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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167 views

Generating random variates from the following pdf

I'm working through some example of probability distributions and I'm struggling to derive the formula for the following pdf $f(x) = \frac{1}{0.02}e^{-\left\lvert x \right\rvert/0.01}$ My undersanding ...
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Binomial distribution index

I stumbled against this problem and found it really hard and help would be much appreciated Let $X_{1},X_{2},......,X_{n}$ be a series of independent Bernoulli variables with $P(X_{i}=1)=\theta$ and $...
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12 views

Standardized PDF of random variable U

If we have a random variable $U$ with a probability density function $f(v)$, and a standardized random variable $\hat{U}$, such that: $$\hat{U} = \frac{U - \langle U \rangle}{\sigma_u}$$ Where $\...
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0answers
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Problem calculating expectation using law of total expectation

I'm confusing myself with conditional expectation and could really use your help! I am trying to calculate an expectation that arises in the context of doing variational inference. However, the ...
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1answer
24 views

Independence of variables between X and Y^X

If X and Y are independent, then are X and Y^X independent? Does the realisation of X have to be the same as the X in the power of Y? I think this question sounds silly but I'm trying to clear a major ...
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20 views

Mutual information between two functions of a random variable?

Let $X$ be a random variable with density $P(X)$. Let $f(x), g(x)$ be two deterministic functions. How can one compute the mutual information, $$I(f(X), g(X))=\int P(x)\ln \frac{P(f(x),g(y))}{P(f(x))P(...
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3answers
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Is heights of humans actually a discrete random variable? [duplicate]

Suppose the human population consisted of $N = 3$ people, each with a specific height. Let $X^N$ be the random variable representing the heights of this population of $N$ people. Since $X^N$ can only ...
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22 views

How to isolate $m$ from the probability formula $\Pr( |x_1 -x_2| > j\cdot m)$? [closed]

How to isolate $m$ from a probability? Is it possible? Central limit theorem? $r = \Pr( |x_1 -x_2| > j\cdot m)$
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1answer
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If $X,Y \sim D$, is it possible to find the joint pdf of $X,Y$ without knowing whether $X,Y$ are dependent?

I am learning probability and have a (probably dumb) question. It's not a homework question, just something that confuses me. If someone tells me that $X,Y\sim D$ where $D$ is some known distribution (...
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+50

Hypothesis test for discrete random vector with few samples

Consider a random vector $X \in \mathbb{R}^{d}$ with support $\text{supp}(X) = \{1,2,3,4\}^d$, and let $P_X$ denote its known probability mass function. Note that $\lvert \text{supp}(X) \rvert = 4^d$. ...
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Distribution of the combination of two Chi-squared distributions [duplicate]

The random variable $A$ has a $\chi^2$ distribution with $p$ degrees of freedom. The random variable $B$ is independent of $A$ and has a $\chi^2$ distribution with $q$ degrees of freedom. Show that $(...
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Lots of modes of convergence for random variables but why is there nothing on convergence of probability density functions?

There are numerous modes of convergence for random variables. But why do I never read anything about convergence of probability density functions? It seems like this would also be an important notion ...
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1answer
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PMF of $aX_1 + bX_2$ (Bernoulli)

Let $Y_1 = aX_1 \sim \text{Bernoulli}(p)$ and $Y_2 = bX_2 \sim \text{Bernoulli}(p)$, what is the PMF of $Z = Y_1 + Y_2$ for $a > 0$, $b > 0$ and $a \neq b$? Can somebody check my result? $$p_{...
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2answers
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Need help understanding how only variable A can be correlated to the absolute value of A-B

I'm currently working with the dataset of a study I'm conducting. The data is comprised of serially drawn samples from patients where we've measured the cell counts of those samples and compared them ...
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Implications of continuity of a distribution function

Consider a random vector $(X,Y)$ with distribution function $F:\mathbb{R}^2\rightarrow [0,1]$ and let $\mu_F$ be the associated measure. Take any $(a,b)\in \mathbb{R}^2$ and consider the set $$ \...
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1answer
29 views

Continuous distribution when there are flat regions

Consider a distribution function $F:\mathbb{R}\rightarrow [0,1]$ definining the positive, finite measures $\mu_F$ determined by $$ \mu_F((a,b])\equiv F(b)-F(a) $$ for each $a,b\in \mathbb{R}$ with $b&...
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Bounding the norm of the difference between two related probability densities

Suppose we have a continuous random variable $X$ and two continuous functions $f$ and $g$ such that $f(X)$ and $g(X)$ are continuous random variables. Let $p_A$ be the probability density function of ...
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1answer
26 views

the law of total probability with extra variables

Suppose $X$ and $Y$ are two discrete random variables. The law of total probability states that: $$ p(x) = \sum\limits_y {p(x,y) = } \sum\limits_y {p(x|y)p(y)} $$ Now suppose we have another random ...
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2answers
556 views

If all the marginal distributions are continuous, then the joint distribution is continuous?

