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Questions tagged [random-variables]

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

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Connecting two interpretations of the negative binomial distribution

In my probability course, my professor derived the negative binomial distribution by reasoning about the probability that the time of the $k$-th success, $T_k$, takes some value $n$. If $p$ is ...
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How to sample two values from a random variable X with the lesser to be a random variable Y?

The variable X has pdf $$f(x) = \frac18(6 - x)$$ for $$2 ≤ x ≤ 6$$ A sample of two values of X is taken. Denoting the lesser of the two values by Y, use the cdf of X to write down the cdf of Y. Hence ...
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Convergence in probability of running maximum

Suppose we have a sequence of integrable random variables $(X_n)$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ such that $n^{-1}X_n\to 0$ in probability as $n\to\infty$. Suppose further ...
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1answer
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find P( X$\lt$ Y) given X and Y are two independent exponential RV

The question is as follows: You arrive at the post office and as you enter, each of the two clerks, Jim and Jack, starts serving a client. The amount of time needed by Jim to serve his client is X ...
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1answer
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Is a Poisson r.v.'s parameter a rate $\mu$ or a count $\mu t$?

Let's say I want to model the arrivals of some quantity of interest, say customers coming to a store. I know that on average, $\mu$ customers arrive in on hour. My understanding is that if $N$ is the ...
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1answer
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Let $(X,Y)$ be a uniformly chosen point of a region $A$, given $A$ compute $EX$

Let $(X,Y)$ be a uniformly chosen point of a region $A \subset \mathbb{R}^2$ Given we have the following joint pdf: $$ f(x,y) = \begin{cases} \dfrac{a}{ \text{area of}~ A} & (x,y)\in A \\ 0 &...
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Characteristic function of exponential distributed random variable

Given: $$f_X(x) = \lambda e^{-\lambda x},\; x\in X$$ Wanted: The corresponding characteristic function $\phi(ju)$. \begin{align} \phi(ju)&=\mathbb{E}(e^{j^2ux})\\ &= \lambda \int^{\infty}...
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let z1 and z2 be independent standard normal RV, find the pdf of $e^{3Z_1+2Z_2}$

Let Z1 and Z2 be independent standard normal random variables. Find the following The probability density function of $e^{3Z_1+2Z_2}$ The given solutions is as follows: But what doesn't make sense ...
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1answer
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given exponential Cumulative distribution function, finding another Cumulative distribution function with functionl connection

There is given $X$ a random variable with exponential cumulative distribution such that $X~Exp(1)$ so the exponential Cumulative distribution function is: $P(x\le t)= F_x(t)=(1-e^{-t} , 0\le t) \...
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How to minimize piecewise binary function with absolute value threshold

Let $f: \Bbb R^n \rightarrow \Bbb R$ and $\mathbf{g}:\Bbb R^n \rightarrow \Bbb R^n$. Let $\mathbf{a,b}$ be random variables with values in $\Bbb R^n$. $$f\mathbf{(a,g(b))} = \begin{cases} ...
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Joint distribution of absolute difference and sum of two independent exponential distributions

If $X\sim \rm{Exp}(1)$ and $Y\sim \rm{Exp}(1)$ are two independent random variables. What is the joint distribution of $U = |X - Y|$ and $V = X + Y$? I used the Jacobian transformation to obtain ...
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How do I evaluate the following combination of random variables? Is it martingale?

I'm about to analyse the following expression $$Z_n:=\prod_{k=1}^n \left(\frac{\frac{Y_k}{\prod_{i=1}^k X_i}}{\sum_{j=1}^k \frac{Y_j}{\prod_{i=1}^j X_i}} \right),$$ where $Y_j$ for all $j\in \mathbb{...
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Division of two independent uniformly random variable [duplicate]

Given two independent random variable X and Y which both have uniform distribution over[0,1] I want to calculate PDF of $Z =\frac{X}{Y}$ and here is my solution: $\int_{-\infty}^{\infty}zf_X(yz)f_Y(y)...
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1answer
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ls it possible to construct discrete r.v.s given expectation and variance?

Suppose there is a discrete r.v.s X, all we know is: E(X) = 10 and VAR(X) = 2500 Any general way to find PMF of X? Thanks.
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Special case of Bertrand Paradox or just a mistake?

