Questions about maps from a probability space to a measure space which are measurable.
0
votes
0answers
17 views
Generate Constrained Vector of Random Numbers?
I'm having trouble creating a random vector $\vec{V}$ starting with a standard 0:1 randon number generator subject to the following set of constraints: (given parameters $D$, $L$, and $\theta$)
The ...
1
vote
2answers
114 views
how to prove that $\Bbb E(XY) =\Bbb E(X) \Bbb E(Y)$?
I have to prove that $\Bbb E(XY) =\Bbb E(X) \Bbb E(Y)$, for any pair $X, Y$ of independent random variables on $(\Omega, \mathcal{F}, P)$ which are in $L^1(\Omega)$.
I would say that the claim is the ...
0
votes
1answer
45 views
Sequence of independent random variables, mean = 0
Can someone give me a sequence of independent random variables (or an example of it, with explanation, if possible) with mean 0 such that:
$\frac{1}{n} \sum X_i \rightarrow - \infty $
Thank you.
1
vote
1answer
92 views
Almost Sure Convergence in $L^{p}$
Let $(X_n)_{n\geq 1}$ be a sequence of i.i.d. random variables, on the same probability space, with law given by $\displaystyle \mathbb P(X_1=(-1)^{m}m)=\frac{1}{(cm^2\log m)}$ for $m\geq 2$ where $c$ ...
1
vote
1answer
44 views
Inequality between two Random Walks
Let's consider two Random Walks,
$$x^{(1)}_t = x_0 + \sum_{i=1}^{t}\xi^{(1)}_i,$$
$$x^{(2)}_t = x_0 + \sum_{i=1}^{t}\xi^{(2)}_i.$$
The random variables $\xi^{(1)}_i$ are i. i. d. They take values on ...
1
vote
2answers
51 views
Definition of atomic $\sigma$-field.
Reading an article in probability theory I faced with phrase atomic $\sigma$-field. I tried to search for the definition, but google doesn't give any meaningful result. As a result I'm looking for the ...
3
votes
0answers
60 views
Expected value with a kronecker product and Gaussian distributional assumption
What is the expected value, $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I) $ is a random variable? The kronecker product ...
0
votes
0answers
48 views
How to prove that $Y=\ln(X)$ approximately Normal when $X$ is a Normal random variable with $\mu\gg\sigma$
I wanted to prove that PDF of $Y=\ln(X)$ tends to a Normal distribution with $\mathcal{N}(\ln(\mu_{x}),\sigma^{2}_{y})$ when $X\sim\mathcal{N}(\mu_{x},\sigma^{2}_{x})$. It is also important to note ...
0
votes
0answers
24 views
Calculation of the error function.
I have the next two signals:
$X(t)$ and $G(t)$ and a random process $Y(t)=G(t)X(t)$ where $X(t)$ and $G(t)$ are wide sense stationary with expectation values: $E(X)=0, E(G)=1$.
Now, it's also given ...
3
votes
2answers
72 views
Kolmogorov's maximal inequality and convergence of random series.
Let $(X_n)_{n\ge 1}$ be a sequence of mutually independent random variables, on the same probability space, with expectation 0 and finite variance. Let $S_n = \sum_{l=1}^n X_l$. Prove that for any ...
2
votes
1answer
49 views
Hitting time level and Bernoullis
Let $(X_n)_{n\ge 1}$ be a sequence of i.i.d. Bernoulli random variables, on
the same probability space, with parameter $1/2$, and let $\tau_n$ be the hitting time of level n by the partial sums, i.e. ...
-2
votes
1answer
30 views
Statistics Binomial with Probability Distribution function
Let $X$ be a binomial random variable with $n=2, θ=\frac14$. Find the probability distribution function of $Y=(X^2)+2$.
0
votes
1answer
52 views
Does Not Converge in Probability?
Let $\left(X_n\right)_{n\geq 1}$ be a sequence of i.i.d. real random variables, with $\mathbb E(X_1)=0$, var$(X_1)=1$. Let $S_n=X_1+\cdots+X_n$.
