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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Relationship between two randomly-generated variables

Using stata, I generate two random variables and regress them with each other. ...
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uniform distribution, showing that two variables are uncorrelated and not independent

Let (X,Y) have uniform distribution on the four points(0,1),(0,βˆ’1),(1,0),(βˆ’1,0). How can I show that X and Y are uncorrelated but not independent? Could someone just point me in the right direction ...
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Need help in probability [closed]

Two machines M1, M2 are used to manufacture resistors with a design specification of 1000 ohm with 10% tolerance. Resistors of M1 are found to follow normal distribution with mean 1050 ohm and ...
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How to decompose a CDF into discrete and continuous parts?

I understand that any C.D.F may be represented in the form $$F(x) = p_1F^d(x) + p_2F^c(x)\,,$$ where $F^d(x)$ represents discrete c.d.f , $F^c(x)$ represents continuous c.d.f and $p_1+ p_2=1$. ...
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moment generating function M(t) of a discrete random variable [closed]

Let the moment generating function M(t) of a discrete random variable X be given by (5 marks) 𝑀(𝑑) =1\6+1\3cosh 3𝑑 + π‘˜ cosh 5𝑑. i) Find the probability mass function of X. ii) Find the mean ...
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Distance in Square between Randomly Selected [duplicate]

I am trying to find the expected Euclidean distance between independent, randomly-selected variables in the unit square and I have some technical questions. For co text, I know if we were selecting ...
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Assume $X,Y$ are two independent random variables. Let $Z=f(X,Y)$. If $Z$ is independent of $X$, $f(X,Y)$ is constant in $X$. Is this true?

Let $X\in \mathbb{R}^n$ and $Y\in \mathbb{R}^m$ be two independent random vectors. Then, say that we have a third real valued random variable $Z=f(X,Y)$, with $f$ being measurable. Say that we know ...
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generating a uniform random variable from the first digit of an exponential random variable?

in "introduction to probability models", Ross talks about simulating with the rejection method, and he needs an exponential random variable, and a uniform random variable (used only for checking ...
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sample space for a feature in machine learning [duplicate]

In machine learning data set each feature is considered as a random variable. Random variable is a function which maps the outcomes in sample space to a real value. Now I am trying to understand since ...
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Does the covariance of i.i.d. random vectors/multivariate random variables have any zero terms?

If we have i.i.d. random variables, $X$ and $Y$, then $\text{Cov}(X,Y)=0$. But let's say we have i.i.d. random vectors $\boldsymbol{X}$ and $\boldsymbol{Y}$, where $\boldsymbol{X}=(X_{1},...,X_{p})$ ...
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Truncated CEF of normally distributed RV. Is sample analogue a consistent estimator of the 'population' truncated CEF?

If I have a random variable that is normally distributed, and truncated such that I only see $y$ if $y\geq 0$, and I want to do some calculations with the truncated Conditional Expectation Function in ...
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Expectation of A given A,B Random variables

So I am self-teaching myself some stuff about random variables and expectations for a course that I am going to take in the upcoming semester, and I found some resources online for properties that I ...
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If x is correlated with y, is there formula to use that alone to write x as a function of y, or to relate the two variables?

for example, if $\text{corr}(x,y)\not=0$, can you always decompose $x$ to be: $x={\text{cov}(x,y)/\text{var}(x)} + \epsilon$ where $\epsilon$ is just all other remaining parts of $x$ uncorrelated with ...
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Expectation of any function of X, a nonnegative integer valued random variable [duplicate]

How to show that if X is a nonnegative integervalued random variable with distribution F,then $$E(X)=\displaystyle\int_0^\infty \overline{F}(X)dx$$ and $$E(X^n)=\displaystyle\int_0^\infty n*X^{n-1}\...
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How to model ratings on a single scale, while also incorporating observations from other scales?

