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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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The joint distribution of Y=AX and Z=BX given a projection matrix A and residual maker matrix B, and a random vector X with known pdf?

This question follows on from a previous question I asked which was answered. It turns out my question lacked some important details, which was revealed by the answer posted on that thread. This is ...
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Probability mass function for the first non-increasing sample from a random sequence

Consider a sequence of random numbers drawn IID from some distribution $g(x)$. How would I determine the distribution of the value of the first sample from that sequence which is not greater than all ...
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What does the pmf of a discrete random variable look like if it can take on the value $\infty$?

What does the pmf of a discrete random variable look like if it can take on the value $\infty$? Consider a stopping time $\tau$ of a Markov chain, which is a random variable that takes values in the ...
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How to derive the joint distribution of Y=AX and Z=BX given a random vector X with known pdf?

Given a random vector $X \in \mathbb{R}^k$, with a known pdf given by $f_X$. If $Y, Z \in \mathbb{R}^k$ are defined by $Y = AX$, $Z = BX$, where $A,B \in \mathbb{R}^{k\times k}$ are different, given, ...
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How to split a variable into two variables [closed]

I have a time series data (from 1948 to 2005) of USA on employment and value of output Data is industry-vise with two codes = 0 and 1. So, for each code the data on employment and value is available ...
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Couldn't understand conditional distribution and density function

The random variable X has a distribution function as shown in the graph below and $ Y=X^2$. Find a. $P(1/2 ≤ X ≤ 3/2)$ b. $P(1/2 ≤ X ≤ 3/2 | Y ≤ 1)$ g. $P(X+Y ≤ 3/4)$ I couldn't understand how ...
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How can I compute expected return time of a state in a Markov Chain?

I was watching a YouTube video regarding the calculation of expected return time of a Markov Chain. https://www.youtube.com/watch?v=X_Ll0-Ytu7U&vl=en I haven't understood the calculation of $m_{...
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Mean and Variance of weighted sum of n random variables? [duplicate]

Suppose we have n jointly distributed random variables $x_i,i=1,...,n,$ with mean and variance $E(x_i)=\mu_i$, $Var(x_i)=\sigma^2_i$ and covariance $Cov(x_i,x_j)=\sigma_{ij}.$ Then the weighted sum of ...
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Product of a linear function of IDD variables [duplicate]

if $r_i$ are IID variables with $E[r_i]=μ$ and $f$ is a constant, is the following correct? $E\left[\prod_{i=1}^n(1+f\space r_i)\right] = \prod_{i=1}^nE\left[1+f\space r_i\right] = \prod_{i=1}^n(1+...
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Sufficient condition for existence of maximal/optimal coupling

Let us consider two random variables $X$ and $X'$ defined on the same measurable space $(E, \xi)$. Is there a sufficient condition on the distributions of the random variables $X$ and $X'$ for ...
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Conditions for independence of a random vector to a random variable

Given a random vector $Z=(X_1, \dots, X_n)$ and a random variable $Y$, and that $X_i$ and $Y$ are independent for $i=1, \dots, n$, under what conditions are $Z$ and $Y$ are independent? The answer to ...
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1answer
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What is the difference between multivariate random variables and sample random variables? [closed]

Multivariate random variables consists of more than one random variable which may be independent , eg. Height , weight , age can be called three random variables and we can write their joint ...
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Transformation of random variables and Jacobian

When transforming 2+ continuous random variables, you use a Jacobian matrix and compute the determinant. Do you also compute the Jacobian for discrete random variables?
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How to normalize mutual information between to real-valued random variables?

How can I normalize mutual information between to real-valued random variables using Python or R? ...
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Distribution of N objects into C bins that are then sorted?

Let's say we have $C$ bins and $N$ indistinguishable objects. For each object we choose one bin at random where each bin is equally likely (with probability $1/C$). Let $B_k$ be the number of objects ...
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What does it exactly mean if a random variable follows a distribution

Imagine there's a random variable such as $ε$. Then we say that $ε$ is i.i.d and follows a normal distribution with mean $0$ and variance $σ^2$. What does this mean? Is this not a variable anymore? ...
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How can observations of random variables be IID, if they are not themselves random variables?

Suppose we perform some experiment which results in an outcome $\omega \in \Omega$. A random variable $X$ maps $\omega$ to a real number, and the (discrete) distribution $P(X)$ maps $X$ to $[0, 1]$. ...
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Notation for drawing once from a distribution and reusing values [closed]

Is there a succinct way to denote a one-time draw from a distribution? This is essentially what I want to achieve: ...
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1answer
26 views

Kernel for a markov process?