I've been working on a question and it seems I have obtained a paradoxical answer. Odds are I've just committed a mistake somewhere, however, I will elucidate the question and my solution just in ...
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PMF of function of random variables

Let X be a Geom($\frac{1}{2}$) random variable, and define Y=$X^{-1}$ What is the p.m.f. of Y ? attempt: pmf of a Geom RV in general form is $p(1-p)^{k-1}$ There is this similar question, not ...
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Shortcut to finding the distribution of a specific random variable

Question: A dice is rolled 3 times. Let X denote the maximum of the three values rolled. What is the distribution of X (that is, P[X = x] for x = 1,2,3,4,6)? You can leave your final answer in terms ...
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Calculate Mean from the Moment Generating Function ( m.g.f / mgf) of Y = 2X +3

The question is as follows: No calculators. Let X be a random variable with moment generating function $M_{x}(t) = \frac{e^{(e^t-1)}}{2e^{-t} -1} \;\;\;\;\;for \;\; \;t<log(2)$ given Y = 2X +3 ...
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Expressing quadratic form of normal variables in terms of chi-squared variables

On Wikpedia and in the references therein [1,2], it is stated that any quadratic form of normally distributed random variables can be expressed as the sum of many independent non-central chi-squared ...
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“Normalized” covariance matrix of a Gaussian random vector

Let $X\sim\mathcal{N}(0,I_{d})$. I would like to compute the the following quantity: \begin{equation} \mathbb{E}\bigg[\frac{XX^{\top}}{\|X\|_{2}^{2}}\bigg]. \end{equation} Letting $B=\frac{XX^{\top}}{...
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Bayesian LASSO: A step within the Gibbs sampler

I'm intending to implement a Bayesian LASSO inside the Gibbs sampler I use to estimate a multivariate time-series model, but I have a doubt about how to draw this step. The prior is a Double-...
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Quick terminology question - “Standardizing” Random Variables

This is a really boring question I apologize. I'm not sure what to call this process, I call it "standardization" however perhaps non beginners would think I'm being sloppy with terminology. For ...
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Intuition behind expected value formula? [duplicate]

I just learned the expected value formula for a positive random variable: $E[X] = \int_0^\infty P(X>x)dx$. I can follow the proof but I'm a little unclear as to what the intuition behind this ...
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Variance analysis of the rank of groups of variables

So I am looking to analyse the variance in two groups of random variables. There are 2 sets of 3 random variables A, B and C and X, Y and Z. If I calculate the numerical rank of these variables ...
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1answer
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Is this Poisson Process problem worked out correctly?

Calls are received at a company call center according to a Poisson process at the rate of five calls per minute. (a) Find the probability that no call occurs over a 30-second period. (b) Find the ...
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Example of higher random vector moments

While reading about random vectors, I learned that... $$ E\left[\vec{X}\right] = \left[\begin{array}{cccc} E\left[\vec{X}_1\right] & E\left[\vec{X}_2\right] & \cdots & E\left[\vec{X}_m\...
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1answer
33 views

Geometric distribution/ Formula

I have a geometrically distributed random variable $X$. I want to calculate $P(X \geq k)$. I get that $P(X \geq k)= (1-p)^{k-1}$. Now I want to show that $P(X \geq k+n \mid X\geq n) = P(X \geq k)$. ...
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Expectation of non-negative random variable vs Truncated random variable

I want to show that E[X|X>t]>=E[X], where X is discrete non-negative r.v, and t is positive deterministic value. Some ideas? Edit: >= instead of >
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Prove that a random variable defined in terms of lim sup of independent random variables is constant

I'm asked to do the following: Let $(X_n)$ be a sequence of independent random variables. Show that the following random variable is constant almost surely: $$X = \begin{cases}\exp\{-\limsup_{n \to \...
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Distribution/Variance of correlated squared normal random variables

If $X_{1}, X_{2}, \ldots, X_{N}$ are identically distributed normal random variables with mean $0$ and variance $\frac{(N+3)D\sigma^{2}}{N}$, then I want to calculate the distribution, or at least the ...
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28 views

Random variable and probability calculations

X is a random variable that follows a binomial property, with E(X) = 36 and $\sigma$(X) = 3. Calculate p(X = $10$) I know that the variance is V(X) = 9 and I know that I have to use the formulas of ...
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the difference between independent random variables and independent events, a strange instance

I know the conclusion that if event $A$ and $B$ are independent, $P(AB)=P(A)P(B) $. And furthermore, the two sigma algebra $\sigma(I_A)$ and $\sigma(I_B)$ should be independent, where $I_A:\Omega \...
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Inequality regarding sample mean

I was looking at the book "Asymptotic Theory of statistics and probability, DasGupta A., 2008" and in one point of a proof they use an inequality which I have not been able to understand. Given that $...
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Expected value-random variables

