Questions tagged [random-variables]

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

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For the probability triple $(\Omega, \mathcal{F}, \Bbb{P})$, a random variable $X$, and a function $g$, is $g(X)$ automatically measurable?

For the probability triple $(\Omega, \mathcal{F}, \Bbb{P})$, a random variable $X: \Omega \to D$, and an arbitrary function $g: D \to E$, is $g \circ X$ also measurable and thus a random variable? I ...
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Solving for cross-covariances between two random vectors

Let $\mathbf{x}$ and $\mathbf{y}$ be Gaussian random vectors with zero mean and covariances $\mathbf{C}_{\mathbf{xx}}$, $\mathbf{C}_{\mathbf{yy}}$, respectively. Define the sum of these processes as $\...
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Almost sure convergence for lipschitz functions

Let $x_n \to x$($x_n$ sequence of random variables) s.t $\sum \mathbb{E}|f(x_n) - f(x)| < \infty$. For any $f$ Lipschitz and bounded. Then $x_n \to x$ almost sure. My attempt: As series converge, ...
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Expected value of a random variable on interval $[a,\infty)$

I'm having hard time understanding this property: If $X$ is a random variable defined on the interval $[a,\infty)$ and $f(x)=0$ for $ x<a$ then $E[X] = a + \int_a^\infty[1-F(x)]dx$. This is valid ...
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"randomizing" a sequence of random variables

I have a (kind of) follow up question on this question. Let $I \subset \mathbb{R}$ and for any $a \in I$ let $X_a$ be a real-valued random variable on $(\Omega_a, \mathcal{F}_a, \mathbb{P}_a)$. ...
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Help on "randomizing" deterministic functions

I'm completely stuck on a problem and maybe someone of you out there has some thoughts on how I could proceed. The setting is the following: Let $f: \mathbb{R}^2 \to \mathbb{R},(x,y) \mapsto f(x,y)$ ...
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Almost Sure Convergence of a Random Variable

I'm having trouble understanding almost sure convergence of random variables. The definition given on wikipedia is that a sequence $\{X_n\}$ of random variables defined on a probablity space $\left(\...
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Exemple where tower property of conditional expectation is NOT verify

Question: Let $\Omega=\{a,b,c\}$. Give an example for $X, F_1, F_2$ in which $E(E(X|F_1)|F_2) \neq E(E(X|F_2)|F_1)$ My answer: I am not at all sure of my answer. If you have any shorter and nicer ...
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Deriving the discretized equation of the Geometric Brownian Motion EDE

I am trying to obtain the discretized equation for the Geometric Brownian Motion EDE, $$ d S_{t}=\mu S_{t} d t+\sigma S_{t} \eta_tdt \tag{1} $$ I am looking for the discretization for the case where $\...
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Understanding random variables as functions

First of all, I have read What is a function and I have understood it basically and it is clear to me that in order to caluclate statistics "things" have to be transformed or mapped to ...
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M/M/1/10 queueing process with two different classes

I'm looking at a problem where we have calls queueing under two different classes, new calls and handovers. The number of calls arriving follow a Poisson process with $\lambda_{1} = 125$ per hour ...
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Calculating the transformation of 1/X where the pdf of X is 1/x

I feel very silly asking this, since it seems so simple but I'm almost certain I have done something wrong. I have a random variable $A$ with the following pdf: \begin{equation} f_{A}(x_{A}) = \begin{...
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Expected value of distance in one-dimensional random walk

Let $S_n$ be a one-dimensional random walk; that is, the sum of $n$ i.i.d. coin toss variables $X_1, ..., X_n$ (meaning that $P(X_1=1)=P(X_1=-1)=1/2$). My question is to compute the expected value of $...
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What positive distribution has the form $x^{-\ln(x)}$

What is the name for the continuous distribution $f(x)\propto x^{-\ln(x)}$ for $x\geq 0$? More precisely, I'd like the name for the family of distributions of the form $f(x)\propto x^{-c \ln(x)}$, ...
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Clarifying weird integral notation

I would like your help to understand the following notation for integrals which I have never seen. Consider an integral $$ \int_{a}^b 3 \text{ }d X $$ where $X$ is a random variable. What does this ...
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What positive distribution has the form $e^{-x-\frac{1}{x}}$

What is the name for the continuous distribution $$f(x)\propto e^{-\left(x+\frac{1}{x}\right)}$$, where $x\geq 0$? Does it even have a name? My interest in it is because it is the simplest ...
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Probability: are realizations of random variables what is actually observed?

