Questions tagged [random-variables]
Questions about maps from a probability space to a measure space which are measurable.
0
votes
1answer
25 views
How to calculate E[Xi Xj]?
This question is from an example in the book of Bertsekas. (p240 of 1st edition). I would like to know why $$E[X_{i} X_{j}] = P(X_{i} = 1\text{ and }X_{j}=1)$$ and $$E[X_{i}] = P(X_{i}=1)$$. please ...
1
vote
1answer
17 views
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 ...
0
votes
0answers
19 views
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 $X_t$ ...
1
vote
1answer
30 views
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 ...
0
votes
1answer
35 views
Find two random variables X and Y such that P(X<Y)=2/3
The problem is in the title (sorry if that's not kosher) but I have absolutely no clue how to get started. What would be a good approach to take with this one? I don't believe I have even seen a $\Bbb ...
0
votes
2answers
32 views
How to use the law of total variance
I know that the law of total variance states
$$Var(X)=\Bbb E[Var(X|Y)]+Var(\Bbb E[X|Y])$$
But how does one treat $Var(X|Y)$ and $\Bbb E[X|Y]$ as random variables? For example, say we know that $$\Bbb ...
0
votes
1answer
24 views
Finding expected value when conditional distribution is known
If the distribution of $Y$ conditional on $X=x$ is known, and the distribution of $X$ is known, what would be the general process for finding the expected value $\Bbb E[Y]$? IS there a general ...
0
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0answers
31 views
Is random variable a way to search for actual input numbers?
The formal definition of a random variable is
Random variable over a sample space is a function from sample space to
$R$
I want to get intuitively what actually it is doing. In this context, I ...
3
votes
2answers
77 views
What is the formalism that allows Random Variables to be treated algebraically like real or complex numbers?
We all know that if we have a variable x, then there is a meaning to - for example -
$$y=e^x$$.
And we all know how to manipulate that algebraically and to do calculus. For example, if
$$y_1=e^{...
0
votes
1answer
29 views
Is the sum of two (non-real) random variables necessarily a random variable?
Please note that I'm working with the following definition of random variable, which allows for a codomain other than $\mathbb{R}$.
Definition: Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a ...
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0answers
22 views
Proving expectation and variance of a function of a random variable tends to a fix point
Given $f:\mathcal{X} \rightarrow \mathbb{R}$ is a continuous function and $\mathbb{E}_{Q(X)}[X] \rightarrow x^\star$ ($x^\star$ is a fix number), $\mathbb{V}\text{ar}_{Q(X)}[X] \rightarrow 0$. How can ...
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votes
0answers
23 views
Bound for the expectation of a function of a Gaussian random vector
A paper I am currently reading seem to be using the bound
$$ E_h\left[\inf_{\|z\|_{\infty}\leq t} \|z-h\|^2\right] \leq n\frac{1}{\pi}\frac1t e^{-t^2/2},$$
where $n$ is the dimension of the vector, ...
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votes
0answers
14 views
Proving convergence of expectation and variance given Rényi's $\alpha$-divergence tends to 0
I denote $p, q$ as density function of $P, Q$.
Given $Y, X$ are random variables and
\begin{align}
\int q(x)\mathbb{D}_{\alpha}[p(Y\mid X=x)\,||\,p(Y\mid x^\star)] \,dx \rightarrow 0
\end{align}
...
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votes
0answers
13 views
range of correlation of the linear combination of two random variables
$X_1, X_2, Y$ are normal random variables,
If we know $\operatorname{corr}(Y, X_1) = \rho_1$, $\operatorname{corr}(Y, X_2) = \rho_2$, $\operatorname{corr}(X_1, X_2) = \rho_3$
What's the range of the ...
2
votes
3answers
85 views
Why is $y$ separated into two intervals?
So, here's a question and a solution to part b). I do not understand why they make $y^{1/2}$ belong to interval $[0,1)$ and then separately to the interval $[1,3)$.
-1
votes
0answers
21 views
Joint PDF of two dependent exponential random variables [on hold]
I need help in following
Let's assume we have two identically distributed exponential random variables $X$ and $Y$. What will be the joint pdf of those random variables, i need to consider two cases, ...
-1
votes
2answers
36 views
How to find the mean and variance of minimum of two dependent random variables?