On $\Omega=[0,1]$ we consider $\sigma$-algebra Borel sets and Lebesgue measure. Let $Y(x)=x(1-x)$. Show that for any integrable random variable $X$: $$E(X\mid Y)(x)=\frac{X(x)+X(1-x)}{2}\quad \text{...
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Let $X$ a random variable (such that $E(X)<+\infty$, proving : $\lim_{t \rightarrow -\infty }tP(X<t)=0$

Let $X$ be a random variable, and $E(X)<+\infty$. I'm stuck on proving $\lim_{t \rightarrow -\infty }tP(X<t)=0$. But I proved $\lim_{t \rightarrow +\infty }tP(X>t)=0$ and here's my method :...
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How would one go about proving that two random variables are uncorrelated but not independent? [duplicate]

Suppose $\theta$ is a Uniform random variable on [0, 2$\pi$]. Let $X$ be $cos(\theta)$ and $Y$ be $sin(\theta)$. We have to show that $X$ and $Y$ are uncorrelated but not independent. My solution: $...
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1answer
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If $X$ is an exponentially distributed variable with mean $ \lambda$, $Y=−3\ln(X)$ has Gumbel distribution?

Let X be a random variable which follows an exponential distribution with parameter $\lambda$ ($\lambda>0$), find the distribution of the random variable $Y = −3\ln(X)$. So this is my answer for ...
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1answer
50 views

Calculation of expectation of Poisson Process.

This is a problem related to Poisson Process where $\lambda = 2$. $ E(N_3N_4) \\ = E[N_3(N_4-N_3 + N_3)] \\ = E[N_3(N_4-N_3) + N^2_3)] \\ = E[N_3 - N_0(N_4-N_3) + N^2_3)] \\ = E[N_3 - N_0(N_4-...
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Conditional distribution of two random variables.

Suppose I have two continuous random variables $X$ and $Z$, $X$ and $Z$ are not independent. While I know that $Z \sim \mathcal{N}(0, 1)$, I have no information for the distribution of $X$. Now one ...
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Are the two random variables independent?

Consider the following two random variables, In first case you record the number of people arriving at a queue, for a random amount of time. Note, here the arrival of people in the queue is random (...
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A die roll and Binomial distribution.

Note: This is not an accurate picture. This picture is given to give a sense only. Suppose, I assign probabilities to a die according to Binomial distribution. I.e. 1 and 6 have least probability of ...
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Estimate number of permutations with unknown number of variables

I need to estimate the number of possible permutations of a problem. A way to do this, I suspect, is to randomly generate a decent amount of problems, and then find out how many were repeated. If ...
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1answer
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What does it mean if someone says “this experiment has a X distribution”?

Kindly, help me to clear my understanding of probability distribution. As far as I understand, probability-distribution means a table/list which lists the probabilities of each outcome of an ...
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1answer
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What could be an example of 'time' in a stochastic process where 'time' is not a natural time?

I need any example of a stochastic process where 'time' is not a natural/real-world element what we generally call the time, and measure it using clocks/watches. What could be an example of 'time' in ...
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What does it mean by 'time' in a random process?

I understand that A random process is a series of random variables which is composed of two components: time values Both time and values can be either discrete and ...
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How to relate between Random Variable, Probability mass/density function and Probability distribution

While studying probability I came across these terms Random variable, probability function, and probability distribution. How to relate these these 3 terms in unfair coin toss problem where head ...
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Bernoulli random variables and two player games

With hockey playoffs upon us, I have been thinking about ways of modeling the results of athletic events. Suppose team A wins 50% of its games against average teams, and team B wins 50% of its games ...
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32 views

Calculate the Mutual Information between $X$ and $X^2$, where $X$ is uniformly distributed.

Let us consider two random variables, $X$ and $Y$. Let $X$ be uniformly distributed in $[-1,1]$ and let $Y=X^{2}$. Is it possible to calculate the Mutual Information between them? E what is the ...
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Proof of “Sum of random variables is a random variable”

There is already a question about it where in a proof is given to show that sum of two random variables X and Y, is a random variable. Calling Z=X+Y, the idea is to somehow show that Z is a measurable ...
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sum of 3 correlated jointly random variables

Suppose we have $3$ jointly normally distributed random variables $X_1, X_2, X_3$ with mean $0$ and variances $\sigma_1^2, \sigma_2^2, \sigma_3^2$. Suppose their correlations are $\rho_{12}, \rho_{23},...