Could anyone just explain to me what does it mean by mathematically, $P_n(x, dy)$ is the law of $X_n$ here in the page $46$. https://statweb.stanford.edu/~cgates/PERSI/papers/iterate.pdf Thanks for ...
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How does the inverse transform method work in discrete r.v.?

In this question How does the inverse transform method work? it's mentioned the general procedure to generate r.v. U <- runif(1e6) X <- qnorm(U) X How ...
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Which curve to select for finding the CDF of a function of a continuous joint distribution?

I came across a question which required to find the CDF of a function of a continuous joint distribution: $W=XY$. The following is the joint PDF: $$f_{X,Y}(x,y)=\begin{cases}\frac{xy}{4000}&,\, ...
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How to find CDF of a function of continuous joint distribution from PDF of joint distribution?

Here's what I think I should proceed: Make a level curve for the function keeping the constraints given in the PDF of joint distribution in mind. Find the area of interest keeping the constraints ...
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1answer
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Favorite sequence in 10 flip tosses

I have the following question. Somebody likes to flip coins. In particular, this person is delighted to get a sequence HTTH. Assuming the person flipped coins 10 times, what is the probability that ...
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Level curves and functions of pair of Random Variables?

I came across the following question: I tried solving it, the following is my 1st attempt:(2nd method at the end) 𝑃[𝑊≤𝑤]=𝑃[𝑋𝑌≤𝑤]=𝑃[𝑌≤𝑤/𝑋] And then I simply double integrated keeping the ...
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Why do we take a function of x as the limit of integration over y while calculating the marginal pdf of x?

I searched through similar questions but couldn't find one answering my question. I know the following is the way of finding marginal pdf from joint pdf. $$ f_x(x)= \int_{-\infty}^{\infty} f_{x,y}(x,y)...
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1answer
33 views

Why is the relationship between max(X,Y) and X and Y the way it is?

My textbook says that the above follows from the observation: $\{W\leq w\}=\{X\leq w\},\{Y\leq w\}$, How do we prove the above observation?
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1answer
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What is the distribution of conditional expectation of a function f(X) of the random variable X? i.e. E(f(X)|X)

I have a continuous random variable X with a known PDF. I want to find the distribution of f(X) where ...
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1answer
33 views

Finding the uniformly most powerful test for hypothesis

Let $\mathbf{X}=(X_1,...,X_n)^T$ is a simple sample where $X$ belongs to exponential distribution family $\mathcal{P}=\{ f(x;\mu,\sigma \}, -\infty<\mu<\infty, 0<\sigma<\infty.$ Density is ...
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How to check these sequences generated by i.i.d random variables are martingales?

Let $\{Y_n\}_{n\geq 1}$ be a sequence of independent, identically distributed random variables. $P(Y_i=1)=P(Y_i=-1)=\frac12$ Set $S_0=0$ and $S_n=Y_1+...+Y_n$ if $n\geq 1$ I want to check if the ...
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PDF of the minimum of a geometric random variable and a constant

I have $X \sim Geo(p)$, such that $$p(x) = (1-p)^{k-1}p, \ \ x = 1,2,3, \ldots$$ and Y is a constant random variable which assumes the value of the constant integer $t$, such that $$P(Y=t) = 1, \ \ ...
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How to evaluate $\mathbb{P}(XY=a)=\mathbb{P}(X=\frac{a}{Y})$, when $X$ or $Y$ take multiple values?

Assume $X,Y$ are discrete and have a finite outcome space. How to evaluate $\mathbb{P}(XY=a)=\mathbb{P}(X=\frac{a}{Y})$, when $X$ or $Y$ take multple values? Does one write out the result for each ...
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1answer
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Expectation of a function of random variables

I'm trying to simplify the following expectation so that I can later solve a maximization problem: $max_k E[(A - kB)^2]$, where $A$ ~ $N(0,\sigma^2_1)$, $B = A+ \epsilon$ and $\epsilon$ ~ $N(0,\sigma^...
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Give upper bound for random variable

Problem: Suppose $X$ is a random variable such that $E[2^X] = 4$. Give an upper bound for $P(X ≥ 3)$. Justify your answer. Attempt: I know the following equations $P(X \ge 3) \le \frac{E[X]}{3}$ $E[...
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What is the mode of the convoluted probability density function?