Prove that $\displaystyle ...
0
votes
1answer
46 views
Discrete random variable with infinite expectation
Consider a discrete random variable taking only positive integers as values with
$$\mathbb{P}[X=n]=\frac{1}{n(n+1)}.$$
(a) Show that $\mathbb{E}[X]=\infty$.
(b) Show that $\mathbb{P}[X ...
1
vote
1answer
53 views
Law of Large Numbers Deduction
Let $\left(X_n\right)_{n\geq 1}$ be i.i.d random variables on $\left(\Omega,\mathcal A, \mathbb P\right)$, $X_1$ with mean $\mu$, and $$
L(\lambda) =
\begin{cases}
\log\mathbb E\left(e^{\lambda ...
1
vote
1answer
51 views
Large Deviations Question
Let $\left(X_n\right)_{n\geq 1}$ be i.i.d random variables on $\left(\Omega,\mathcal A, \mathbb P\right)$, $X_1$ with mean $\mu$, and
$$
L(\lambda) =
\begin{cases}
\log\mathbb E\left(e^{\lambda ...
0
votes
0answers
17 views
How to test whether there is an association between two data fields by testing a hypothesis?
The table below cross classifies Education by Employment Confidence and is
based on a sample 1363 randomly selected adult respondents in China.
Highest degree Employment Confidence Total
...
0
votes
0answers
32 views
Using an appropriate hypothesis to test whether two means are different
Manager examined potential differences between two models of bicycles. The
mean life of the bicycles is of primary concern. The followings table provides the
available date which measured in ...
0
votes
1answer
46 views
if RVs X and Y are indicators of independent events, does that imply their complements do too?
$ p(X⋂Y)=p(X)p(Y)=>p(X^c⋂Y^c)=p(X^c)p(Y^c)?$
im having trouble deciding weither the above statement is true or not, my intuation is that its true, can any one prove or contradict it?
btw, this is ...
4
votes
1answer
58 views
Subsequence of Sequence of Random Variables and Convergence in Probability
Let $\left(X_n\right)_{n\geq 1}$ be a sequence of i.i.d. real random variables, with $\mathbb E(X_1)=0$, $\operatorname{var}(X_1)=1$. Let $S_n=X_1+\cdots+X_n$.
Prove that for any subsequence ...
2
votes
1answer
47 views
Sum of Sequence of Random Variables
Let $\left(X_n\right)_{n\geq 1}$ be a sequence of i.i.d. real random variables, with $\mathbb E(X_1)=0$, $\operatorname{var}(X_1)=1$. Let $S_n=X_1+\cdots+X_n$.
Prove that for any $A>0$, ...
2
votes
2answers
52 views
Can we simplify an expression of random variables? (can we treat random variables as real numbers?)
Suppose that we have an expression of random variables including $X-X$ or $2X-X$ or $XY-XY$ and so on. can we treat random variables as real numbers? That is, can we delete $X-X$ or replace $2X-X$ by ...
1
vote
1answer
37 views
Proof about how to get a uniform random variable from a generic one
Consider real random variable $X \in \mathbb{R}$. I know that if I consider r.v. $U = F_X(X)$ where $F_X(x)$ is $X$'s CDF, we get a uniform r.v. in $[0,1]$. So the following holds:
$$U \sim ...
0
votes
1answer
47 views
Standard deviation of function of two RVs
I've stumbled upon a problem that basically reduces to having two random variables
$$X \sim N(\mu_X,\sigma_X)$$
$$Y \sim N(\mu_Y,\sigma_Y)$$
and defining the third as
$$Z = \sqrt{X^2 + Y^2}$$
Although ...
0
votes
1answer
44 views
What is the correlation function in multivariable/vectoral case?
I know that the correlation function between random variables $X$ and $Y$ is defined as
$$
\rho_{X,Y}=\mathrm{corr}(X,Y)={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ={E[(X-\mu_X)(Y-\mu_Y)] \over ...