Consider a random vector $X\equiv (X_1,...,X_L)$. Assume that each $X_l$ is continuously distributed with support $\mathbb{R}$, for $l=1,...,L$. Does this imply that also $X$ should be continuously ...
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Existence of a vector with desired distribution

Suppose that there exists a random vector $\eta\equiv (\eta_1, \eta_2, \eta_3)$ continuously distributed on $\mathbb{R}^3$ and with full support. Can we always find a vector $\epsilon\equiv (\...
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1answer
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Help needed with Repeated Measures ANOVA - how to correctly specify nested effect?

I need some advice regarding nested effects. My data set has the following variables: HeatingRate = dependent variable Individual (n = 4) = fixed Temperature gradient (n = 3) = fixed Trial (n = 3 per ...
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1answer
14 views

Showing that a discrete random variable has the same moments as a Normal Distribution

Suppose I define $X$ to be normally distributed with $\mu = 0, \sigma^2 = 1$, so that $X$ has the pdf $f_{X}(x) = \frac{1}{\sqrt{2 \pi}} e^{-x^2 / 2}, \quad -\infty < x <\infty.$ Let discrete ...
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0answers
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Approximating mean/covariance of truncated/folded/censored normal distribution

Given a normally distributed $X$, what is the best way to approximate the covariance matrix and mean vector of $\tilde{X} = \max(0, X)$? I am interested in the censored distribution, but the truncated ...
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0answers
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How to interpret the standard deviation of a discrete random variable

How would you interpret the standard deviation of a discrete random variable and state it simply for someone with limited statistical background? For example, Assume I am betting on an event with a 21%...
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0answers
19 views

Generalization of the Payley-Zigmund inequality

The Payley-Zigmund inequality states that for a positive random variable $Z$ the following holds \begin{equation} \operatorname{P}( Z > \theta\operatorname{E}[Z] ) \ge (1-\theta)^2 \frac{\...
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0answers
63 views

Expectation of a Function of a RV Problem

Problem. Let $X$ be a random variable (either continuous or discrete) that takes nonnegative values. Prove or provide a counterexample to the following statement: There does not exist an X such that $...
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1answer
39 views

A function of random variables $X_1, …, X_k$ that goes from $\mathcal{R}^k$ to the reals is measurable with respect to $\sigma(X_1, …, X_k)$

I'm reading Resnick's "A probability Path" and doing exercise 3 on page 85. The statement is: Suppose $f : \mathcal{R}^k \rightarrow \mathcal{R}$ and $f \in \mathcal{B}(\mathcal{R}^k) / \...
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2answers
223 views

The interpretation of a random variable

A variable is modelled as a random variable, without reference to the question whether it is truly random in reality. For example, when the outcome of a flip coin is modelled as a random variable, no ...
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31 views

Attempt at proving two random variables have the same distribution

Sorry if this has already been asked, I searched and wasn't able to find something similar in the first pages that showed up, most of the questions were more advanced. Given a random variable $X$ that ...
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1answer
20 views

Combined variance estimate for samples of varying sizes

I'm working on my master thesis, and something's come up where I don't know if I'm "allowed" to do this and call it good science, I've scoured the internet to no avail of finding my answer, ...
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0answers
29 views

Joint distribution of two equal random variables

Suppose that X2 is continuously distributed with density g(x2) > 0 on [0, 1). Let X1 = X2. Derive the joint distribution function F (x1, x2) and the marginal density fX1 (x1). I'm not sure about ...
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1answer
25 views

Convergence in probability with double limits

Suppose you have a sequence of random variables $ \left\lbrace X_{i}\right\rbrace_{i=1,...,n}$ which converges in probability to a random variable $X$, shown by $ X_n \ \xrightarrow{p}\ X$ as n goes ...
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1answer
54 views

Multiplying by a constant (T random variable)

Suppose that $T$ has a distribution $t(n-1)$. If we were to multiply $T$ by $\frac{1}{\sqrt{n}}$, what would be the distribution of $\frac{T}{\sqrt{n}}$?
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1answer
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Transfer Learning: data in the source domain and the target domain are required to be independent and identically distributed

In instance-based transfer learning, it is said that data in the source domain and the target domain are required to be independent and identically distributed. When it says that the data "are ...
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1answer
106 views

Existence of a random vector such that the differences of its components satisfy some restrictions

