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

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2
votes
1answer
41 views

Bounds on $E[(f(X)-g(X))^2]$

I am looking for good upper bounds on $E[(f(X)-g(X))^2]$. For example, here are two bound that I derived: $$ E[(f(X)-g(X))^2] \le E[4 \max(f(X)^2, g(X)^2)] $$ where I used $(x-y)^2 \le 4 ...
1
vote
0answers
16 views

Finding the conditional distribution of a normal RV given another normal RV

I'm having trouble with this question from a past qualifying exam: Question Suppose $Z \sim N(\mu,\sigma^{2})$, $W \sim N(0,1)$ and $V \sim N(0,1)$ are mutually independent normal random variables. ...
0
votes
0answers
16 views

What is $\operatorname{Var}(x^y) x$ is normally distributed random variable?

I am trying to understand the $\operatorname{Var}(x^y)$. I thought that it is $E(x^2)^y-[E^2(x)]^n$ but I understand that I was wrong, or wasn't I?
0
votes
4answers
57 views

Conditional distribution of order statistics

Let $X_{(1)},...,X_{(n)}$ be the order statistics of a set of $n$ independent uniform $(0,1)$ random variables. Find the conditional distribution of $X_{(n)}$ given that ...
1
vote
2answers
31 views

Change of Variables in Second Moment

$X$ is a non-negative continuous random variable with pdf $f(x)$. $G(t) = \int_{t}^\infty f(x) dx$. Show that $$E[X^2] = 2 \int_{0}^\infty tG(t) dt$$. I tried to write out E[X^2] (second moment of ...
1
vote
1answer
29 views

conditional probability has binomial distribution

Let $X_1, X_2$ be two independent random variables with $X_i \sim \mathrm{Pois}(\lambda)\,$ for $i=1,2$, where $\lambda>0$. Let $k,n \in \mathbb{N}$ and $0\leq k \leq n$. Define ...
0
votes
1answer
63 views

independent random variables probability and measure theory

Let $X$ and $Y$ be independent random variables and suppose that $P(X + Y = c) = 1$, where $c \in \mathbb{R}$. Prove that $X$ and $Y$ are both almost deterministic random variables, that is $P(X = a) ...
1
vote
1answer
26 views

Existence of a random variable given a cdf

For every real function F which can be a CDF (so has the properties that $F(+\infty)=1$, $F(-\infty)=0$, and F is non-decreasing and right continuous), does there exist a random variable on a ...
0
votes
1answer
27 views

Expectation of a function of a normal random variable

Suppose $X\sim\mathcal{N}(0,1)$. I would like to find $\mathbb{E}[\frac{1}{\alpha+\beta X}|A<X<B]$ where $A, \alpha, \beta>0$. How should I go about it? Finally, if the answer is that there ...
5
votes
2answers
33 views

correlation between $\sum_{i=1}^{98}X_i$ and $\sum_{i=3}^{100}X_i$

Let $X_1,...,X_{100}$ be iid $N(0,1)$ random variables. The correlation between $\sum\limits_{i=1}^{98}X_i$ and $\sum\limits_{i=3}^{100}X_i$ is equal to (A) $0$ (B) $\dfrac{96}{98}$ (C) ...
0
votes
0answers
22 views

Order statistics difficult problem

The $n+1$ random variables $X_i$ ($1\le i\le n+1$) are independent and identically distributed with cummulative distribution $F$. Let $Y_k$the order statistics of $X_1,...,X_n$ and let $Z_k$ the order ...
1
vote
1answer
21 views

Exponential order statistics

Let $X_1,...,X_n$ exponential random variables with parameter $\lambda$ and let $X_{(1)},...,X_{(n)}$ the order statistics of the random variables. I know that $X_{(1)}$ is exponential with parameter ...
2
votes
1answer
15 views

An independent sequence of square-integrable random variables with convergent sum of variances converges stochastically

Let $(X_i)_{i\in\mathbb{N}}$ be a sequence of independent and square-integrable random variables with $\operatorname{E}\left[X_i\right]=0$ and ...
0
votes
0answers
50 views

Expected value of a function of truncated normal

I need to find the expected value of the following type of an expression: $$\mathbb{E}[\frac{1-\alpha}{1-\alpha-\frac{X}{\beta}}]$$ where $\alpha$ and $\beta$ are constants and $X$ is a random ...
1
vote
3answers
27 views

Let $X$ ~ $Exp(1)$, and $Y$ ~ $Exp(2)$ be independent random variables. Let $Z = max(X, Y)$. calculate $E(Z)$

Here's a question I'm trying to solve: Let $X$ ~ $Exp(1)$, and $Y$ ~ $Exp(2)$ be independent random variables. Let $Z = max(X, Y)$. calculate $E(Z)$ I'm can't understand how to deal with ...
0
votes
2answers
25 views

If $X_1,X_2…$ are independent, uniform random variables do there exist an infinite amount of $Y_n = X_nX_{n+1} < \frac{1}{8n}$?