If I am aware of the distributions of both $V$ and $U$, is there general guiding principle in terms of the position of the mode of the distribution of $\varepsilon =V-U$. As I am not specifying the ...
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155 views

Definition of Statistic

I keep seeing conflicting definitions of a statistic. Is a statistic a random variable such that it is a function of the random variables of a random sample? Or is it the value of the function of the ...
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using attribute outcomes to define variable limits

I am working on developing a specification for peel strength of a package (looking for a minimum value, higher is better). The limit of where it is considered "good" is determined by a leak test (pass/...
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Finding $P(X>d)$ when $X\mid \Lambda$ is Pareto and $\Lambda$ is Gamma distributed

Supposedly $X$ have a Pareto distribution with parameters $\Lambda$ and $\theta$. Let $\Lambda$ have a Gamma distribution with parameters $\alpha$ and $1$ (i.e., scale parameter $= 1$). Calculate (...
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1answer
24 views

The covariance of 2 independent and identically distributed

I am currently researching a paper and they have the following set-up: " $(\epsilon_{1}, \epsilon_{2})iid \sim N(\mu, \xi)$. captures the collective biases that in-vestors may have about d, is ...
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Derive the CDF of the sum of two independent random variables [duplicate]

Some notational remarks before presenting the question: $k<\infty$, with $k\in \mathbb{N}$. $\lambda\equiv (\lambda_1,...,\lambda_k)$, $\lambda_j\in [0,1]^k$ $\forall j$, and $\sum_{j=1}^k\...
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1answer
110 views

Sampling from Gaussian mixture models, when are the sampled data independent?

Suppose I generate a Gaussian mixture model with $N$ Gaussian distributions $p(x) = \sum\limits_{i = 1}^N w_i \mathcal{N}(x;\mu_i, \Sigma_i)$ where $w_i$ are the weights. Now I sample some points $\...
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1answer
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Expectation and Variance of dot product of a random vector and random linear combinations of vectors from the same distribution?

Let's say we have a multivariate distribution $D$ which generates random $n$-dimensional vectors $x$ for us ($x \in R^n$). We know that the dimensions of vector $x$ are correlated, and that each ...
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1answer
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Difference between finiteness and boundedness of a random variable

In a stochastic processes class, we're studying a theorem which required that a random variable $T$ have finite mean. The notes presented a counterexample where a R.V. $T$ was such that $P(T<\infty)...
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1answer
48 views

Finding unconditional pdf of a Pareto-Gamma mixed distribution

Let X have a Pareto distribution with parameters Λ and θ. Let Λ have a gamma distribution with parameters α and 1 (i.e., scale parameter = 1). Find the unconditional pdf of X. I tried finding the ...
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What does it mean that a variable is resolved (in time or space)?

I have been for some time using the terms time-resolved or spatially-resolved when it comes to describing a random variable. When I paused to ponder what do I mean by that, I realised that I can only ...
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Asymptotic distribution using the Delta method

Let $X_1, \ldots, X_n$ be i.i.d. normal random variables, where $X_i \sim N(\theta, \theta^2)$ with an unknown $\theta > 0$. We could for example estimate $\theta$ using the average $$\hat\theta_n ...
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Continuous random variable that is constant for a fraction of the domain

Consider a random variable $X$ defined on $[0,1]$, which is $ X(\omega)= \begin{cases} 0.5, \omega \in [0,0.5]\\ \omega, \omega \in (0.5,1] \end{cases}$. Intuitively, this definition makes sense to me....
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22 views

Generate random sample of X1 and X2

If X2 is dependent on X1, how to generate the random sample of (X1,X2)? One scenario is that we know the prior distribution of X1 and functional relationship between X1 and X2, how to generate the ...
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43 views

PDF transformation for y=|x|

Suppose I have the random variable X with a pdf: $$f(x)=exp(-(x+1)) u(x+1)$$ where u is the unit step function; such that u = 0 for x<-1 and u=1 for x>-1 $$y= |x|$$ for $$-1<x<1$$ ...
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1answer
44 views

Conditional expectation of two independent RV

The expectation of the product of two independent random variables $X$ and $Y$ is the product of the expectations: \begin{align} E(XY) = E(X)E(Y) \end{align} Let's add another random variable $Z$ in ...
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Is this formula for the Law of Iterated Expectations correct?

I saw two versions of the law of iterated expectations, this one: \begin{align} E(E(Y\vert X)) = E(Y) \end{align} and this one: \begin{align} E(E(Y\vert X_1, X_2)\vert X_1) = E(Y \vert X_1) \end{align}...