probability that the Largest of Three Independent Uniformly Distributed Random Variables Exceeds the Sum of the Other Two

I am currently studying probability and statistics and I came across a problem that I am having trouble with. I have some understanding of random variables and their distributions, but this particular ...
prob1 yuma's user avatar
1 vote
1 answer
32 views

Limit of expectation of product of random variables - using conditional expectation

I'm working on this probability problem and am quite stuck on how to approach the following proof: $ X_n, n = 1, 2, ... \text{and } Z \text{ are random variables defined on} (\Omega, \mathcal{F}, P)$ ...
Chemistryman's user avatar
1 vote
3 answers
96 views

Calculating Density Function and Conditional Expectation of Independent Exponential Random Variables

I am currently studying probability and statistics and I came across a problem involving exponential random variables that I'm having trouble with. I have some understanding of exponential ...
prob1 yuma's user avatar
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7 views

Generating Random Numbers using Rising and Falling Exponential Function

I am working with this normalized pdf, $$ f(t) = (\tau_d - \tau_r)^{-1}(-e^{-t/\tau_r} + e^{-t/\tau_d}),~~t \geq0 $$ And $\tau_d > \tau_r$. Are there any methods for generating random values $t \...
abnowack's user avatar
  • 155
5 votes
2 answers
329 views

Sum of iid random variables

Iid random variables $Y_1, Y_2, \dots, Y_{50}$ take only the values $0$, $1$ and $2$ with probabilities $\mathbb{P}(Y_i = 0) = \mathbb{P}(Y_i = 1) = \frac{4}{9}, \mathbb{P}(Y_i = 2) = \frac{1}{9}$. ...
ABlack's user avatar
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1 answer
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If X and Y are unbounded rvs that have the same moment generating function do they have the same distribution

$X$ and $Y$ are unbounded random variables, if $X^n$ and $Y^n$ are integrable for all n, and $E(X^n) = E(Y^n)$ for all $n\ge0$ must $X$ and $Y$ have the same distribution? My intuition says no. I’ve ...
edster101's user avatar
0 votes
1 answer
42 views

Using the same random variable multiple times in an expression

If $X \sim \mathcal{N}(0,1)$ and $Y \sim \mathcal{N}(0,1)$ with $X$ and $Y$ independent, we would say that $(X+Y) \sim \mathcal{N}(0, 2)$ and $2X \sim \mathcal{N}(0, 4)$. But would $(X+X) \sim \...
Vityou's user avatar
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If $X(t)$ is Ergodic, what about $X^2(t)$?

Given: ${X(t)}$ is W.S.S, Gauss with expected value $=0$, which has $R_{XX}(\tau)$ So $C_{XX}(\tau)=R_{XX}(\tau)$ $\int_0^{\infty}R_{XX}\left(\tau\right)d\tau<\infty$, since $X(t)$ is Ergodic. ...
Analysis_Complex_Study's user avatar
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1 answer
25 views

discrete random variable probability question

I have homework in probability and hope it's not against the rules to post them here, I read them and didn't find anything about HW questions so sorry if it is. I'll describe the question first and ...
Ellie's user avatar
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12 votes
3 answers
3k views

How many rolls are sufficient to ensure, with probability 99%, that the sum is greater than 100?

I roll a pair of fair dice $n$ times, and calculate the sum of all $2n$ faces which come up: Suppose each roll of each die is independent of other rolls. How many rolls are sufficient to ensure, with ...
yanruijie's user avatar
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4 votes
3 answers
67 views

Independent random variables with $X^2 + Y^2 =1$

Does there exists independent non-constant random variables with $X^2 + Y^2 =1$? I think not because intuitively if there is a relation between them it must mean they are dependent but I can't think ...
Invincible's user avatar
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7 votes
1 answer
406 views

What probability distribution function is this?

This is sort of a followup to this question (I'll mention everything relevant in this post though so no need to click link). Main Question: I was trying to study a random variable $Y$. I will ...
hamburglar's user avatar
4 votes
1 answer
49 views

What does "behaving like independent random variables" mean?

