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

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0
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1answer
25 views

Confusion on the definition of variance on a random variable

For $X(s)$, I interpret that as the value the random variable gives based on the input of an element in the sample space. $E(X)$ is the sort of mean for all of the expected values in the sample space. ...
0
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0answers
16 views

Covariance matrix of random vector of vectors

I am a beginner in statistics and tried to research my question online without much success. Motivation: I am working on an undergraduate project in cosmology. My problem involves several ...
1
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1answer
26 views

Confusion on random variable linearity of expectations

I'm confused on why $E(X) = E(X_1) + E(X_2)+· · ·+E(X_n) = np$. How can we multiply the number of elements by $p$ if we do not know if each element in the sum has the same probability? Or just overall ...
0
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0answers
11 views

Probability of two related continuous intervals, open interval equinumerosity with the continuum

If the open interval (a, b) is equinumerous with the cardinality of the continuum, why is it that the probability of two related intervals can have different values? Take $$x=x_0cos(wt)\qquad ...
2
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0answers
35 views

Proving two Gaussian random variables are independent given the third: a necessary and sufficient conditon for inverse of covariance matrix

In my probability class I was given this problem that truly has me stumped: Let $ X=(X_1,X_2,X_3) $ be a Gaussian random vector with mean vector zeros, with the 3x3 co variance matrix: $ ...
0
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0answers
21 views

Can every random variable be written as a Borel-measurable function of a uniformly distributed random variable?

I am trying to find a proof for the claim that every random variable can be written as a Borel-measurable function of a uniformly distributed random variable. Can anyone point it out to me? Thanks
1
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0answers
29 views

Find distribution of a random variable sequence

Let $X_n$ be the sequence of random variables which have their values from $(0, n]$ for $n > 0$. The cumulative distributive function of $X_n$ is $F_{X_n}(x) = 1 - (1 - x/n)^n$ for $0 < x \leq ...
1
vote
2answers
12 views

Double Partial Derivatives of sum of variances of dependent random variables

I have the following function $$f(α)=Var[αX+(1−α)Y]=Var(αX)+Var[(1−α)Y]+2α(1−α)Cov(X,Y)$$ Partial derivative of this function w.r.t α leads us to following result ...
0
votes
1answer
6 views

Cross-Covariance matrix from two covariance matrices

Let $x=(x_1,...,x_n)^T$ and $y=(y_1,...,y_n)^T$ be two random vectors, with covariance matrices $E_{xx}$ and $E_{yy}$, respectively. Could I compute the cross-covariance matrix $E_{xy}$ using ...
2
votes
1answer
80 views

Random permutations composition

I'm trying to prove a theorem that seems very intuitive. However, I seem to be missing a piece of the puzzle. If: $\pi$ is a random permutation ($S_n$), $\pi_1, \pi_2$ - random permutations with ...
3
votes
0answers
78 views

Trace of power of random matrix / sum of random variables with semicircle distribution

I want to calculate the expectation value for the trace of the $m$-th power of the $n\times n$ adjacency matrix $A$ of a large Erdos-Renyi random graph (without self-coupling, i.e., all diagonal ...
0
votes
1answer
37 views

Why do we have to use pre-image in the formal definition of random variable?

There is this definition of random variable: Let $(\Omega,\mathcal{F}), (\Omega',\mathcal{F}')$ be two event spaces. We say that a function $X:\Omega\to\Omega'$ is a random variable from ...
0
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0answers
9 views

cross-covariance matrix of two random vectors given two covariance matrices

Let $x=(x_1,...,x_n)^T$ and $y=(y_1,...,y_n)^T$ two random vectors, with covariance matrices $E_{xx}$ and $E_{yy}$, respectively. How could I compute the cross-covariance matrix $E_{xy}$, using ...
9
votes
4answers
2k views

Two rifleman shooting at a target where the game ends when two hits are observed.

The problem is as following: Person A and B are shooting at a target. Independently of who is shooting, the probability that the shot results in a hit is $p$, and each shot is independent ...
1
vote
1answer
36 views

Integrating the bivariate normal distribution

Let $X$ and $Y$ have the bivariate normal density function, $$ f(x, y) = \frac{1}{2 \pi \sqrt{1 - p^2}} \exp \left\{ - \frac{1}{2(1 - p^2)} (x^2 - 2pxy + y^2) \right\} $$ for fixed $p \in (-1, 1)$. ...
1
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0answers
33 views

Calculating a variance of a division of two sums.