I have 40 words that were each rated on 25 different scales. I want to know if properties of the 40 words predict their ratings on any of the scales. I have chosen to analyze this at the level of ...
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Estimation of first moments of the sum of two hidden gaussian distributed random variable

I am stuck with a problem at work. I hope you guys can help me. I observe a random variable $Y$ and I know it is the sum of two gaussian distributed random variable $X_1$, $X_2$. But I can't observe ...
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Expectation of (sum subtract the expectation of sum)

Let's say we have random variables $\mathbf{X}$, and we have $P(\mathbf{X}\in [a, b])=1$, we have $\mathbf{S}_n = \mathbf{X}_1 + \mathbf{X}_2, +\dots + \mathbf{X}_n$. If $\mathbf{X}_1, \mathbf{X}_2, ...
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Expectation of division of random variables [closed]

Is this true? E(Xn/Yn) goes to E(Xn)/E(Yn) in probability even if Xn and Yn are not independent?
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Transformation of a random variable with a gamma distribution

Suppose $X_i \stackrel{i.i.d}{\sim}$ Exp$(1/\theta)$ which implies $\sum_{i =1}^{n} X_i \sim$ Gamma $(n, 1/\theta)$. But, then, the book that I am reading says that $(2/\theta)\sum_{i =1}^{n} X_i \...
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Is the independence of this sequence of random variables not implicitly given when we define their probability distributions?

In this post, the user asks whether the following random variable converges to $0$ almost surely: $X_n = \begin{cases} 0, & \text{with probability 1 - $\frac{1}{2n}$,} \\ n, & \text{with ...
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Distribution of the second minimum of a set of random variables

Given $n$ i.i.d. random variables $X_1,...X_n$, what is the distribution of the second smallest value ? I know from this question that CDF of the minimum value is $1 - (1-F(x))^n$ where $F(x)$ is the ...
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Distributions over sequences of non-independent Bernoulli random variables

I'm modelling the generation of some sequences using random variables for generation of their components. For instance, the probability of generating the sequence $ABC$ should depend entirely on $P(A) ...
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Calibration of correlation

Let's say there are two random variables (for example, two time series data of S&P500 and a stock) and their correlation is 0.95. What is the best way to reduce the correlation between these two ...
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Variance of the autocovariance function at a particular lag

For a uniform random number(u) having mean $\mu$, the autocovariance at a lag $\tau$ is given by $$C(\tau)=\frac{1}{N-\tau} \sum_{i=0}^{n-\tau} (u_i-\mu)(u_{i+\tau}-\mu)$$ For the uniformly ...
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Cross validation on unbalanced datasets using a simulation approach on a subset of data

Within my field I often end up using linear regression to look at two variables, normally how some factor (e.g. size) changes through time. I'm increasingly coming across datasets that are unbalanced ...
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If a discrete random variable is independent to other two discrete random variable, is it independent to the sum?

Suppose X,Y,Z are discrete random variables and X is independent to both Y and Z. Is X independent to Y + Z? I know this is not necessarily true but I am struggling to find a counterexample. It ...
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A variance measure for a variable that takes both positive and negative values with the mean close to zero

I've been using relative standard deviation as a measure of variance for my variables, but some of them take both positive and negative values and their mean is close to zero, so it's not a good ...
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Let A and B be two random variables, both independent from another random variable C. Is A*B also independent from C?

Let A and B be two random variables both independent from another random variable C. If A is independent from B, is A*B also independent from C? And if A and B are no independent from each other?
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Which distributions preserve its family after linear transformation?

For example, if $X$ obeys a multivariate gaussian distribution and let $Y = AX + B$ where $A$ is the matrix for the transformation and $B$ a constant vector. Then $Y$ still obeys a gaussian ...
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What is the covariance of the product of two independent random vectors with known covariances? [duplicate]

Suppose I have two independent random vectors, $X$ and $Y$, both of size $n\times 1$. We have that $\mathbb{E}[X] = \mu_X$ and $\mathbb{E}[Y] = \mu_Y$. We also have that $\mathrm{Cov}(X) = \Sigma_X$ ...
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Omit a random effect when its variance is >0 but <1?

On two threads, here and here, it suggests that random effects in GLMMs can be omitted if their variance approximates 0. This raises the question, below which level of variance can you reasonably omit ...
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What is the name of the property for two random variables having disjoint supports?

Let $X$ and $Y$ be two real-valued random variables. There is a specific name for the property when the support for $X$ is disjoint from the support for $Y$. I can't remember it, and I can't seem to ...
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Mutual information between $X$ and $f(X)$

Let $X$ and $Y$ be two random variables, where $Y=f(X)$ is a deterministic function of $X$. Furthermore suppose $X,Y$ are continuous and that $f$ is smooth. Is the mutual information between $X$ and $...
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How to estimate the distribution of a continuous random variable to find its expected value?