Assume that $X$, $Y$, and $Z$ are identical independent Gaussian random variables. I'd like to compute the mean and variance of $S=\min\{P, Q\}$, where :
$$Q=(X-Y)^2$$ and
$$P=(X-Z)^2$$
Any help is ...
0
votes
0answers
22 views
For which random variable is the spectral density a PDF?
The spectral density of a stationary random process is the inverse Fourier transform of the autocorrelation function of that random process. This spectral density is a probability density function for ...
0
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0answers
19 views
Example of a pair of random variables
I have this example of random variables
let $ \ \ \ f_{XY}( x,y) = \begin{cases}
cx \ \ \ \ 0 \leq x \leq 1 \ , \ 0 \leq y \leq 1 \\0 \ \ \ \ \ other
\end{cases}
$
a) determine the $c$ ...
1
vote
1answer
19 views
Distance of squared random variables and upper bound
I got two sequences of random variables, $(X_n)_n$ and $(Y_n)_n$, and I know that
$| X_n - Y_n | \leq C a_n $, for some constant $C$ or equivalently $|X_n - Y_n | = \mathcal{O}(a_n)$.
Now I want to ...
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votes
0answers
7 views
Point is randomly selected from area, find the mean of abscissa.
my question sounds like this:
Point is randomly selected from area, which is limited with parabola y
= x2 and straight lines y = 0, x = 0.97, x = 2.78. Find this point the mean of abscissa.
This ...
1
vote
0answers
39 views
Expectation of an inner product in an infinite dimensional Hilbert space
Let $\mathcal{H}$ be a Hilbert space with the Borel $\sigma$-algebra. Let $(\Omega, \mathcal{F}, P)$ be a probability space and $x,y$ two $\mathcal{H}$-valued random variables, i.e. measurable maps ...
1
vote
1answer
25 views
Finding the estimator of π(1-π) from a random sample of n Bernoulli trials.
A random sample of n independent Bernoulli trials with success probability π results in R successes.
Derive an unbiased estimator of π (1 − π).
So, from what I understand (correct me if anything I ...
1
vote
1answer
29 views
Finding density function of random variable
Choose an uniformly distributed random variable $U$ on the unit interval $[0,1]$. Then, what is the probability density function of $Y= \ln(U+ 1)$?
I know the density function is the derivative of ...
0
votes
1answer
43 views
Transforming sum of n exponential distribution to a Poisson distribution
Let $X_1,...,X_n$ be i.i.d exponential random variable with mean $\lambda$
$S=X_1+...+X_n$
So by finding the mgf of S, we get that $S \sim \operatorname{Gamma}(n,\lambda)$
The problem I am stuck ...
0
votes
1answer
12 views
Find mean and variance from mgf where t is denominator
For continuous random variable X,
pdf: $f_{X}(x)=2(1-x), x\in[0,1]$
mgf: $M_{X}(t)=\frac{2(e^t-t-1)}{t^2}$
Problem is to find mean and variance from mgf, I tried using $\frac{d}{dt}M_{X}(0)$ and $\...
0
votes
2answers
17 views
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 ...
2
votes
1answer
54 views
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 ...
0
votes
0answers
22 views
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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votes
0answers
14 views
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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votes
1answer
37 views
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 ...
1
vote
1answer
49 views
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 ...
0
votes
0answers
8 views
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 ...
0
votes
0answers
19 views
binomial distribution problem prove or disprove
Let $Y_n\sim B(10, 2p) \wedge 0<p<\frac{1}{20}$.
prove or disprove:
$\lim_{n\rightarrow\infty}p(|Y_n-20p|>\epsilon) = 0$
my try:
$p(|Y_n-20p|>ϵ)=p(Y_n-20p>ϵ)+ p(Y_n-20p≤-ϵ)
=p(Y_n&...
0
votes
1answer
14 views
Existence of random points
Let $\mathbb{S}^k$ be the $k$-dimensional unit sphere and let $\sigma(\mathbb{S}^k)$ be its surface area. Suppose we have a regular area partition $\{S_i\}_{i=1}^n$ of $\mathbb{S}^k$ with constant $c&...
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votes
0answers
42 views
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 ...
1
vote
1answer
31 views
$X$, $Y$ 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, ...