Let us fix any three numbers in $[0,1]$ and summing up to $1$. I denote them by $p_1, p_2, p_3$. Could you help to show that, for every possible vector of reals $U\equiv (U_0, U_1, U_2)\in \mathbb{R}^...
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0answers
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replace variable in a linear model with new variable with same covariance that yields the same least-sqares parameter estimate

Consider the following linear model, which explains the relation between a $d$-dimensional set of explanatory variables $\{\mathbf{X},D \}$ and a 1-dimensional effect variable $Y$ ($\{\mathbf{X},D \}$ ...
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0answers
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Sampling the hitting time of a Brownian motion with drift

Consider a Brownian motion with drift $\mu > 0$ and variance parameter $\sigma^2$. Then the pdf of the first hitting time to the value $a > 0$ is $$ f(t) = \frac{a}{\sigma\sqrt{2\pi t^3}}\exp\...
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5answers
465 views

Dependent and not identically distributed random variables

I am trying to deepen my knowledge in probability and I am having hard times to understand dependent and not identically distributed random variables. Can someone maybe provide me a real world example ...
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1answer
43 views

Generate Uniform Random Variates with Constant Norm [duplicate]

How can one generate $k$ uniform random variates centered at zero, $X_1, X_2, ..., X_k$, given a constant Euclidean norm, $c =\sqrt{X_1^2+X_2^2+...X_k^2}$?
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1answer
22 views

What is the entropy of a riskless random variable?

Variance and standard deviation are often used as proxies for risk and volatility. I make the analogy to information theory as follows, correct if it's wrong: a random variable $x\in \mathbb{R}$ that ...
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1answer
71 views

Correlation is a symmetric measure, but scatter plot matrix shows asymmetric dependence

The correlation matrix demonstrates that correlation is a symmetric measure: $\rho(X,Y) = \rho(Y,X)$ since the lower off-diagonals are mirror images of the upper off-diagonals. The scatterplot matrix ...
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1answer
61 views

Simulating realizations of joint Bernoulli distribution

Let $X$ and $Y$ be Bernoulli random variables with success probability $p$ and $q$ respectively, i.e., \begin{align*} X = \begin{cases} 1 & \text{with probability $p$} \\ 0 & \text{with ...
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2answers
67 views

Can I see i.i.d. variables as just one?

I'm trying to understand the part of this book, page 276 which explains about sample mean: In statistical inference, a central problem is how to use data to estimate unknown parameters of a ...
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1answer
39 views

Converge of Scaled Bernoulli Random Process

Suppose a random sequence is defined by $X_n := n B_n$, where $B_n$ is a Bernoulli sequence such that $\mathbb{P}(B_n = 1) = 1/n$. I am interested in the convergence properties of this random process ...
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1answer
39 views

What is the mean and variance for this Normal distrbution? [duplicate]

Suppose, we have $X_1,X_2,$ and $X_3$ i.i.d random variables (RV's) from $N(\mu,\sigma^2)$. Let $Y=\frac{X_1-X_2}{2}$, and $W=X_3-Y$. Then is $W\sim N(0,2\sigma^2)$ or $W\sim N(0,\sigma^2)$?
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0answers
64 views

Derivation of skewness and kurtosis algebra of random variables

In algebra of random variables, the symbolic rule for computing variance of random variable $X\in\mathbb{R}^{n\times p}$ multiplied by a coefficent vector, $a\in\mathbb{R}^p$, is $$\text{Var}(X\cdot a)...
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1answer
14 views

Confusion on independent variables and prediction

Suppose $X_{i}$ is an i.i.d random variable for $i = 1, 2, \dots, n$. Since $X_{i}$ and $X_{j}$ are independent for all $i \neq j$, the mutual information $I(X_{i}, X_{j}) = 0$. If $X_{i}$ has ...
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1answer
33 views

Expected vallue calculation of i.i.d. random variables

Suppose $X_1,X_2,\ldots,X_n$ are a sequence of i.i.d. random variables with mean $\mu$ and variance $\sigma^2$. Define the sample mean $\bar{X} := \frac{1}{n} \sum_{i=1}^{n} X_i$, which we know is an ...
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0answers
18 views

Minimization of absolute value of residuals of a discrete random variable [duplicate]

Let $X$ be a continuous random variable with cdf $F_{X}$ and pdf $f_{X}$, median $m≡F^{−1}X(\frac{1}{2})$, and $f_{X}(m)>0$. It is a well known result that $m= arg min_{b}E[|X−b|]$. Does this ...
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0answers
22 views

Conditions for joint probability ratio to converge to zero when one variable is known to be less likely

In an experiment of mine, I noticed a seemingly regularity, but which I am struggling to verify formally. Consider you have 4 types of industry in a local economy, A, B, C and D. If you pick an ...

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