I have a list of scores obtained from a piece of analysis, and I would like to find its expected value. My understanding is that, because the scores can take on any value as opposed to an integer, ...
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What does $cov(x_1,x_2) >> 0, cov(y_1, y_2) >> 0$ and $cov(x_1+y_1, x_2+y_2) = 0$ tell us about $x_1, x_2, y_1, y_2$?

I was posed this problem where we know: $$ cov(x_1,x_2) >> 0 \\ cov(y_1, y_2) >> 0 \\ cov(x_1+y_1, x_2+y_2) = 0 \\ $$ What does this tell us about the structure of $x_1, x_2, y_1, y_2$? ...
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Why do we use the term “uncorrelated” to describe linear dependence/independence?

Terminologically, "uncorrelated" to me means that 2 things have no relationship, not necessarily constrained to linear relationships. However, in statistics, we seem to confine "uncorrelated" to mean ...
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Random variable is parameter for distribution of another random variable

What would you do to find probability when a uniform random variable is the parameter for the distribution of another uniform random variable. ie: $Z \sim Unif(0,1)$ $Y \sim Unif(0,Z)$ And we are ...
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Probability - Find $P(X_n \ge max(X_1, X_2, \dots, X_{n-1}))$ [duplicate]

We have a set $X_1, X_2$, . . . , $X_n$Β iid standard normal random variables. How would I go about finding $P(X_n \ge max(X_1, X_2, \dots , X_{n-1}))$
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Individual identity as random effect in survival GLMM?

I am currently working on a population, monitored between 2016 and 2019, for which I need to see if my biometric variables influence survival. In my population, I only know if each individual survived ...
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Random Forest Notation Confusion

I've been trying to mathematically understand the random forest algorithm but I've been confused with the notation. In these slides (page 17), it is stated: Let $\mathbf{x} = (x_1, ..., x_d)$ be a ...
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What does one mean by a 'random variable' in the real world?

I want to understand what it means for process to be random or for an object to have random properties. Contrary to what I used to believe and to popular belief J Schmidhuber argues that true ...
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Is there any significance to derivatives of the moment generating function about non-zero values?

The $n^{th}$ derivative about zero of the moment generating function gives the expected value of the $n^{th}$ power of the random variable. $E[X^n] = M^n(0)$ What about the derivatives about ...
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1answer
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Expected value of a sum of random variables raised by $e$?

There is a function $y$ defined $$y=\exp(-\boldsymbol{\alpha}'\mathbf{b})\:\:;\:\:\:\:y\in(0,\infty)$$ where $\boldsymbol{\alpha}$ is a vector of random variables and $\mathbf{b}$ is a vector of non-...
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Any other way to calculate range of transformed variable?

I have been doing questions related to transformations of 2d random variables. In a question I have to find a range for $$u = (x-y)/2 $$ and $$v = y $$ where $x,y > 0$ So is it necessary that ...
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Simulating values from a random variable that is a sum of other random variables

$X$ is $\mathcal N(0,4)$, $Y$ is $\mathcal N(0,5)$, $Z = X + Y$ I need to simulate 1000 values for each of these variables, $X$,$Y$,$Z$. I have simulated 1000 values for both $X$ and 1000 values for ...
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How to show that $X_n/n$ approaches a constant as $n \to \infty$ if $X_n \tilde\ \chi_{n-p}^2$

Page 18 here states that if $X_n \sim \chi_{n-p}^2$ with fixed $p$, then $X_n/n$ approaches a constant. How do I show this?
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CDF for the linear combination of p Gumbel random variables

I'd like to know if there is an article or any insights to derive the exact CDF for the linear combination of p Gumbel random variables, as it was shown for the PDF in this article. Let $Z = \...
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What does the assumption: “The independent variable is not random.” in OLS mean?

What does the assumption: "The independent variable is not random." in OLS mean? How can you verify that hypothesis?
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The Mean of a Discrete Random Variable

I understand how one gets the mean of a random discrete random variable, but not necessarily how the process makes logical sense. For instance, how does the multiplication of x by P(X = x) give the ...
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2answers
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Let $A, B$ be random variables. Does $P(A|B)=1-P(\overline{A}|B)$?

Say I have discrete random variables $A$ and $B$. Is it true that $P(A|B)=1-P(\overline{A}|B)$? My intuition is conflicting; it makes sense that if $P(A|B)=p$, then it is only possible that $P(\...

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