1
vote
1answer
32 views
$X_n \overset{a.s.}{\longrightarrow} X$ and $X_n \overset{L^1}{\longrightarrow} Y$ implies $X = Y$ a.s.?
If I have a sequence of random variables $\{X_n\}_{n \geq 0}$ such that
$$X_n \overset{a.s.}{\longrightarrow} X \quad\textrm{and}\quad X_n \overset{L^1}{\longrightarrow} Y$$
then is it always true ...
0
votes
1answer
37 views
Question regarding random variable in product probability space
I am struggling at solving product probability space questions, I am wondering if anyone could me with the following question. Let $x_{i}$ be a random variable at probability space ($X_{i}$, ...
1
vote
1answer
54 views
Maximum/minimum of two random variables is a random variable
Suppose $X,Y$ are random variables. I'm trying to understand why $\max\{X,Y\}$ and $\min\{X,Y\}$ are also random variables. The proof in the book that I'm using states that for each $t$,
$\{ ...
1
vote
1answer
35 views
Variance of Matrix Trace
Given a random variables $X \in \mathbb{R}^n$, and a constant real matrix $Z$, how can the variance given by $Var[Tr(ZXX^T)]$ be calculated? Note that $Z$ is p.s.d and $X$ is $N (0,C)$.
0
votes
0answers
49 views
Property of Sum of Random Variables
Let $\left(X_n\right)_{n\geq 1}$ be a sequence of i.i.d. real random variables, with $\mathbb E(X_1)=0$, $\operatorname{var}(X_1)=1$. Let $S_n=X_1+\cdots+X_n$.
Prove that for any $A>0$, ...
1
vote
0answers
53 views
Definition of $x\left<Y\right>$ notation in probability theory
I am working on the basics of probability theory in Koller's Probabilistic Graphical Models - Principles and Techniques. Unfortunately I am having trouble understanding a formal definition (possibly ...
2
votes
1answer
40 views
Independence of two products of random variables
Consider the following problem:
$$z_1 = a_1 x_1$$ $$z_2 = a_2 x_2$$ where $a_1, a_2$ are i.i.d. (regardless of their distribution; in the actual case study it is a symmetric Bernoulli distribution ...
0
votes
0answers
14 views
Distinct objects and random variable problem [duplicate]
so this what I got for this question I dont think i am correct can someone help me out
-1
votes
3answers
73 views
Variance of transformed random variable
The relationship of two random variables is given by
$$ X = \Phi(Y) \Leftrightarrow Y = \Phi^{-1}(X),$$
where $\Phi(\bullet)$ is the standard normal cdf and $\Phi^{-1}(\bullet)$ the inverse of the ...
0
votes
0answers
18 views
is there a Kalman filter for distribution function?
The standard Kalman filter uses a series of measurements observed over time, to decomposite the signal and noise.
However, when I'm modeling the distribution (pdf or cdf) of a variant, is there a ...
3
votes
2answers
53 views
Let $\{X_n\}$ be i.i.d integrable r.v.s, show that $\frac{1}{n}\max_{1\leq j\leq n}|X_j|\to 0 \quad \mbox{a.e.}$
This problem is an exercise in Probability theory,independence,interchangeable, martingale(Chow), exercise 4.1.10.
Let $\{X_n,n\geq 1\}$ be independent identical distributed integrable random ...
0
votes
1answer
54 views
How is conditional density function with two given conditions ($f_{X\mid Y,Z}(x\mid y,z)$) defined?
Let $X$, $Y$ and $Z$ be random variables.
Given this conditional density function with two conditions; $Y=y$ and $Z=z$:
$$ f_{X\mid Y,Z}(x \mid y, z) = f_{X\mid Y,Z}(x \mid Y=y, Z=z) $$
I have a ...
