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

learn more… | top users | synonyms

0
votes
1answer
15 views

Calculate expectation of Cumulative distribution function of a normal distribution

I have to calculate the expectation of the Cumulative Distribution Function of a normally distributed random variable X, which has variance 1 and mean 0. I calculated the integral of the CDF (taken as ...
1
vote
0answers
17 views

Expectation of the maximum of random variables

I'm trying to get $E(\max \{ a-X, b-X-Y, 0 \})$, where $X$ ~ $N(0,\sigma^2)$, and $Y$ ~ $N(\mu, \gamma^2)$, and $X,Y$ are independent. I've been trying to figure this out by doing, $E(\max \{ a-X, ...
1
vote
2answers
32 views

the definition of random variable

If we supposed that X is a random variable, is X - X a random variable? Could the outcome of an event is only 1? Cause X-X has only one outcome, and the possibility of it is 1. How about X + X?
0
votes
0answers
17 views

Two different random processes for generating polynomials

Consider two processes for generating random complex polynomials: choosing the roots uniformly and independently throughout the unit disc, and choosing the coefficients uniformly and independently ...
1
vote
0answers
35 views

Mean Value of a Random Process

Consider a random process $X(t) = Z(t)\sin(wt-Q)$. Here $Q$ is a random variable taking values $q$ in $[-\pi/2,\pi/2]$ with PDF given by $$p_1^Q(q) = \frac{\cos(q)}{2}$$ $Z(t)$ is some random ...
0
votes
0answers
28 views

What is the joint distribution of order statistics and samples?

If samples $X_1, X_2, ... X_t$ are picked independently and identically from the uniform distribution $[1,2, ..., P]$, what is the joint distribution ...
-1
votes
0answers
25 views

determining $\sigma(X)$ and $\sigma(Y)$ [closed]

I am having troubles determining $\sigma(X)$ and $\sigma(Y)$: We have defined, $\sigma(X) = \{ X \in B: B \in \mathcal{B} \}$ but I honestly have no idea what this definition means as I have never ...
1
vote
0answers
48 views

average of reciprocal of sum of random variables

I have a slightly more complex problem, but I believe the technical nature is captured in the following: For the $s$-indexed sequence, $X_s$, of iid positive-valued random variables with finite ...
0
votes
1answer
42 views

Central Limit Theorem; Exponential Distribution

I'm trying to prove the Central Limit Theorem for the exponential distribution and I'm running into problems. This is what I've done so far: Given $S = X_1+X_2+...+X_n$ where each $X_i$ is an ...
2
votes
1answer
27 views

Is f(X)=X+a exponentially distributed when X is exponential R.V and a>0?

Suppose $X$ is an exponential R.V with parameter $\lambda$. Then, I am interested to find the distribution of $f(X)=X+a$, for $a>0$. To this end, I have computed the following: $$\Bbb ...
0
votes
2answers
18 views

How to find a random variable with well-behaved maximum

I'm looking for a continuous random variable with the following properties It is not bounded towards $+\infty$. The expected value of the maximum of x-many draws out of that random variable has a ...
1
vote
0answers
52 views

Find the probability mass function of a poisson distribution

A random sample $X_1, X_2, . . . , X_5$ is taken from a Poisson distribution with parameter λ for some λ > 0. Find the joint probability mass function in as simplified a form as possible for ($X_1, ...
0
votes
0answers
22 views

Calculation of a Autocorrelation function and Power spectral density

A sample of a random process is given as: $$ x(t) = Acos(2\pi f_0t) + Bw(t) $$ where w(t) is a white noise process with 0 mean and a power spectral density of N0/2, and f, A and B are constants. ...
1
vote
1answer
19 views

Probability density of transformed random variable

Let $X$ be a random variable whose probability density function is $f(x) = xe^{x-2}$, if $1 < x < 2$ and $0$ elsewhere. Let $F(x)$ be the cumulative distribution function of $X$. Find the ...
0
votes
1answer
20 views

Elements of the range of a random variable that are transformed into the same element

Let $X$ be a random variable and $Y = g(X)$. Then, the range or support of $Y$ can be written as $R_Y = \{g(x) \mid x \in R_X\}$. My question is whether there is a name (or standard notation) for ...
0
votes
0answers
24 views

using markov and chebyshev inequalities in the same problem

The problem states as following: Let's say the width in the fabrication process of an industrial metal piece has a mean of 10 and an standard deviation of 1. Find the probability for when the width is ...
1
vote
1answer
30 views

Joint distribution of multivariate normal distribution

So the question asks: Let $X = (X_1, ... ,X_{2n})$~ $ N (0, ∑)$ (multivariate normal distribution with mean vector $(0,..., 0)$ and covariance matrix $∑$ ), where $n≥ 1$. Find the joint distribution ...
1
vote
1answer
32 views

