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Questions tagged [random-variable]

A random variable or stochastic variable is a value that is subject to chance variation (i.e., randomness in a mathematical sense).

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What is the median of the minimum or maximum of multiple samples?

Suppose I have a variable with a known distribution, and suppose I sample that variable k times and record the minimum. If I repeat this many times, will the median of the minimum converge to a ...
bridget's user avatar
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17 views

Singular values of $A^{-1}$ where $A$ has standard normal entries

Is anything known about behavior of singular values of $A^{-1}$ where $A$ has standard normal entries? Empirically, $k$th singular value appears to decay as $1/k$, is this well-known? colab
Yaroslav Bulatov's user avatar
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41 views

Distribution of a product of random variables

I have two independent distributions $X$ and $Y$. $X$ is defined by the piecewise CDF $$F_X(x) = \begin{cases} F_X^1(x) & x \in (-\infty, a_1)\\ F_X^2(x) & x \in [a_1, a_2)\\ F_X^3(x) & x \...
rkim's user avatar
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0 votes
1 answer
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Are all random variables estimators? [duplicate]

My hand-wavey understanding is a random variable is a function from a domain of possible outcomes in a sample space to a measurable space valued in real numbers. We might denote a random variable from ...
Estimate the estimators's user avatar
2 votes
4 answers
632 views

Confusion about the probability of a continuous random variable at a given point

A person randomly chooses a battery from a store which has 80 batteries of type A and 260 batteries of type B. Battery life of type A and type B batteries are exponentially distributed with average ...
Samar's user avatar
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1 vote
1 answer
38 views

Is the variance of the mean of a set of possibly dependent random variables less than the average of their respective variances? [closed]

Is the variance of the mean of a set of possibly dependent random variables less than or equal to the average of their respective variances? Mathematically, given random variables $X_1, X_2, ..., X_n$ ...
HappyFace's user avatar
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1 answer
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Calculating the joint pdf of linearly dependent random variables $X$ and $Y=X$

Let $X$ and $Y$ be two random variables and $p_{(X,Y)}(x,y)$ be the joint pdf of $(X,Y)$. Suppose that $(X,Y)$ transformed to $(X,X)$. We want to calculate the joint pdf of transformed random ...
Naveen Kumar's user avatar
1 vote
1 answer
46 views

When running a Bayesian mixed effects regression, if a random effect estimate has 95% CIs that include zero, should it be disregarded?

Consider a Bayesian mixed effects regression. I am interested in the correlation between two of the random slopes. However, the 95% CIs for the correlation value include 0. Should I disregard the ...
Dave's user avatar
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1 vote
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I don't think this is conditional dependence, so what is it?

I am looking for the name of the following phenomenon. There are three random variables, $X,Y,Z$. We have $P(X,Y) \neq P(X)P(Y)$ and $P(Y,Z) \neq P(Y)P(Z)$. In other words, $X$ and $Y$ are dependent, ...
Wapiti's user avatar
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2 votes
1 answer
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With $X$ and $Y$ being two independent $\text{Bernoulli(1/3)}$ rvs, show whether $U = |Y-X|,~V = X+Y$ are independent or not

Let $X$ and $Y$ be two independent $\text{Bernoulli(1/3)}$ random variables. Define random variables $U$ and $V$ as $$U = |Y-X|, \hspace{5mm} V = X+Y$$ Are $U$ and $V$ independent? I am new to the ...
Samar's user avatar
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0 answers
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What transformation to adjust empirical data variance 'smoothly'?

Let's assume we have empirical data for an outdated process that generated it. We have reason to believe the process of today is similar (e.g. same distribution) but with a different variance. General ...
yeahman269's user avatar
4 votes
1 answer
51 views

What is the difference between a matrix normal distribution and the multivariate gaussian distribution?

$\newcommand{\vec}{\operatorname{vec}}$Consider a set of $N$ matrices $X_1, X_2, \ldots, X_N$. I want to estimate the distribution of these matrices represented by the mean and covariance. I address ...
Ralff's user avatar
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5 votes
1 answer
328 views

Can a finite decimal number be a discrete variable?