1
vote
3answers
89 views
Let $ X_1,X_2,…,X_n$ be i.i.d. $N(\theta_1, \theta_2)$, please prove that $E[(X_1-\theta_1)^4] = 3\theta_2^2$
If $X_{1}$, $X_{2}$, ..., $X_{n}$ is sampled from $N(\theta_1, \theta_2)$, how can I prove that $E [(X_{1} - \theta_1)^{4}] = 3 \theta_2^{2}$?
I started off this question finding the completely ...
2
votes
0answers
21 views
Fast fourier transforms of random binary data
I am a physicist who is trying to make sense of FFTs and binary data.
Say I have a series of random binary data, which is measured with a repetition rate of 400Hz (interval time of 0.0025s). I have a ...
0
votes
0answers
31 views
Inequality for expected values
Let $x=(x_1, \ldots, x_n)$ be real valued vector. Let $\pi(\cdot)$ be a permutation on the set $\{1, \ldots, n\}$ with a uniform distribution.
Prove the following inequality
$$
E \left|\sum_{i=1}^n ...
0
votes
2answers
35 views
Using join probability distribution
Say I'm given a probability distribution of two random variables $A$ and $B$. What does it mean to calculate the join probability distribution of $3^{(A-B)}$?
The distribution is in fact discrete.
0
votes
2answers
84 views
probability density function with Gaussian distributed random variables
Create $100$ samples each of two Gaussian distributed random variables $X$ and $Y$ of your choice. Form a random variable $Z$ according to $z= \sqrt{x^2+y^2}$. Using these samples, estimate and plot ...
2
votes
4answers
205 views
Convergence in probability of the product of two random variables
Suppose $\{X_n\}$ and $\{Y_n\}$ converge in probability to $X$ and $Y$, respectively.
Will $X_n Y_n$ converge in probability to $X Y$?
I know the answer is yes.
If we treat $(X_n,Y_n)$ as a random ...
1
vote
1answer
119 views
Calculate the probability density function of $Y = 2 X + 3$
Let $X$ be normal with mean 1 and variance 4. Let $Y = 2X + 3$.
(a) Calculate the probability density function of $Y$.
(b) Find $P(Y \geq 0)$.
-1
votes
1answer
73 views
Find the median of the exponential random variable with parameter λ
The median of a random variable X is a number µ that satisfies
Find the median of the exponential random variable with parameter λ.
-2
votes
1answer
113 views
If X is a continuous random variable which takes non-negative value, prove that the expectation of X can be calculated using the following integral
If X is a continuous random variable which takes non-negative value, prove that
the expectation of X can be calculated using the following integral:
2
votes
1answer
37 views
Do Convergence in Distribution and Convergence of the Variances determine the Variance of the Limit?
Suppose we have a sequence $(X_n)_{n\in\mathbb{N}}$ that satisfies:
$X_n \rightarrow_d X$, for $n\rightarrow \infty$, where $\rightarrow_d$ denotes convergence in distribution;
$\mathrm{Var}(X_n)$ ...
4
votes
2answers
126 views
Random sum of random variables
Say you sum i.i.d. variables $X_i$ a total of $Y$ times. If you know the distribution of random variables $Y$ and $X_i$, what is the calculation you have to do to get the distribution of the sum?
0
votes
2answers
60 views
Cumulative distribution function & expectation
Let a be a real number and f:
$$f:\mathbb{R}\rightarrow \mathbb{R}, \ f(x) = \begin{cases} a3^x & \text{for } x < 0\\ 1& \text{for } x =0 \\ a3^{-x} & \text{for } x > 0\end{cases}$$
...
1
vote
1answer
76 views
$\lim \sup\{X_n\geq x\}$ vs $\{\lim \sup X_n \geq x\}$
Let $(X_n)$ (n is a natural number) be a sequence of real valued random variables.
For any real number $x$, let's define:
$E_x = \limsup \{ X_n \geq x\} $, $F_x = \{\limsup X_n \geq x\} $
If $x$ is ...
