# Questions tagged [random-variables]

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

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### Are “independent events” in probability really independent? [closed]

This is a hard and deep question. I understand very well the concept of independence. But, let us take two events: Event A (I throw a dice) and event B (some star explodes in an near galaxy). Are ...
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### Find the limit cdf for a sequence of random varibales whose characteristic functions do not converge.

X is a random variable with $P(x=-1)=P(x=1)=\frac{1}{2}$, $Y_n=nX$ is a sequence of random variables, n=1,2, $\dots$ $S_n = \frac{1}{n}\sum_{i=1}^{n} Y_i$ is the partial sum of $Y_n$. I am asked to ...
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### Almost sure convergence of random variables with same mean and the difference goes to zero on the product

Let $X_n$ be a sequence of independent real valued random variables on the same event space, with the same (finite) mean $\mu$. Suppose that for almost every couple of points $(\omega,\omega')$ in ...
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### Understanding the Monte Carlo Equation which returns value to the expected function

I was reading an article related to Monte Carlo Method. The link of the article is: Monte Carlo Lecture I have following questions: 1)In the equation related to the attached image, we are assigning ...
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### Distribution of $\Big(Y_1+Y_2\Big)^2$ and $\Big(Y_1-Y_2\Big)^2$ where $Y_i \sim N(0,1)$

Does anyone know what is the distribution of $(Y_1+Y_2)^2$ and $(Y_1-Y_2)^2$ where $Y_i \sim N(0,1)$ are independent variables? I have tried to go through the joint pdf, but when trying to change ...
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### An upper bound for $\sum_{n=1}^{\infty} n^{r-2} P\{|\sum_{k=1}^{n} \sum_{i=-\infty}^{\infty} a_{i} X_{i+k}|>\varepsilon n\}$

Let $1 \leq r<2$ and let $\left\{X, X_{i},-\infty<i<\infty\right\}$ be a sequence of pairwise i.i.d. random variables. Let $\left\{a_{i},-\infty<i<\infty\right\}$ be a sequence of real ...
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### Brownian motion increments - are they random variables or random processes

If $W_t$ is a Brownian motion process and $0 \le t_1 \le t_2$ then is the increment $W_{t2} - W_{t1}$ a random variable or a random process? My lectures say "random variable" but I believe it makes ...
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### Almost sure convergence of a sum of independent r.v

Let $S_n:=\sum_{i=1}^nX_i$ where $X_1,X_2,...$ are indepentent r.v.'s such that: $P(X_n=n^2-1)=\frac{1}{n^2}$ and $P(X_n=-1)=1-\frac{1}{n^2}$ Show that $\frac{S_n}{n}\rightarrow-1$ almost ...
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### How do I obtain the pdf of a random variable, which is a function of random variable.

A random variable, $X$, has a value of zero with probability $1/3$, and follows a uniform distribution over $[-1, 1]$ with probability $2/3$. How can I derive the pdf of $X$? In my opinion, $X$ can ...
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### In what sense are independent random vectors “independent”? [duplicate]

Let's say we have independent random vectors $\boldsymbol{X}$ and $\boldsymbol{Y}$, where $\boldsymbol{X}=(X_{1},...,X_{p})$ and $\boldsymbol{Y}=(Y_{1},...,Y_{q})$. What is it that makes them ...
It's a problem from 'High Dimensional Statistics' (MIT's lecture notes) (Problem 1.3,page30)Let $X_1,X_2$ ... be an infinite sequence of sub-Guassian random variables with variance proxy \$\sigma_{i}^...