# Questions tagged [random-variables]

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

7,295 questions
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### Difference of normal random variables

I have two random variables $$X_{s+t} \sim N(0, s+t)$$ $$X_s \sim N(0, s)$$ where $s \leq t$. How do I show that... $$X_{s+t} - X_s \sim N\left(0,s + t + s -2\sqrt{s(s+t)} \right)??$$ I understand ...
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### Quadratic variation of sum od random variables

Let $N = (Nt)_{t>0}$ be a Poisson process and consider random variables $Z_n$, $n \in N$. Compute the quadratic variation $[X]_t$ where $X_t =\sum_{n=1}^{N_t} Z_n$. What I did was plugging $X_t$ ...
1answer
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### Joint PDF transform using jacobian

Seriously I dont have any idea what is this thing called. I know how to find Joint PDf of two variables.. But i dont know how to transform it in other variables ? Do they require Jacobians? Here is ...
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### What is the average number of matches when randomly picking letters

Suppose we have three pieces of paper. On the first one you have the letter A, on the second on the letter B, and on the third one the letter C. Now suppose I'm going to randomly pick each one from a ...
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### Suppose $X_1, \dots, X_n, Y$ are independent random variables. Prove that $X = (X_1, \dots, X_n)$ and $Y$ are independent variables.

Suppose $X_1, \dots, X_n, Y$ are independent random variables. Prove that $X = (X_1, \dots, X_n)$ and $Y$ are independent variables. My attempt: Fix $A \in \mathcal{R}$ (a Borel subset of the real ...
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### Lognormal distributed random variable excercise

Let Y be a random variable distributed Lognormally, Q1 be it's first quartile, Q3 it's third quartile and M be its median: prove that- M - Q1 < Q3 - M I have managed to figure out that Q3 > M > ...
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### Convergence on Geometric distribution [on hold]

Suppose that $Xn$ ∼ $Geo( {λ/n+λ} )$ with $n = 1,2,...$ where λ is a positive constant. Show that $Xn/n$ converges on when n → ∞, and determine the parameter of the limit distribution.
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### Expected value in a linear combination

I have a random variable Y, that is defined by: $$Y = aX_1 + bX_2$$ Where we know $X_1$ and $X_2$ are independent. How do I write out $EX_1$ and $EX_2$ in terms of only a, b, EY, and VarY? I ...
1answer
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### Inserting random numbers from 1 to $n^2$ in a matrix of size $n \times n$

I have two matrices of size nxn with random numbers that are in range of $1$ to $n^2$. I'm trying to calculate the probability of : the numbers 1 and 9 are present in the same indices in the two ...
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### Approximation of mean of a rational function of random variables

Let $\xi_i$ with $i\in\{1,\dots,n\}$ be iid random variables and let $Q(x,y)$ be a rational function. I need to compute one $x$ that satisfies $$\frac{1}{n}\sum_{i=1}^n Q(x,\xi_i)=0.$$ This is a ...
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### Understanding i.i.d. random variables from product measure space perspective

I have a very weak background in measure theory, and I am having some troubles understanding i.i.d. random variables from a measure theoretic perspective. Let $X$ be a random variable defined on a ...
1answer
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### $X$, $Y$ i.i.d r.v's. Prove that $\mathbb{E}[X\mathbb{1}_{\{X+Y \in B\}}] = \mathbb{E}[Y\mathbb{1}_{\{X+Y \in B\}}]$

Let $X, Y$ be i.i.d random variables with finite expected values. I want to justify that $$\int_{\{x+y \in B\}}x\mu(dx)\mu(du)=\int_{\{x+y \in B\}}y\mu(dx)\mu(du).$$ I would appreciate any hints, ...
2answers
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1answer
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### The probability of two samples of a waveform to be separated by certain amount

I am trying to mathematically model the probability of observing a waveform crossing based on two of its samples. Im particular, I want to know the effect of the sampling frequency and phase on this. ...