If I have an array {0.1, 0.2, 0.3, 0.4, 0.5}, is this a discrete array? Are the values in it discrete values? (I have this doubt because many materials show that discrete values are usually integers) ...
Cathy's user avatar
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5 votes
1 answer
104 views

$E[(X+Y)^{a}] > E[(X)^{a}]$?

Assume I have two strictly positive i.i.d. random variables, $X$ and $Y$. Under what conditions is the following inequality true? $$E[(X+Y)^{a}] > E[(X)^{a}], \hspace{2mm} a \in (0,1)$$ Should have ...
econ_ugrad's user avatar
0 votes
1 answer
39 views

Goodness of fit test binomial random variables without "compensation"

Let $N_{1}, N_{2}\in\mathbb{N}^{+}$ and random variables $X_{1}\sim B(N_{1},p_{1}),X_{2}\sim B(N_{2},p_{2})$ with null hypothesis $H_{0}:p_{1}=p_{1,0}, p_{2}=p_{2,0}$. I would like to apply a GoF test ...
CorrieElba's user avatar
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0 answers
16 views

Combining Variation and Uncertainty from Replicate measurements

I have 3 measurements from 3 independent experiments {m_1, m_2, m_3}. I have another 3 measurements that are used to scale the m measurements {n_1, n_2, n_3} from the same experiment (different from m)...
mAthletic's user avatar
2 votes
1 answer
120 views

Sum of asymptotically independent random variables - Convergence

Let $\theta_N=\frac{1}{N}\sum_{i=1}^N \pi_i\cdot g_i$ where $0<\pi_i<1$ and $0<g_i<1/\pi_i$ such that $\theta_N\overset{N\rightarrow \infty}{\rightarrow}\theta$. If $X_i\sim Ber(\pi_i)$, I ...
Pierfrancesco Alaimo Di Loro's user avatar
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0 answers
13 views

Does a random variable that includes the summation of independent samples from different distributions obey Central Limit Theorem? [duplicate]

I am learning from the book of statistics by sheldon M ross and it's a great book. However, I landed upon a small query that book failed to address me. According to CLT , sum of random variables when ...
CREATIVITY Unleashed's user avatar
6 votes
4 answers
991 views

Is it incorrect terminology to say "confidence interval of a random variable"?

I have seen claims that "population paramter is not a random variable" when discussing confidence intervals. eg here Be sure to note that the population parameter is not a random variable. ...
Shreyans's user avatar
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Expectation of $u^\top(u+Ax)$, when $A$ and $u$ are nonlinear functions of $x$

Let $x\in\mathbb R^d$, and $s=\operatorname{softmax}(x)$. Let $y$ be a fixed one-hot vector such that $$u = s-y \\ v =(\operatorname{diag}(s) - ss^\top)x$$ I am interested in the inequality $u^\top (u ...
Phoenix's user avatar
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2 votes
1 answer
43 views

Random Variables in the Neural Net diagrams

When I look at this discussion on linear regression: What is a random variable and what isn't in regression models It says: If we have the population regression function: $$Y_i = \beta_0 + \...
a12345's user avatar
  • 65
2 votes
1 answer
162 views

What do these definitions have to do with random variables?

(This is a full rewrite of the original question I asked. The original was very poorly phrased - I'm hoping this rewrite gets at the central question more directly.) (Take two of the rewrite because I ...
guest131071's user avatar
0 votes
0 answers
46 views

Expectation of the product of two random variables

I recently tried to derive a formula that I saw in a paper. The scenario was a follows: Let $X\in\lbrace 0,1\rbrace $ a.s. be a binary random variable and $Y$ be a continuous random variable. Let $a,b\...
stats19's user avatar
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0 answers
28 views

In the Law of the Unconscious Statistician, is $p(Y=y_1) = p(X\in g^{-1}(y_1))$ a convention or can it be derived? [duplicate]

There's something I am not really sure about the Law of the Unconscious Statistician. Let $X$ be a random variable and $Y=g(X)$ be another one. When we fix $y$, say, $y=4$, we have that $p(Y=4)=p(X=4)+...
niobium's user avatar
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1 vote
0 answers
33 views

