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Questions tagged [random-variables]

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

A random variable $X: \Omega \to E$ is a measurable function from a set of possible outcomes $\Omega$ to a measurable space $E$. The technical axiomatic definition requires $\Omega$ to be a sample space of a probability triple. Usually $X$ is real-valued.

The probability that $X$ takes on a value in a measurable set $S \subseteq E$ is written as :

$$P(X \in S) = P(\{ \omega \in \Omega|X(\omega) \in S\})$$

where $P$ is the probability measure equipped with $\Omega$.

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Independence and expectational independece?

Let X and Y be two random variables and $E[f_1(X), f_2(Y)] = E[f_1(X)] *E[f_2(Y)]$ then can I conclude that $f_1(X)$ and $f_2(Y)$ are independent?
hanugm
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Bound on the Expectation of the ratio of two random variables not necessarily independent

Suppose we have two sequences of random variables $\{X_n\}$ and $\{Y_n\}$ with $X_n \ge Y_n \ge 1$ such that $\mathbb{E}[X_n - Y_n] < c$ (for some constant $c$). Is it possible $\mathbb{E}[X_n/Y_n]$ is unbounded?
KrypticKoala
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Are the sets of outcomes of the underlying experiment mapped to specific values of a random variable necessarily disjoint?

If a discrete random variable $X$ is defined as a mapping of subsets of a sample space $\mathcal{S}=\{s_1,s_2,s_3\ldots\}$ to particular values $x_i$ which together make up its range $R_X=\{x_1,x_2,x_3,\ldots\}$, then must the mapped subsets be…
Undertherainbow
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Covariance of Increasing Functions, how do I use the hint?

Given that $f,g$ are monotonically increasing functions and $X$ is a random variable, how do I show that $$\mathrm{Cov}(f(X),g(X)) \geq 0? $$ I've seen: covariance of increasing functions and tried to use the hint that given $X,Y$ iid, then…
OneGapLater
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Finding K for a cumulative distribution function.

I have the following CDF for some fixed number $k$: When $x$ is smaller than $1$ then $F(x) = 0$ When $x$ greater or equal to $1$ then $k - \frac{1}{⌊x⌋}$ applies. However, I cannot figure out how to find $k$. I believe that $k$ cannot be larger…
Libor Zachoval
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Variance of a Discrete Random Variable

Can someone help me wrap my mind around this: When calculating the variance of a sample, we divide the squared distance of each data point from the mean by the sample size - 1. But how come we don't need to divide the sum by anything when we are…
Hannah Tang
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discrete random variable PMF - Confusion about alternative answer

I am reading solution for this problem in Dimitri P. Bertsekas's book and I see this question post in this forum as well. "suppose You rented a house and realtor gave you 5 keys, one for each of the 5 doors of house. unfortunately all keys look…
Amos Ku
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Bounds of sum of independant random variables

If X is a random variable with the following densitiy $f_X(x) = \lambda e^{-\lambda x}$ for $ x \ge 0$ else $0$ (Exponential distribution) Y is a random variable with density $f_Y(y) = 1$ if $0 \le y \le 1$ else $0$ (Uniform distribution with bounds…
Stefan B
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Is random variable a way to search for actual input numbers?

The formal definition of a random variable is Random variable over a sample space is a function from sample space to $R$ I want to get intuitively what actually it is doing. In this context, I had the following idea A random variable is a…
hanugm
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Why does the indicator function fulfill the random variable definition?

Why does the indicator function fulfill the random variable definition? Def. of random variable: Pre-image of random variable It's intuitive that the pre-image of $1_A=1, \omega \in A$ is in $F$. Since $A \in F$. However, since $1_A=0$, when $\omega…
mavavilj
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Calculate the expected time spend playing the game. Hint: Use Wald's Equation.

Alice and Bob play each other in a checkers tournament, where the first player to win four games wins the match. The players are evenly matched, so the probability that each player wins each game is $1$ $2$, independent of all other games. The…
mskanyal
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Differentiable homeomorphism between continuous random variables

First, let me apologize if the question is not formulated appropriately. I'm a bit rusty in math lately. Let $X$ be a continuous random variable in $\mathbb{R}^d$, and $N\sim\mathcal{N}(0, \mathbb{1}_d)$ be the standard multidimensional normal…
Álvaro Parafita
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Permutation on a uniform distribution is uniform

Let $f: D \rightarrow D$ be a permutation, and suppose $X$ is uniformly distributed, i.e. $X$ is a uniform random variable with support $D$. Then $f(X)$ is also a random variable with support $D$, and also uniform. How would I formally show or…
user308485
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Random sample and random variables

I am new to the statistics and my problem is that i do not understand what a random sample is. Is it a set of constant numbers, like $X=\{c_{1},c_{2},...,c_{n}\}$. Is it a set of random variables, with every random variable represents the results of…
Laur155 155
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Distribution of exp with Rayleigh arg

I am wondering what the distribution of $$ y = exp(i\phi)$$ where $\phi$ is Rayleigh distributed and $i=\sqrt{-1}$. My thought process to solve was to use LOTUS but I am unsure how to handle the $i$ as the arg into the Rayleigh distribution.
Avedis
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