The following is from Gonek's paper: The situation is much more interesting if the Riemann hypothesis holds. In that case we may rewrite (6) as $$ \sum_{0 < γ \le T} x^{iγ} « Tx^{-1/2+ Ɛ} + x^{1/2+...
Ali's user avatar
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0 answers
46 views

Uniform distribution problem involving two buses

I am trying to solve the following problem: A bus on line A arrives at a bus station every 4 minutes and a bus on line B every 6 minutes. The time interval between an arrival of a bus for line A and a ...
Camilo Diaz's user avatar
2 votes
1 answer
79 views

Understanding the use of Indicator function

in my research work I came across following expression: $A = \mathbb{P}[X\geq \tau_s, D\leq \frac{c}{2}(T_c-\frac{cT_c}{2B_s}), Y\geq \tau_c]$---(1) where $\mathbb{P}$ denotes probability, $X,Y$ are ...
Heretolearn's user avatar
3 votes
1 answer
46 views

An upperbound for the variance of mean estimator in higher dimensions

I am trying to learn from the paper [paper] http://proceedings.mlr.press/v139/karimireddy21a/karimireddy21a.pdf [proofs] http://proceedings.mlr.press/v139/karimireddy21a/karimireddy21a-supp.pdf , ...
Neustart's user avatar
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1 vote
0 answers
21 views

GLMs where response variable is calculated from multiple data points in a time series?

Hopefully this isn't too broad of a topic: I'm a student research assistant working with matched gene expression counts data in a time series. As a simplified example, say I have two sets of time ...
lineardepression's user avatar
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0 answers
21 views

Probability of i.i.d. uninform random variables assuming a particular ordering.

I'm doing the following exercise. Let $X_0, X_1, X_2, . . . , X_k$ be uniformly and independently distributed on [0, 1]. Declare that the $j^\text{th}$ event succeeds if and only if $X_j < X_0$. ...
fresh_start's user avatar
2 votes
0 answers
23 views

Convergence in distribution of random variables with symmetric distribution

I found the following question in the probability course materials: let $ (X_n)_{n\geq 1} $ be i.i.d. random variables with symmetric distributions such that $\sigma^2=\mathbb{E}[X_1^2]<\infty$. ...
sgvozdic's user avatar
1 vote
0 answers
68 views

Proof of Polynomial Behavior in Sequence of Random Variables

Problem statement: Let $X_0, \xi_{i, j}, \epsilon_k$ (i, j, k ∈ N) be independent, non-negative integer random variables. Suppose that $\xi_{i, j}$ (i, j ∈ N) have the same distribution, $\epsilon_k$ (...
Martin.s's user avatar
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0 votes
1 answer
68 views

Distribution of difference of two random variables

The problem is following: Suppose we have a line segment with a length $L$ and we randomly choose (with uniform distribution) $n$ points on this segment, so we divide the main line segment into $n+1$ ...
XaveryXavier's user avatar
1 vote
1 answer
38 views

Calculate expected value of a function of IID samples

Suppose $X$ is a random variable and $E[X] = \mu$. We define random variable $T$ that for every IID sample $S = \{x_1,..., x_n\}$, then $T(S) = \frac{1 + \sum\limits_1^n x_i}{n}$. Although it is ...
hasanghaforian's user avatar
0 votes
2 answers
32 views

Expectation and variance of Y

Let $X$ be an exponential random variable with mean $1$. Let $Y$ be a uniform random variable over the interval $(0,X)$. We were asked to calculate the mean and variance of $Y$. The mean of $Y$ is ...
Tan Yong Boon's user avatar
1 vote
1 answer
48 views

Related to Laplace Transform and Expectation operator

in my research work I came across the following expression: $P_1 = \mathbb{E}_{I}\bigl[\exp(-\tau\cdot f\cdot I)\bigr]$---(1) where $f,\tau$ are constant and $I$ is random variable. $P_2 = \mathcal{L}...
Heretolearn's user avatar
1 vote
1 answer
32 views

Normally distributed R.V. depends only on numeric parameters.