I have got the following model: $$Y_i=\beta X_i+\epsilon_i, \hspace{1cm} i=1,...,n$$ where $X_i$ are independent $N(\mu, \tau^2)$ random variables and $\epsilon_i$ are i.i.d. $N(0, \sigma^2)$. I need ...
0
votes
3answers
57 views

Given $f_X(x)$ find the pdf $f_Z(z)$ when $Z = X^2$

I have a question on probability density function. I cannot get the value for interval so I'm not sure if I did it the right way: My answer: Thank you!
0
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0answers
8 views

Uniform random samples inside bounded region

In an $n$ dimensional space I have a region bounded by pairs of hyperplanes: \begin{equation} b_j \le \sum_{i=1}^n a_{ij} x_i \le c_j, \quad\forall j=1,\ldots,m. \end{equation} We can include in those ...
0
votes
1answer
76 views

Probability density function given by $f(x) = (3/16) (4 − x ^2 )$, for $0≤X≤ 2$. Find $E(X^3)$

The question asks for the answer to be in 2 decimals. It also does not specify whether the random variable $X$ is continuous or discrete, however I'm assuming it's continuous due to the function ...
0
votes
1answer
18 views

Compute the covariance of 2 random variables and answer if the variables are independent

In an urn there are 5 white balls and 4 black balls. One ball is extracted from the urn and is replaced with a ball of the opposite collor. Then a new ball is extracted. $X$ represents the ...
0
votes
1answer
20 views

Distribution function of $Y$ using $F_X(.)$

How to determine the distribution function of random variable $Y = X^3-6X^2$ in terms of $F_X(.)$. Since this function is not one-to-one, I couldn't find the solution. How can I solve this? Any ...
0
votes
1answer
34 views

Find the range of the Probability Mass Function in which $Y=X^2$ [closed]

A probability mass function of a random variable $X$ has the range $R = \{ -1, 0, 1 \}$, where $$f(-1)=0.2, \quad f(0)=0.6, \quad f(1)=0.2$$ Now we set $Y=X^2$, what will the range and probability ...
0
votes
0answers
17 views

Gaussian Random Variable, Density Function

I have a question that goes like this: Let $X$ be a standart Gaussian random variable. Determine the density function of the random variable $$Y = \ln(X)u(x-1)$$ where $u(x)$, is the unit step ...
0
votes
1answer
17 views

Unit random process

My friend asked me to help with the problem on the random processes, but I am stuck as well, because I don't understand the notation $X_t = 𝟙_{[U,1]}(t), t \in [0,1]$ Could anyone explain this one ...
2
votes
0answers
33 views

Convergence of random variable divided by constant that goes to inifinity

I have read in some book the following Lemma: For any random variable $X$ and for any sequence $c_n$, with $\displaystyle\lim_{n\to\infty} c_n = \infty$ the following is true: ...
1
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2answers
40 views

Clarification on convergence in probability

Wikipedia definition A sequence $X_n$ of random variables converges in probability towards the random variable $X$ if for all $\varepsilon > 0$: \begin{equation} \lim_{n\to\infty}\Pr\big(|X_n-X| ...
2
votes
1answer
25 views

Bounding the variance of a sum of independent random variables

Suppose $\{X_i\}_{i=1}^n$ is a sequence of independently distributed random variables that take values in $[0,1]$. Let $\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$ denote the average of the sequence. ...
3
votes
1answer
33 views

Expected value of a series of random variables

There are $K$ checkout counters in the mall, and there are $N$ shoppers in the queue waiting for a checkout counter. Initially all counters are empty. Whenever a counter is empty, the next shopper in ...
3
votes
4answers
64 views

How does one calculate the expected value?

Suppose a fair coin is flipped twice. Define two random variables $X$ and $Y$ , where $X$ counts the number of heads, and $Y$ counts the number of tails, in the two flips. Evaluate $\Bbb E(X), \Bbb ...
0
votes
1answer
54 views

Conditional distiribution of $X\mid X+Y=c$ with $X,Y$ iid $\sim \exp(1)$ distributions [closed]

I am supposed to find $$f_{X \mid X+Y=c}, \quad \mbox{ for }c>0$$ given that $X,Y \sim \exp(1)$ and independent. I have worked back and forth with convolution formula and Bayes theorem but without ...
1
vote
2answers
41 views

Independent, Identically Distributed Random Variables

Let $(X_n)_{n∈\mathbb{N}}$ a sequence of i.i.d. random variables uniformly distributed on the interval $[0, 1]$. Show that $$\limsup_{n\to+\infty} \frac{X_{2n}}{X_{2n+1}}=+\infty$$ a.s. I tried ...
0
votes
1answer
15 views

deriving $\operatorname{var}(X)=\mathbb{E}(L)(\operatorname{var}(D))+\mathbb{E}(D)^2(\operatorname{var}(L))$

Assume $X$ is continuous random variable representing demand during lead time with density function $f(x)$ and mean, variance, standard deviation $\mathbb{E}(X),\operatorname{var}(X),\sigma_X$ ...
2
votes
1answer
14 views

Is the support of a discrete random vector the Cartesian product?

