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

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0
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1answer
28 views

Probability that one normal Random Variable will fall within a given range of another.

I'm struggling with the following problem: (ed: Don't be lazy. Just type it out. ) A certain small freight elevator has a max. capacity $C$, which is Normally distributed, with mean ...
0
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0answers
15 views

Seeking for an example of Bayesian estimation of two unknown parameters

I searched the web, taking advantage of several search approaches; however, due to redundancy of the existing information about Bayesian estimation of one unknown parameter of random variables (either ...
0
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0answers
11 views

Probability distribution of derivative of function of random variable

The calculation of probability distribution of a function of random variable is a well established theory and there are general rules on how to go from the distribution of r.v. to the distribution to ...
1
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0answers
9 views

Lower bound for (function of) density of well-behaved random variable

Suppose we have a non-negative random variable $\tilde{\theta}$ such that $\mathbb{E}\tilde{\theta} = a > 0$, with finite variance $\sigma^2$. Let its CDF be given by $F(\theta) := ...
2
votes
2answers
399 views

Probability of inequality between random variables

In order to prove a theorem in my research, I would like to use a lemma on basic probability theory, but I don't know if it is correct. For three random variables $X,Y$, and $Z$ not necessarily ...
1
vote
1answer
22 views

Rewriting probabilities as expectation

Consider the stopping time $\tau_a:=\lbrace{t>0| W_t >a\rbrace}$, where $W_t$ is a Brownian Motion. Define: $X_t:=W_{\tau_a+t}-W_{\tau_a}$. We have that $X_t$ is a Brownian Motion independent ...
1
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1answer
37 views

Continuous distribution and independence [on hold]

Problem: In a room, there are 4 boys from high income families, 6 girls from high income families and 6 boys from low income families. How many girls from low income families also need to be present ...
0
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1answer
36 views

If independent r.v. converge in probability to a constant, do they converge almost surely?

I've seen several examples when a sequence of r.v. converge in probability but not almost surely, yet none of them had the sequence to be independent. Would additional conditions of independence and ...
0
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0answers
22 views

Is it possible to exchange a sum in a conditional expectation

Let $X_1, X_2, \ldots \geqslant 0$ and $Y$ be RVs over $\mathbb{R}^n$. Then is it true that $\mathbf{E} \left[ \sum_{i = 1}^{\infty} X_i \mid Y \right] = \sum_{i = 1}^{\infty} \mathbf{E} [X_i \mid ...
0
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2answers
46 views

what is the expected value of $x^TAx$? [on hold]

Assume $x\in \mathbb{R}^N$ is a random variable vector (like a noise sequence). You now want to calculate the following term: $E\{x^{T}Ax\}$, where $A$ is a constant matrix. How can this expression ...
0
votes
2answers
49 views

Prove (or disprove) that $\mathbb{E}[X]\geq 0$ for positive random variable.

Let $X$ be a random variable such that $X\in[0,1]$. I was wondering if $\mathbb{E}[X]$ must be $\geq0$. Since $X$ is a positive random variable, we can apply the Markov-inequality: for each positive ...
0
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2answers
34 views

Eigenvalues of $\mathbb E\pmatrix{2X&X\\ 1-X&2X}$. [on hold]

Let $X$ be a random variable between $0$ and $1$, such that: $\mathbb{E}[X]=\frac{1}{2}$. We have a matrix: $$A=\left( \begin{array}{cc} 2X & X \\ 1-X & 2X \\ \end{array} \right)$$ ...
3
votes
2answers
70 views

If the variance is $0$ is it constant?

We know that the variance of a constant is $0$. Is the converse also true? Can we say that if the variance of some random variable is $0$ it is a constant?
0
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0answers
8 views

RBF transformation on a Normally Distributed Random Variable

I have a random vector $\mathbf{X} \sim \mathcal{N}(\mathbf{m,\Sigma})$ which is transformed by a Gaussian Radial Basis Function into the random variable $\mathbf{Y} = K(\mathbf X)$ where $K = ...
-3
votes
1answer
21 views

Uniform distribution and real values [on hold]

If the random variable $k$ is uniformly distributed in $(0,5)$, What is the probability that the roots of the equation $4x^2+4xk + k + 2 = 0$ are real?
3
votes
2answers
89 views

Given a variable $X$ with a PDF, what is the PDF of $\sqrt{X}$

I feel this is simple and I'm overlooking something really basic. Let's say a have a variable $x$ which obeys the exponential distribution. So if collect 100000 occurrences of $x$ and plot its ...
1
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2answers
66 views

Distribution of Summation of two discrete random variables

Here, $X$ and $Y$ are two non-negative independent discrete integer-valued random variable and the support set of $k_1$ and $k_2$ are $ \{ 2,3,...,7 \}$ and $ \{ 5,6,...,12 \} $ respectively. We ...
0
votes
2answers
203 views

If X,Y and Z are independent, are X and YZ independent?