According to Wikipedia, yes. In probability and statistics, a realization, observation, or observed value, of a random variable is the value that is actually observed (what actually happened). https:...
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Probability Algebra With a Wordle Example

Define $X$ as the proposition "you will win today's Wordle," $A$ as expressing the information returned by the coloring of your first guess (in a vacuum), $B$ as expressing the information ...
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The covariance between the sum of N independent random variables and N.

Problem Let $X_1,X_2,...$ be independent random variables with $E(X_i)=a, Var(X_i)=b$ for $i\geq1$, and $N\geq0$ an integer-valued random variable with $E(N) = c, Var(N) = d^2$ independent of the ...
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Bernoulli trials: Probability of S consecutive successes followed by F consecutive failures

I am reading a probability text book and I am having difficulty understanding a paragraph and how the equation was setup. Can anyone please explain it in more detail or show me how this was derived? ...
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Log-normal distribution: Why is this a density function?

I want to prove that $$f(x) = \frac{1}{\sigma x \sqrt{2\pi}} \exp(-\frac{\ln(x)^2}{2\sigma^2})$$ for $x>0$ and $f(x)=0$ for $x\leq 0$ is a density function, with $\sigma > 0$. So what I have to ...
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If a random variable has an atom at zero, does it have a density?

Let $Y$ be a random variable with distribution function $$ F_Y(x) = \begin{cases} 0 &\quad x<0\\ p &\quad x=0\\ p + (1-p)F_X(x) &\quad x>0 \end{cases} $$ where $X$ is a continuous ...
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What is the smallest unseen number in an iid sample? (From "A number NOBODY has thought of - Numberphile")

In this Numberphile video, the question: "What is a number nobody has thought of?" is addressed. The method is as follows: Estimate a number $N$ as the number of times humans have thought ...
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Can we use mathematics and logic to estimate probability of extremely absurd events?

I'd like to detail my question over the example below. Let's say I have a random pixel generator which has $1024 \times 768$ screen resolution. It also has $24$ bit color which means $2^{24}= 16,777,...
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Product Two Independen Random Variables, Correct Argument?

Let's say we have a continuous RV $X \in \mathbb{R}$ and a discrete RV for example $Y \in \{1,2,3\}$, independent, and we are interested in the distribution of the product of the two. Please note ...
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There exists a non-negative random variable such that $P( \{ \omega \in \Omega: E(X)-2\sigma<X(\omega)<E(X)+2\sigma \})=\frac{3}{5}$. True or False?

Problem State whether given statement is True or False. There exists a non-negative random variable such that: $P( \{ \omega \in \Omega: E(X)-2\sigma<X(\omega)<E(X)+2\sigma \})=\frac{3}{5}$ My ...
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What does $X^{−1}$ mean in in random variables? [closed]

What is the meaning of $X^{−1}$ as in $X^{−1} (B) ⊂ A$ in random variables?
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Variance of product of Gaussian random variables

Suppose I have $r = [r_1, r_2, ..., r_n]$, which are iid and follow normal distribution of $N(\mu, \sigma^2)$, then I have weight vector of $h = [h_1, h_2, ...,h_n]$, which iid followed $N(0, \...
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Probability Generating Function of $X^2$ in terms of $G_X(z)$.

I am struggling with an infinite sum in the computation of the PGF of $X^2$ in terms of that of $X$. Suppose you define the random variable $Z=X^2$, where $X\in\mathbb{N}$. We know that $G_X(z)=\sum_{...
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Fano's Inequality without conditioning

Suppose $X\sim P$ is a random variable taking values on an alphabet $\mathcal{A}=\{1,\dots,m\}$, such that $p:=P(1)>P(k)$ for $k\neq1$. The minimum-probability-of-error predictor of $X$ is $\hat{X}=...
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Example of random variable with infinite entropy

Let $X\sim P$ on $A=\{2,3,\dots\}$, where $P(k)=\frac{C}{k(\log k)^2}$ for $k\geq2$ with $C$ some normalising constant. Show that $H(X)=\infty$. My attempt so far: I have shown, by direct computation ...
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Is this function continuous (when $U$ has no flat regions)?

Apologies for that useless modifier in the brackets in the title -I had to add that to avoid "duplicate titles". It is most natural that there will be multiple questions with the title "...
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Calculating Probabilities and Using Binomial Distribution

A recent survey of residents in Texas concluded that 55% of Austin city residents and 46% of Houston city residents broke a bone at some point during their childhood. a. Let’s say Austin has 5200 ...
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Expected number of rounds needed until a winner is declare in Rock, Paper, Scissors.