Finding a Pivotal quantity

0 I am working on this problem for class, where the setup is the following: Let X be a single observation from the beta(θ,1) pdf. (b) Find a pivotal quantity and use it to set up a CI having the same ...
Harry Lofi's user avatar
1 vote
1 answer
38 views

Transformation of a Random Variable

I am working on this problem for class, where the setup is the following: Let X be a single observation from the $beta(\theta,1)$ pdf. (a) Let $Y=-(logX)^{-1}$. Evaluate the confidence coefficient of ...
Harry Lofi's user avatar
0 votes
0 answers
22 views

Finding the set for random variable transformations

I'm reading through the book "All of Statistics", and in section 2.12, regarding Transformations of Several Random Variables, the author lists three steps for finding the transformation. I ...
David Morton's user avatar
2 votes
1 answer
132 views

Expected value of the product of three random variables

For two dependent random variables we have: $$Cov[X, Y] = E[XY] - E[X]E[Y]$$ So that $E[XY] = E[X]E[Y] + Cov[X, Y]$ In case of three arbitrarily correlated random variables $(X, Y, Z)$, is it possible ...
Stefano Lombardi's user avatar
2 votes
0 answers
62 views

Decision theory when distributions don't have a first moment? [closed]

Lets say that we are presented with two gambling opportunities and would like to decide between them in a decision-theoretic framework. For gamble 1, the cost is $1$ and the payoff is $X_1$ where $X_1 ...
QMath's user avatar
  • 451
1 vote
1 answer
72 views

For the binomial distribution, is the proportion of successes distributed the same as the number of successes?

Given a known population proportion and an IID population, the number of successes given n trials should be: $$ X \sim B(n, p) $$ It seems intuitive that the probability of observing a certain ...
ZPears's user avatar
  • 11
1 vote
1 answer
47 views

Comparing sequence of unfair coin flips with different predicted probabilities

Suppose we have $n$ independent coins, each of which has an unknown and different probability of coming up heads. We have a magical machine that guesses the probabilities of each coin coming up heads. ...
differentiableapple's user avatar
2 votes
1 answer
28 views

Independence of real and imaginary part of the product of two independent normal variables

Let $X_1,X_2,Y_1,Y_2$ be iid standard normal variables $N(0,1).$ Let $X=X_1+iX_2,$ $Y=Y_1+iY_2$ and $Z=XY.$ We have : $Z=(X_1Y_1 - X_2Y_2) + i(X_1Y_2 + X_2Y_1).$ From https://en.wikipedia.org/wiki/...
fbrx's user avatar
  • 23
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0 answers
9 views

Sampling: when is multivariate random variable interpretation dissimilar to repeated realization of single random variable interpretation

After reading many different posts on this site regarding the relationship between random variables and samples, I still have one lingering question (my apologies if I've missed any post explicitly ...
S.C.'s user avatar
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0 votes
0 answers
21 views

Expected standard errors depend on the underlying distribution in regression

I am investigating the effect of distributions of predictive variables ($x$) on their standard errors in regression. I thus programmed a little simulation to see how the standard errors behave. I used ...
POC's user avatar
  • 668
0 votes
1 answer
181 views

What is the variance of convolution of two random variables?

Consider two random variables $Z$ and $W$. Given the variances of $Z$ and $W$, how can we compute the variance of their convolution $Z \circledast W $? As an example, please consider the case of noise ...
user409495's user avatar
0 votes
0 answers
43 views

How to calculate the expectancy of the ratio of non-independent random variables?

How can I calculate this expectancy: $$ E \left [ \frac{\sum_{t=1}^T{Z_tX_t}}{\sum_{t=1}^T{Z_t^2}} \right ] $$ where $Z_t \sim N(0,1)$ and $X_t \sim N(0,1)$ are independent? Any tricks? Is it ...
PaulG's user avatar
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1 vote
1 answer
80 views

properties of a expectation for a non-negative random variable

Say I have a non-negative discrete random variable $X$ (values of $X$ can be mapped to integers $(0, 2^n -1)$ for $n \in \mathbb{Z}$) and an associated distribution $P(X)$. Given a non-negative scalar ...
Manas Sajjan's user avatar
1 vote
0 answers
8 views

Undirected graphs and implications of independence (Wasserman chapter 18)