If yes: I know that f(X) and g(Y) are independent if X and Y are independent and f and g are "measurable".* If that is to be used, is g(Y) = YZ measurable? If not, how else to approach this? If ...
4
votes
6answers
61 views

Finding $P(X < Y)$ where $X$ and $Y$ are independent uniform random variables

Suppose $X$ and $Y$ are two independent uniform variables in the intervals $(0,2)$ and $(1,3)$ respectively. I need to find $P(X < Y)$. I've tried in this way: $$ \begin{eqnarray} P(X < Y) ...
0
votes
1answer
22 views

Probability involving a moment generating function

Suppose that X1 and X2 are independent and identically distributed discrete random variables. The moment generating function of X1 + X2 is: M(t)= 0.01e^(-2t) + 0.15e^(-t) +0.5925 + 0.225e^(t) + ...
0
votes
0answers
27 views

What is the PMF of the Hamming weight of a multinomial random variable?

Assume that $X$ is a random variable following a multinomial distribution of parameters $n$ (number of trials) and $p=(p_1,\dots,p_k)$ (event probabilities). Hence, ...
0
votes
1answer
290 views

partial differentiation of function of expectation of random variable

We have $E(U)=\int_0^B V f(V) dV + B \int_B^\infty f(V)dV$; Here $V$ is random variable. $E(U)$ stands for expectation of $U$. We have $Z=f(E(U))$ i.e. $Z$ is function of $E(U)$. Can we write ...
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0answers
20 views

Conditional expectation of discrete uniform random variables with one fixed

Came across a problem that I worked on sometime ago having the following structure: Given an opaque container (or locomotive with so many passenger cars, etc) that has--with equal probability--1 to N ...
0
votes
1answer
28 views

Simple Probability Inequality with Stopping Times

Suppose $U_1,...,U_n$ are independent random variable with $\mathbb{E}[U_i]=0$. Define $Z_k:=\sum_{i=1}^k U_i$. Set $T:=\inf \lbrace k \in N \mid |Z_k|>2\alpha \rbrace$. Clearly $\lbrace T=k ...
0
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0answers
19 views

Probability distribution to failure [closed]

I am going to do a simulation for a manufactruing system, i must consider a scenario as: a $20\%$ probability of failures occurring in $M1$. Q: What is the probability distributions the time to ...
0
votes
1answer
31 views

Prove that limsup and liminf of an independent sequence are independent of finite number of terms

Let $X_1, X_2, ...$ be an independent sequence of random variables on $(\Omega, \mathscr{F}, \mathbb{P})$. What I'm trying to prove is: Prove that $X_1, X_2, ..., X_k$ is independent of $\liminf ...
0
votes
2answers
58 views

Are random variables independent of their tail sigma-algebra?

Let $X_1, X_2, ...$ be independent random variables. Define $$\mathscr{T}_n = \sigma(X_{n+1}, X_{n+2}, \ldots)$$ and $$\mathscr{T} = \bigcap_{n} \mathscr{T}_n,$$ the tail σ-algebra of $(X_1, X_2, ...
1
vote
2answers
22 views

Show that $Cov(X,Y) \geq -23$

if $X,Y$ are two random variables and: $Var(X) = Var( Y) = 23$ how can i show that $Cov(X,Y)\geq -23$ can someone give me some hints on how to show it?(not an answer) i know that $Cov(X,Y) = E(XY) ...
0
votes
1answer
32 views

Defining the set of pre-images of a product of random variables in terms of the sets of pre-image of the original random variables

Say I have two random variables $X$ and $Y$. Their respective $\sigma$-algebras are $$\sigma(X) = \{ X^{-1}(B) \mid B \in \mathscr{B} \}$$ and $$\sigma(Y) = \{ Y^{-1}(B) \mid B \in \mathscr{B} \}.$$ ...
0
votes
2answers
54 views

Random increment through a probability distribution function

To Clarify i am trying to generate a random variable from a gamma pdf If $\Delta X$ indicates a random increment and it is said that $\Delta X$ follows a Gamma distribution. What would that mean ...
1
vote
1answer
47 views

Existence of a global maximum of a function defined with the moment-generating function

Can someone give me an idea how to prove the following exercise? Let $Z$ be a real-valued random variable whose moment-generating function $m_Z$, with $m_Z(\gamma)= E\left[ \exp(\gamma Z) \right]$, ...
0
votes
1answer
52 views

For zero-mean r.v. $X$ with var. $\sigma^2$, want to show $E[e^{2X}]\leq e^{2\sigma^2}$.