Independence of some derived random variables

Positive random variables $X_1, X_2,$ and $X_3$ have joint probability density give by $$f_{\lower{0.5ex}{X_1,X_2,X_3}}\!(x_1,x_2,x_3)= \begin{cases}48~x_1~x_2~x_3 & : \textsf{if ...
0
votes
2answers
27 views

Theorem about Expected Value of a Random Variable

Well, we have to demonstrate this theorem but I don't have any hint. I need to verify this: Let $X$ be a discrete random variable with values in the nonnegative integers and such that $E(X)$ and ...
1
vote
1answer
36 views

Does $E(Z\vert X) = 0$ mean $Z$ does not include a non-zero constant?

Let $Z, X$ be random vectors. If $E(Z\vert X) = 0$ then is it impossible for $Z$ to include a non-zero constant? That is, it is impossible for $Z$ to look like $$ \begin{pmatrix} 1 & z_{12} & ...
0
votes
2answers
27 views

Probability of random variables inequality

Let $X,Y$ be two independent continuous random variables. I have seen places that state that $$ P(X\leq Y ) =\int\limits_{-\infty}^\infty \left(\int\limits_x^\infty f_Y(y) \, dy\right)f_X(x) \, dx $$ ...
0
votes
1answer
42 views

Approximate the mean of a function of two random variables

I have the next two random variables function: $f(r,\theta) = \left(\frac{r^{2}}{D^2+2Dr\cos{\theta}+r^2}\right)^2$, where $D$ is a constant and $r$ and $\theta$ have PDFs: $f_{r}=\frac{2r}{R^2 - ...
1
vote
0answers
41 views

Sample space of infinite coin tosses experiment

In one of the online statistics/probability web sites http://www.statlect.com/asymptotic-theory/mean-square-convergencee, the following definition is given for the mean - square convergence of the ...
0
votes
2answers
37 views

What is the domain and range of the sum of two random variables?

Let $(\Omega, \mathcal{F}, P)$ be a probability space s.t. $\Omega = \{0,1\}$. Let $X_1: \Omega \rightarrow \{0,1\}$, $X_2: \Omega \rightarrow \{0,1\}$ be two random variables over $\Omega$ (i.e., ...
1
vote
1answer
33 views

Probability question using no-memory property of exponential distribution

A customer must be served first by server 1, then by server 2, and finally by server 3. The amount of time required for service by server i is an exponential random variable with rate µi , for ...
1
vote
2answers
27 views

What is the variance of a variable given itself?

Given an event $X$, what is $\operatorname{var}[X\mid X]$. In addition, what would $E[\operatorname{var}(X\mid X)]$ be? I have been told that $\operatorname{var}[X\mid X] = 0$, but I don't understand ...
-1
votes
0answers
3 views

Bayes and James-Stein estimators for the means of independent Gaussian random variables

I have tried to solve thas question but I cant find the way to solve it. this is the question: Bayes and James-Stein estimators for the means of independent Gaussian random variables thanks for any ...
0
votes
0answers
12 views

Coupling Brownian Motions

I want to simulate three freight rate indices which are naturally correlated. The freight rate dynamics ($X$) can be modeled as a geometric Brownian motion: $dX_{t} = \mu X_{t}dt + \sigma ...
3
votes
1answer
34 views

Poisson: $P(N(4)-N(2)=5|N(4)=8)$

Let {N(t), t ≥ 0} be a Poisson process of rate 2. Determine: a) $P(N(4)-N(2)=5|N(4)=8)$ Attempt: The conditional poisson distribution is uniformly distributed between the interval [0,4]. Therefore, ...
0
votes
2answers
53 views

What does the “$+$” typically denote when summing random variables?

Let $X_1$ and $X_2$ be two random variables. When in the literature (for example, in the context of the law of large numbers) one sees statements along the lines of $$ X = X_1 + X_2 $$ ...does the ...
0
votes
1answer
28 views

Confusion on a random variance theorem

I'm confused on how $E(X)^2\sum_{s∈S}p(s) = E(X)^2$ because $E(X) = \sum_{s∈S}p(s)X(s)$. Wouldn't $E(X)^2=(\sum_{s∈S}p(s)X(s))^2$?
0
votes
1answer
23 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
votes
0answers
15 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
vote
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
votes
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
votes
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
votes
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
vote
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
76 views

Random permutations composition

I'm trying to prove a theorem that seems very intuitive. However, I seem to be missing a piece. If: $\pi$ is a random permutation ($S_n$), $\pi_1, \pi_2$ - random permutations with uniform ...
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
votes
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
35 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
vote
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
52 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
votes
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 ...