Random variables $X$ and $Y$ are independent and have normal distribution with $0$ mean and $1$ variance. prove that distribution of r.v. $Z = (X+a)^2 + (Y+b)^2$ depends only from $r=\sqrt{a^2+b^2}$, ...
Jane Doe's user avatar
  • 117
0 votes
0 answers
11 views

Related to integration transformation from Cartesian plane to Polar coordinates

I am working in area of wireless communication that involves extensive use of probability, random variables and stochastic geometry. My system model is as follows: It consists of a main base station (...
Heretolearn's user avatar
2 votes
0 answers
23 views

Given $C \sim \text{Cauchy}$, use Box-Muller to demonstrate that $C - 1/C \sim 2C$

Question: the Cauchy distribution is defined as the ratio of two iid random variables with a normal distribution. Letting $C \sim \text{Cauchy}$, use the Box-Muller representation of normal variables ...
akm's user avatar
  • 394
0 votes
1 answer
37 views

Finding the joint distribution of two dependent variables $X_1$ and $X_2=(X_1)^2$

Let $X_1$ be a uniform RV along the interval $[-a;a]$ and define $X_2=X_1^2$. We can obviously see that $X_1$ and $X_2$ are dependent, upon calculating their covariance we find that it is equal to $0$,...
aza's user avatar
  • 11
0 votes
0 answers
10 views

what is a sufficient condition for the minimum value to be strictly positive with strictly positive probability?

$Q(a,b|X)=(g_1(X)+g_2(X)+1+g_3(X)-ah(X)-b)^2$, suppose $X$ is a continuous random variable with support $\mathcal{X}$, and I want $\underset{a,b}{min} Q(a,b|X)>0$ with strictly positive probability....
ExcitedSnail's user avatar
0 votes
0 answers
30 views

Does this identity relating to Brier scores $B=E[S]-E[S^2]$ relate to any standard quantity or parameter in probability?

Consider a binary random variable $Y$, and another random variable $S$ which represents a "score", such that $P(Y=1|S=s)=s$ (call this the "calibration" assumption). I'm thinking ...
crf's user avatar
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-2 votes
0 answers
31 views

Can we find the solution of $3^m - 2^n = 1$ [duplicate]

For the equation, $3^m - 2^n = 1$ is there any way to find the possible solution. What I feel is that there are two variables in the equation so we must have at least two equations to get the solution....
Shinnaaan's user avatar
1 vote
1 answer
57 views

Almost surely convergence of Bernoulli distribution ($\frac{1}{n}$)

So I have the following exercise: Let $X_n$ independent Bernoulli with parameter $\frac{1}{n}$, so $$ P(X_n=1)=\dfrac{1}{n}, \quad P(X_n=0)=1-\dfrac{1}{n}.$$ Show that $X_n$ converges to $1$ almost ...
Elemer Kit's user avatar
0 votes
2 answers
100 views

Show $E[Y | E[Y | X]] = E[Y | X]$. [closed]

Given Let $X$, $Y$ be random variables on a common probability space $(\Omega, \mathcal{A}, P)$. To show: $E[Y | E[Y | X]] = E[Y | X]$ For this problem, I'm unsure how to rewrite the left-hand side of ...
clementine1001's user avatar
2 votes
1 answer
42 views

Show: $\text{Var}(Y) = \text{Var}(Y - E[Y | X]) + \text{Var}(E[Y | X])$.

Given: Let $X$, $Y$ be random variables on a common probability space $(\Omega, \mathcal{A}, P)$. To prove: $\text{Var}(Y) = \text{Var}(Y - E[Y | X]) + \text{Var}(E[Y | X])$. Attempt: $\text{Var}(Y) =...
clementine1001's user avatar
3 votes
1 answer
50 views

Expected rank of a random binary matrix with Bernoulli probability p?

Let $M \in \mathbb{R}^{m,n}$. The entries are in {$0, 1$} (with the value $1$ having probability $p$, and the value $0$ having probability $(1-p)$). What is the expected rank of $M$? Follow-up: how ...
user3667125's user avatar
0 votes
0 answers
27 views

Can a Student's t-test be used to identify whether or not a single observed data point should be judged as an outlier from a null distribution?

FYI: This question may be more thematically appropriate for Cross Validated's site, but, from previous experience, there is considerably less traffic there compared to Math, and I still feel that this ...
S.C.'s user avatar
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