Let $X$ have zero mean, $E[X]=0$ and finite variance $E[X^2]=\sigma^2<\infty$. I'm trying to show $$ E[e^{2X}] \leq e^{2\sigma^2}. $$ I started out with this related question, but I hadn't quite ...
2
votes
2answers
155 views

Monte Carlo Importance Sampling

I am reading the book on Monte Carlo by Sobol (A Primer for the Monte Carlo Method). In the section on Importance Sampling, he writes: $I = \int_a^b g(x) \: dx$ "to compute this integral, we could ...
0
votes
1answer
27 views

Partition-based entropy of a sequence

The entropy $H$ of a discrete random variable $X$ is defined by $$H(X)=E[I(X)]=\sum_xP(x)I(x)=\sum_xP(x)\log P(x)^{-1}$$ where $x$ are the possible values of $X$, $P(x)$ is the probability of $x$, ...
0
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1answer
28 views

Convergence in proability does not imply convergence a.s. [duplicate]

I know convergence in probability does not imply convergence in measure. I would like to see some simple example. Do you have any ideas please?
1
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2answers
53 views

Bounding the expectation of a function of a zero-mean random variable

I have a random variable $X$ with mean zero, $E[X]=0$, and finite second moment, $E[X^2]=\sigma^2<\infty$. I'm wondering if it's possible to show the following bound: $$ E[(e^{X/2}-1)^2] \leq ...
2
votes
0answers
29 views

Generating cross-correlated stochastic processes

I am looking for a robust way to represent and generate multiple stochastic processes that contain time and cross-correlations i.e. I am looking at stochastic processes $X_t^{1}$, $X_t^{2}$, $\ldots$, ...
0
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0answers
14 views

Generating pseudo-random numbers around a distribution with an uncertain/chaotic mean

I originally asked this question on cross-validated, but apparently it is too mathematical a problem for that site. I want to simulate data collected by an instrument realistically. The problem is ...
0
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2answers
16 views

statistics question about two random variables

Two random variables $X$ and $Y$ have the following joint pdf: $$f(x, y) = \begin{cases} \frac35x(y + y^2) & \text{if }0<x<2\text{ and } 0 < y < 1\\ 0 & \text{otherwise} ...
0
votes
2answers
62 views

How to find distribution of order statistic

Let $X_i$ be iid random variables with common density $f$ and distribution $F$. Let $Y_k = X_{(k)}$ be the k-th order statistics (that is, $Y_1 = X_{(1)} = \min(X_i)$ etc.). Show that the joint ...
1
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0answers
23 views

mean and variance of this Gaussian random variable

I am trying to read through this paper - http://www.malcolmdshuster.com/Pub_2002c_J_scale_scan.pdf Equation 2(b)from the paper says [A] $\nu_k \equiv 2(B_k - b).\epsilon_k - |\epsilon_k|^2 $ where ...
1
vote
1answer
393 views

Expected value vs using method of indicator

I am having a hard time understanding the difference between getting the Expected value by finding the mean E(X)=np and using the method of indicator to find the expected value. For example if we ...
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votes
1answer
86 views

How can I prove if $Y\leq X$ then $E[Y]\leq E[X]$?

If $Y\leq X$ always holds, then $E[Y]\leq E[X]$. How can I prove this (formally)? Also, can the equality happen if we know that $Y=X$ does not always hold? (i.e. $X$ and $Y$ are not exactly the same)
0
votes
3answers
97 views

How is it possible for two random variables to have same distribution function but not same probability for every event?

It is completely out of the world for me to hear that such a case exists. I was shocked and could not develop any intuition as to how it is possible. It also breaks my understanding (intuitive) of the ...
1
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0answers
32 views

Expectation of $\min(X, c)$ for $X$ truncated r.v. and $c$ constant

I have a random variable $X$ and a constant $c\geq 0$. I define the r.v. $Y = \min(X, c)$ and I want to calculate $E[Y]$. I have seen different posts on similar topics, so I am trying to pull all ...
0
votes
1answer
67 views

Probability distribution of bored people

5 people are arranged in a row, a person is talkative with a probability of $p$ and silent with a probability of $1-p$, each is independent. A person is bored if he's talkative and sits between two ...
1
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0answers
40 views

Defining a function over time in terms of a random variable that is undefined at a certain time

Let $X_n$ be a random variable taking on one of three values $a,b$ or $c$ over time. That is, for each $n \in \mathbb{N}$, we have $X_n \in \{a,b,c\}$. Also, for each $n \in \mathbb{N}$, let $F_n$ be ...
0
votes
1answer
22 views

Check if the weak law of large numbers holds true for the following sequence of random variables

Suppose we have $n$ independent discrete random variables, whose distribution is as follows: $X(k)$, where $k$ is any integer from $1$ to $n$, can take any of three values: $-\sqrt{k}$ with a ...
1
vote
1answer
32 views

What is the probability density function of **the multiplication of Gaussian variables**?

Assuming $x_1,x_2,\ldots, x_n$ are $n$ independent variables from standard Gaussian distribution $N(0,1)$. Then we construct a new variable by $y=\Pi_{i=1}^n x_i$. Can anyone show the probability ...
3
votes
1answer
148 views

CLT for independent, but non-identically distributed exponential variables

This problem is practice for my qualifying exam and comes from Resnick, chapter 9. Could anyone comment on my solution(s)? Problem Suppose ${e_n, n\ge 1}$ are independent exponentially distributed ...