If $X_1,X_2...$ are independent, uniform random variables on the interval $[0,1]$, do there exist an infinite amount of $Y_n = X_nX_{n+1} < \frac{1}{8n}$? I want to use the Borel-Cantelli lemma, ...
1
vote
0answers
9 views

Conditional probability in bivariate distribution

Suppose I have a multivariate normal distribution $N$ for the continuous multivariate RV $X = (X_1, X_2)^T$. It is clear that it has to hold $\int N = 1$ i.e. probabilities sum up to one. Suppose I ...
1
vote
2answers
44 views

Distribution of a random measure is determined by the characteristic function

I ham trying to understand a proof from a book I am reading. It says the proof follows directly from the prior theorem and I just can't see that. Let $X$ be a random measure on a locally finite, ...
2
votes
0answers
56 views

Tail bound on the sum of independent identical geometric random variables

Suppose $X_1,\ldots,X_k$ are k independent geometric random variables with the same success probability and let $X=X_1+\cdots+X_k$. Hence $E[X_i] = 1/p$, the expected number of trials needed is ...
2
votes
1answer
14 views

Expected value complex random variable

I want to check that if $X: \Omega \to \mathbb{C}$ is a random variable, then the inequality $| \mathbb{E} X| \le \mathbb{E} |X|$ also holds like in the real case. We can write $$X = \Re X + i \cdot ...
0
votes
0answers
7 views

Understanding definition of multivariate RVs

Wikipedia defines a multivariate RV as a [...] column vector $X = (X_1, ..., X_n)^T$ whose components are scalar-valued random variables on the same probability space $(\Omega, F, P)$ where ...
0
votes
1answer
13 views

Borel measurable function, Borel set, probability

Show that for two independent random variables $X: \Omega \to \mathbb{R}^m, Y: \Omega \to \mathbb{R}^n$, Borel measurable function $g: \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}$ and a Borel set ...
0
votes
1answer
34 views

Integrable variable $X$ , integral of the expected value of $|X|$ over a small set $A$

Let $(\Omega, \Sigma, P)$ be our probability space. Prove that a random variable $X$ is integrable, that is $\mathbb{E}(X) < \infty$ $\iff$ $$\forall \varepsilon >0 \exists \delta>0 : ...
1
vote
1answer
24 views

Expected value, borel measurable function, dependent variables

We are given two random vectors: $X: \Omega \to \mathbb{R}^m$ and $Y: \Omega \to \mathbb{R}^n$ - not necessarily independent, and a Borel measurable function: $g: \mathbb{R}^m \times \mathbb{R}^n \to ...
1
vote
1answer
84 views

3 urns, each with 4 balls. select one ball from each

Three urns are labeled $1,2,3$. Each urn contains $4$ balls labeled $1,2,3,4$. A ball is drawn from each urn such that any ball is equally likely to be drawn. The number on the ball is compared to the ...
2
votes
3answers
94 views

Expected value of number of sorted elements in a permutation

Consider the obvious algorithm for checking whether a list of integers is sorted: start at the beginning of the list, and scan along until we first find a successive pair of elements that is out of ...
1
vote
1answer
109 views

Convolution of Uniform Distribution and Square of Uniform Distribution

I am trying to find the CDF of $Z=X+Y$ whereby $X$ and $Y$ are random variables. Given that the CDF of Z is: $$F_Z(z)=\int F_X\left(z-y\right)f_Y(y)dy$$ Given that $X$ is uniform distribution over ...
0
votes
0answers
14 views

Multiple Items Probability

Right, I have programmed a system that selects items and decides which ones to drop depending on how likely it is to be selected. I have 10 items, each with a 10% chance, however only 5 items can ...
0
votes
2answers
26 views

Inverse of a Poisson distribution function

I have two i.i.d random variables $X_{1}$ and $X_{2}$ following a continuous Poisson distribution function $P(x) = \lambda e^{-\lambda\cdot x}$. I wish to obtain a distribution function of sum of ...
0
votes
1answer
39 views

Deriving the formula for transforming random variables

I try to obtain the formula of transforming the probability density function of a random variable $X$, into the probability density function of another variable $Y$, which are related with a ...
2
votes
1answer
27 views

How to show the sum is a martingale

In the hypothesis of the martingale central limit theorem, my book says that given a sequence of random variables $X_n$ with the condition that $E(X_n \mid \mathcal F_{n-1}) = 0$, then $S_n = ...
0
votes
0answers
27 views

Given random triangle with side's length of X, Y, are random variables such that $X, Y$ ~ $U[0, 1]$. Calculate triangle's area variance.

I've got the following answer: Given random triangle with side's length of X, Y, are random variables such that $X, Y$ ~ $U[0, 1]$. Calculate triangle's area variance. Can you please tell me the ...
0
votes
1answer
17 views

how can I get the marginal distribution of Y when Y|X=x~Poisson(x)?