A game of Rock, Paper, Scissors is usually played with 2 or more (n) people. If when all n people reveal their play and only 1 or all 3 options appear, that round is deemed indecisive, and they just ...
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Joint Probability Mass Function of Rock, Paper and Scissors

Problem A game of Rock, Paper, Scissors is usually played with $2$ people, but it can be extended to an arbitrarily large number of $n$ people, where $n\geq2$. If when 'all $n$ people reveal their ...
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Expectation of Exponential Random Variable

I was a bit confused about this question. You’re clearing out your garage for a garage sale, and you want to get rid of as much stuff as possible quickly. You found a dresser and decided to sell it ...
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X1, X2, X3 are independent and uniformly distributed over the interval (0,1). Decide the distribution of the second greatest of X1, X2, X3.

This problem is from a textbook. I have no idea where to begin. The textbook gives a lead: "The second greatest (variable) is smaller than $t$ if two or three of the $X$-variables are smaller ...
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1 answer
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Calculating Expected Number from Exponential Distribution

I was a bit confused about this question. You’re clearing out your garage for a garage sale, and you want to get rid of as much stuff as possible quickly. You found a dresser and decided to sell it ...
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1 vote
2 answers
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Meaning of 'almost-everywhere constant' random variable

My question stems from page 2 of this paper by Bucy, which states: [A random variable $x$] is almost everywhere constant a.e. $P$. where $P$ is a probability measure. My interpretation of this is as ...
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1 vote
1 answer
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On the convergence in distribution of some random variables

In this paper, they mention an equivalent result of the convergence in distribution of the random variable in page $11$. I don't understand why the convergence in law of $Arg(R_n)$ to the uniform ...
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Construction of a discrete random variable uniformly choosing random variables

Let $(R_{n,k})_{n \in \mathbb{N}, k \le n}$ a sequence of iid discrete random variables. I think that Kolmogorov proved such variables exist in a probability space. I want to construct a sequence of ...
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1 vote
1 answer
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Bounds for summing Continuous Variable

I have a question on how the bounds are picked for the example below: Given that $X_1$ and $X_2$ are independent random variables, and the pdf is $f_{X_i}(x_i) = λe^{−λx}, x_i > 0$ Find $W=X_1+X_2$ ...
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1 vote
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Chernoff-type bounds for a sum of Poissons

Let $X_1,\dots,X_n$ be i.i.d. Poisson variables $\sim\text{Pois}(\lambda)$, with $\lambda<1$. Now, if $X=X_1+\dots+X_n$ then $X\sim\text{Pois}(\lambda n)$ with $\mathbb E(X)=\lambda n<n$. I ...
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double integral showing equal to 1

The question asked is as follows: Given that the nonnegative function $g(z)$ has the property that $\int_{-\infty}^\infty g(z)dz=1$ show that $f(x,y)=\frac{g(\sqrt{x^2+y^2})}{\pi\sqrt{x^2+y^2}}$ for ...
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How to find bounds for double integration over a region?

I don't quite understand how to do double integration for the joint probability density function through looking at a graph. Which comes from this question: Let $X$ and $Y$ have the joint pdf $f_{X,Y} ...
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Is the variance of these two variables the same?

Let $X$ and $Y$ be random variables having joint density function $$ f(x,y) = \begin{cases} x + y & \text{for } 0 \leq x \leq 1, 0 \leq y \leq 1 \\ 0 & \text{other }x, 0 \leq y \leq 1 \end{...
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Sequence of an exponential random variable

Given that $X_n$ is an exponential random variable with parameter $λ=n$. How does $P(X_n≥ε)=e^{-nε}$ ? According to this probability course, the equality holds since $X_n∼Exponential(n)$. Honestly, ...
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-1 votes
1 answer
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Marie has been taking note of the time she leaves for work and the length of her morning commute. She decided to model the number of hours $X$ [closed]

Marie has been taking note of the time she leaves for work and the length of her morning commute. She decided to model the number of hours $X$ after 6:30 a.m that she leaves for work with a $uniform(2)...
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3 votes
1 answer
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Attempt to improve from $L^1$-boundedness to uniform integrability

Let $(X_n)_{n\geq0}$ be a discrete-time martingale, and let $T$ be an almost surely finite stopping time such that $$\mathbb{E}(|X_T|)<\infty,\hspace{2cm}\lim_{n\to\infty}\mathbb{E}(|X_n|1_{\{T>...
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If the law of a random variable $X$ is zero at a point, is then the density function of $X$ also zero at that point?

Let $(\Omega, F, \mathbb{P})$ be a probability space, $X$ is a random variable s.t. $X[\Omega] = [-a, a), a > 0$ and that $X$ has a density function $f_X$. If we know that $\mathbb{P}_X(c) = 0$ for ...
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