In Wasserman's All of Statistics chapter 18, he defines the following undirected graph: Let $V$ be a set of random variables with distribution $\mathbb{P}$. Construct a graph with one vertex for each ...
NovicePatience's user avatar
1 vote
1 answer
23 views

randomisation issue/crisis

I have randomized into three groups using a randomization aid and groups were examined at 2 time points. however in the analysis found that there was a significant difference at baseline in one of the ...
ketchup28's user avatar
0 votes
0 answers
34 views

How to find the variance for $\frac{\sum Y}{\sum X}$ when X and Y are both independent and normal random variables

The original problem is: Given $Y_i = \beta X_i + \epsilon_i$, $i=1,2,...,n$, where $X \sim N(\mu, \tau^2)$ iid and $\epsilon \sim N(0, \sigma^2)$ iid, $X$ and $\epsilon$ are independent. What is the ...
Kevin's user avatar
  • 1
1 vote
1 answer
132 views

Setting random effect and nested random effect correctly in glmmTMB model

I have been having trouble fitting my data of seedsets of flowers. I have gathered data from seven flowering species in four different elevations. Not all the species appear in every elevation and the ...
Dominik Anyz's user avatar
9 votes
2 answers
626 views

Is the realization of random variable also a random variable?

In class, a teacher told me that the realization of a random variable is also a random variable. For example, if I take the a sample mean, and that mean results in the value 35, then 35 is also a ...
Santiago Valdivieso's user avatar
1 vote
0 answers
41 views

Finding probability involving dependent random variables [closed]

Suppose train on line A arrives in time uniformly distributed between 0 and 4mins, train on line B arrives in time uniformly distributed between 0 and 6 mins, and furthermore the time interval between ...
Harsh's user avatar
  • 111
1 vote
0 answers
64 views

Ratio of cubic and quadratic form in random variables is approximately normal?

Let be $x_{1},x_{2},x_{3}$ i.i.d. random variables following a normal distribution with $\mu=0$ and $\sigma=1$. I'm intrigued by the following random variable, which is a ratio of a cubic form and a ...
rgvalenciaalbornoz's user avatar
2 votes
0 answers
52 views

Distribution IID uniform variables given their ranking [duplicate]

Description Let $N\in\mathbb{N}^{+}$ and $X_{n}\stackrel{IID}{\sim}U(0,1)$ for $n\in\{1,...,N\}$. Given $X_{1}\leq X_{2}\leq X_{3}\leq...\leq X_{N}$, I would like to understand $f_{X_{n}}$ by writing ...
CorrieElba's user avatar
0 votes
0 answers
17 views

Parameter estimation of a WSS process

As my research revolves around parameter estimation from signals that evolve in time in a random fashion, I am curious to know what features/ retrievals people normally use to determine the parameters ...
CfourPiO's user avatar
  • 235
3 votes
2 answers
204 views

Calculating $E[(\sum X_i)^4]$

Trying to figure where I'm going wrong with the following. My goal is to calculate var$(\bar X_n^2)$ using $E[(\bar X_n)^4]=\frac{1}{n^4}E[(\sum X_i)^4]$ given that $X_1,...X_n$ are iid with $EX_1=\mu,...
reyna's user avatar
  • 385
2 votes
1 answer
105 views

If $X \sim\textrm{ Bin}(100, 0.5), $ then what is the approximate distribution of $(X/5 - 10)^2? $

I am not able to solve this using transformation since binomial does not have a CDF. The question has 4 options, so I tried calculating the expectation of this and then comparing it to the ...
Anweshan Goswami's user avatar
0 votes
1 answer
39 views

How does a Random Sample relate to Random Processes and Random Variables?

What is the difference between a Random Sample, Random Variable (RV) and Random Process (RP)? As far as I know, a RV is a mapping from an experimental space to the real numbers and a RP is a mapping ...
Ahsan Yousaf's user avatar
0 votes
0 answers
22 views

Seeking Lower Bound for Partition Probability in Random Variable Analysis

I am reaching out to seek assistance with a probability problem involving random variables. For each $p$ in $[1,\infty)$, consider positive random variables $X_{1,p}, X_{2,p}, \ldots, X_{n,p}$ such ...
Diego Fonseca's user avatar

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