3
votes
2answers
75 views
Expectation of dependent Bernoulli sum
I want to estimate the expected value of the following sum of random variables,
$$ Y = \sum_{i=1}^N X_i $$
where each $X_i$ is a Bernoulli random variable. In particular,
$$ X_1 = \begin{cases} 1, &...
0
votes
0answers
24 views
Find variance D[ξ-2η] when ξ and η are independent
My question is:
Random variables ξ and η are independent and with with equal
probabilities acquires values -3, 0, -3. Find variance D[ξ-2η]
I've tried to solve like this:
But I'm not sure if ...
1
vote
1answer
31 views
Does $|X_n|\le\Delta_n+\delta$, with $\Delta_n\overset{p}\to0$ imply $|X_n|\overset{p}\to0$?
Suppose $\Delta_n\overset{p}\to0$, and for any $\delta>0$, we have
$$|X_n|\le\Delta_n+\delta.$$
Can we conclude that
$$|X_n|\overset{p}\to0?$$
Here $\Delta_n\overset{p}\to0$ means that for any $\...
2
votes
1answer
51 views
Deriving distribution from conditional distribution
Hi guys I am having problems deriving $P(X = k)$ if $P(X = k|X+Y = n)$ = ${n}\choose{k}$ $\times$ $2^{-n} $
X and Y are i.i.d. random variables with values in $\mathbb{N_0}$.
After playing a bit ...
0
votes
1answer
37 views
Statement about convergence in probability
If $X_1, X_2, ..., X_n, ...$ are same distributed and independent random variables and $n \mathbb P (|X_1| > n) \to 0$, $a_n = \mathbb E\left[X_1I_{|X_1| < n}\right]$, does this imply that $X_1+...
0
votes
1answer
38 views
Random variables with Gamma distribution and convergence in probability.
The problem states that $\alpha>0$ and for each $n\in\mathbb{N},\ X_n:\Omega \rightarrow \mathbb{R}$ is a random variable on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with the gamma ...
1
vote
0answers
30 views
Asymptotic distance between distributions in delta method
Let $X_n$ be a sequence of real valued random variables such that $a_n(X_n - \mu) \xrightarrow{\text{d}} X$ for $a_n \to \infty$.
Delta method states that $a_n(g(X_n) - g(\mu)) \xrightarrow{\text{d}} ...
0
votes
1answer
26 views
Does $\xi_{ni}\overset{p}\to0$ imply $\frac1n\sum_{i=1}^n\xi_{ni}\overset{p}\to0?$
Suppose for each $i=1,\cdots,n$, we have
$$\xi_{ni}\overset{p}\to0.$$
Can we claim that
$$\frac1n\sum_{i=1}^n\xi_{ni}\overset{p}\to0?$$
Here $\overset{p}\to$ means convergence in probability. That ...
0
votes
0answers
18 views
Multistage random variable
First we sample from a probability space $\Omega$ and observe the value of $X$, then we sample $Y$ from a probability space $\Omega_X$.
In terms of measure theory, what is $Y$?
Let $X: \Omega \to I$ ...
0
votes
1answer
21 views
About the definition of independent random vectors
I just have a question concerning the definition of independent random vectors.
The random vectors $X=[X_1, \ldots, X_n]$ and $Y=[Y_1,\ldots, Y_m]$ are independent means that $X_1,...,X_n,Y_1,...,Y_m$ ...
0
votes
0answers
22 views
stochastically independence
I never took stochastic courses and need a proof for this task to continue my work at another problem. Can somebody help me out?
Let $X$ and $Y$ be stochastically independent real discrete random ...
0
votes
1answer
22 views
How to efficiently generate random points in a difference of two disks?
Consider two disks: $S$ with radius $R_S$ and position $(x_S,y_S)$, and $M$ with radius $R_M$ and position $(x_M,y_M)$. Let's denote the difference between these two sets as
$$V=S\setminus M.$$
Now ...
0
votes
1answer
32 views
Expected value of multiplication of matrices
While reading through Xu et al. (2016) I stumbled upon this proof:
$$
\begin{align}
\mathbb{E}[Wyy^\intercal W^\intercal] & = \mathbb{E}[Wyy^\intercal W^\intercal-W\mathbb{E}[y]\mathbb{E}[y]^\...