I'm confused on the notion of support of a discrete random vector. Consider a discrete random variable $X$ which can take value in the set $\{0,1\}$ with strictly positive probability. Hence the ...
0
votes
1answer
25 views

$L^{\infty}$ convergence for random variable

I am slightly confused with this borderline case regarding $L^p$ convergence. In some probability books, they clearly state that $p<\infty$ whereas the online sources do not impose this ...
0
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0answers
20 views

Proof strategy - How to prove this modeling of time series

The question is based on a paper titled : Forecasting high waters at Venice Lagoon using chaotic time series analysis and nonlinear neural networks On page 2 right above Eq(1), the authors say ...
4
votes
3answers
58 views

If a coin comes up heads, it is tossed exactly two more times. Find $f_Z$ where $Z$ is the number of heads minus the number of tails.

There are three questions for this problem. Please help me explain the answer for number 3. Find $f_X$ where $X$ is the total number of heads. Yep I got this. Ans: \begin{align}f_X ...
3
votes
0answers
52 views

What does $\Bbb E[X|Y]\Bbb E[Y]$ simplify to?

My Goal I am trying to figure out what $\Bbb E[X|Y]\Bbb E[Y]$ simplifies to. My Work So Far I have the following train of thought $$\Bbb E[X|Y] = \sum_i x_i \Bbb P(X=x_i|Y=y_j)$$ $$\Bbb E[Y] = ...
-1
votes
1answer
23 views

Finding the unknown matrix X.

I'm practicing for my upcoming exam, and I can't find a way around this part of the question, Part a) "Given that A= (2 1 -2 5) , find the inverse of the matrix A+I, where I is the identity ...
0
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0answers
20 views

A problem about expectation value with conditional random variable

I have a questions related to the expected value with conditional Random variable. This is the part of EM procedure and $\mathbf y$ is hidden variables. $\mathbf V_{0}, \mathbf \Sigma_{0} $ means ...
1
vote
1answer
41 views

Finding Mean and Distribution of Normal Random Variables

Assume that $X_1$, and $X_2$ are i.i.d. normal random variables with mean $0$ and variance $1$. Let $Y_1$ and $Y_2$ be defined as $Y_1 =8X_1+6X_2$ and $Y_2 = X_1$. $E[Y_1]= 0$ correct? because ...
1
vote
1answer
42 views

examples of functions that are not random variables

I have been reading the definition of the random variable in R. G. Gallager's book, entitled "Stochastic processes: theory for applications". (Please see Definition 1.3.4 on Page 11/69 in ...
0
votes
1answer
26 views

Fair coin probability experiment with strange pmf

I have a question regarding an experiment where 5 fair coins are flipped, but the random variable has a quirk and is throwing me off. Fair in this case means the probability of success is ...
1
vote
1answer
20 views

How to use probability mass functions to find probability mass function and moment generating function of Z?

Let $X$ and $Y$ be discrete random variables such that $p_{(x)}(x) = \frac{1}{3}, x = -1,0,1$ $p_{(y)}(y) = \frac{1}{2}, y = 2,4$ Let's say $Z = X + Y$. I am trying to find the probability mass ...
2
votes
1answer
58 views

How to compute $P(T\ge 3)$, when $T=Y_1+\dots+Y_{200}$ with $Y_i$'s Bernoulli?

Suppose we have a piece-wise function as the following: $$Y_i = \begin{cases} 1, & {\rm if }\; X_i > 0.9\\ 0, & {\rm if }\; X_i \le 0.9\\ \end{cases} $$ where ...
0
votes
3answers
40 views

Prove that $P(\xi_{n+1}>t) = e^{-\lambda t} \sum_{k=0}^n \frac{(\lambda t)^k}{k!}$

Let $\xi_n = \eta_1 + \dots + \eta_n$ where $\eta_i$ are iid random variables exponentially distributed with the rate $\lambda>0$. In the book there is a proof that $P(\xi_{n+1}>t) = ...
0
votes
1answer
14 views

What does it mean to say a RV is fully defined by its distribution

I get that a CDF is used to observe the behaviour of a distribution and therefore the corresponding RV ? But what does it mean to fully defined by its distribution and what does this have to do with ...
0
votes
1answer
31 views

Cdf of $f(x) = 0.075x + 0.2,3 < x < 5$

Let $X$ have the following probability density function: $$f(x) =\begin{cases} 0.075x + 0.2,&\text{for }3 < x < 5\\ 0,&\text{otherwise.}\end{cases}$$ It derives into cdf ...
1
vote
2answers
30 views

Expectation of a transformed random variable

I'm trying to prove the following: Let $X_n$ be a sequence of positive random variables and $g$ be a positive function. Suppose that $E[X_n]\to \infty$ as $n\to\infty$. If $E[g(X_n)]$ exists, there ...
1
vote
0answers
32 views

A question on conditional expectation leading to zero covariance and vice versa

In my probability class I was tackled with this seemingly weird question involving conditional expectation: Let X,Y be two random variables (it is not mentioned whether or not they are discrete or ...
2
votes
1answer
28 views

Notion of Independent Random Variables

Given a probability space $(\Omega,\Sigma,P)$. A random variable $X$ is a function mapping $\Omega \to \mathbf R$. On the other hand, we know that sum of two random variables is still a random ...