X~Uniform[0,1] and Y|X=x~Poisson(x). Since $f_{Y|X=x}=\frac{f(x,y)}{f_x(x)}$, then $f(x,y)=f_x(x)f_{Y|X=x}(y|x)$ and $f_y(y)=\int_{-\infty}^{\infty}f_x(x)f_{Y|X=x}(y|x) dx=\int_{0}^{1}(1)\frac{x^k ...
3
votes
1answer
39 views

Finding Limits and Its Convolution of Weighted Summation of Random Variables

I am trying to find the CDF of $Z=aX+bY$ whereby $X$ and $Y$ are random variables and $a$ and $b$ are positive integers. Given that the CDF of $Z$ is: $$F_Z(z)=\int ...
0
votes
1answer
29 views

Why the mean value of a Gaussian process is usually set to zero?

In most textbooks (e.g. Rasmussen's book on Gaussian Processes for Machine Learning) the mean value of a gaussian process is set to zero. Of course, this does not mean that all the values are expected ...
1
vote
0answers
31 views

How to predict a sequence from random input sequence.

I have $6$ sequences $S_1$, $S_2$, $\ldots$, $S_6$, with each sequence containing $60$ data points made of the numbers $\{0,1,2\}$. e.g. $S_1 = [0, 1, 1, 2, 0, 0, \ldots, 0, 0, 0, 1]$. These numbers ...
0
votes
1answer
21 views

PDF & CDF of a Sum of Weighted Independent Random Variables $Z=aX+bY$

From this question here, I learned that the Cumulative Distribution Function (CDF) of $Z=X+Y$ is: \begin{eqnarray*} F_Z \left( z \right) & = & \int F_X \left( z - y \right) dF_Y \left( y ...
0
votes
0answers
32 views

Interpretation Of 4th-Order Cumulants For Complex Random Variables

I asked about this over at the DSP site several days ago but have not gotten any responses. I'll replace the word "signal" with "random variable" and hope that someone from a pure math background can ...
0
votes
1answer
36 views

Moment generating function related proof

Assuming that a continuous random variable X has a density denoted by $f_X$. Show that if the k-th moment of X exists (for k ∈ N+), that is E(|X|$^k$) < ∞, then E(|X|$^s$) < ∞ for all 0 < s ...
1
vote
1answer
38 views

Deriving the autocorrelation function for the ARMA model

Definitions The ARMA model $$x_n=-\sum_{p=1}^P a_px_{n-p}+\sum_{q=0}^Qb_qw_{n-q} \tag{1}$$ where $w_n$ is zero mean stationary white noise with unit variance. Question To derive the ...
-1
votes
1answer
37 views

Does the probability commute with limit?

Does the probability commute with limit? For example, is it true that for $(N_t)_t$ random variables which take values in the set of natural numbers, $$ \mathbb{P}(\underset{t \rightarrow 0}{lim} ...
2
votes
1answer
39 views

Yule walker equation limited matrix size

Definitions For an ARMA model $$x_n=-\sum_{p=1}^P a_px_{n-p}+\sum_{q=0}^Qb_qw_{n-q} \tag{1}$$ where $w_n$ is zero mean stationary white noise with unit variance. It is straightforward to show that ...
0
votes
2answers
60 views

How to determine if a random variable is $\mathcal F$-measurable?

For example : Consider the state space $\Omega = \mathbb{R}$, the $\sigma$-algebra, $\mathcal{F} = \{(-\infty, 0], (0, \infty), 0, \mathbb{R}\}$ and the random variable $X : \Omega \rightarrow ...
1
vote
3answers
27 views

Discrete random variable. Tossing a coin.

Two coins are simultaneously tossed until one of them comes up a head and the other a tail. The first coin comes up a head with probability $p$ and the second with probability $q$. All tosses are ...
1
vote
1answer
35 views

Probability to iteratively and independently remove $n$ elements until all gone

The problem is as follows: Let S be a set of n elements. At the first stage each element in S is in- dependently removed with probability p. Those elements not removed constitute the set S1. ...
-1
votes
1answer
29 views

A question involving independent random variables and indicator random variables [closed]

Let $X$ and $Y$ be two independent random variables. It is true that $1_{X > 0}$ and $1_{Y > 0}$ are independent? Why yes / not? Thank you!
1
vote
1answer
18 views

compute conditional expectation with respect to sigma algebra

I'm studying stochastic and I'm stuck at the following problem: I ask myself how to compute this conditional expectation: Let $X$, $Y$ be two independent random variables in $L^1$. What is ...
0
votes
0answers
27 views

Independence of random variable through bitwise rotation

Say, I have a property that requires two $w$-bit random variables $X_1$ and $X_2$ to be independent and uniformly distributed when I perform addition modulo $2^w$. Now, instead of taking two ...
1
vote
1answer
32 views

When can I switch sup and functions

I have a sequence of random variables $X_k$. Under which conditions can I say that $$f\left(\sup_{j \le k} X_k\right) = \sup_{j \le k} f(X_k)$$? Would having $\limsup X_k = \lim_{k \to \infty} ...
0
votes
1answer
76 views

$E[x_i^2 x_j^2]$ for white Gaussian noise

If $x_n$ is a discrete time random signal and is white Gaussian noise (ergodic and WSS) so $$E[x_n x_{n+l}]=\sigma ^2 \delta (l)$$ and $$E[x_n]=0$$ Where $n \in \mathbb{R}$ and $l\in\mathbb{R}$ ...