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

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0
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2answers
34 views

Find the limit of the probability of uniform random variable?

Let $X_1 ,X_2 ,X_3 ,…$ be a sequence of i.i.d. uniform $(0,1)$ random variables. Then, calculate the value of $$\lim_{n\to \infty}P(-\ln(1-X_1)-\ln(1-X_2)-\cdots-\ln(1-X_n)\geq n)?$$ My work: Since ...
0
votes
0answers
22 views

Convergence of expectations of a sequence of exponential random variables.

Suppose $\{X_n\}$ is a sequence of exponentially distributed random variables, where $X_n$ has mean $1/\lambda_n$. Suppose that $\lim_{n\to\infty}\lambda_n = \lambda>0$. Let $X$ be exponentially ...
1
vote
2answers
23 views

Differentiating $\int\cdots \int f(X_1,X_2,\ldots,X_n)\varphi_1(x_1,\theta)\cdots\varphi_n(x_n,\theta)~dx_1\cdots dx_n$

Differentiating:$$\int_{-\infty}^\infty \cdots \int_{-\infty}^\infty f(X_1,X_2,\ldots,X_n)\varphi_1(x_1,\theta)\cdots\varphi_n(x_n,\theta)\,dx_1 \cdots dx_n$$ with respect to $\theta$. The result is ...
2
votes
0answers
72 views

Close-form solution for distribution of the stopping time for a path-dependent random process?

A time series $\{x_s\}_{s=1}^{\infty}$ is generated from $N(\bar{x},1/b)\ i.i.d.$. Parameter $\bar{x}$ is drawn from prior distribution $N(\phi_0,1/a)$. Define conditional expectation of $\bar{x}$ as ...
-1
votes
0answers
14 views

Sum of Independent Levy RVs is Levy RV [on hold]

I want to show that the summation of independent Levy random variables X and Y with scaling parameters a and b is a Levy random variable with scaling parameter c = (a^(1/2)+b^(1/2))^2 using ...
0
votes
2answers
21 views

Does this hold in every case, and if only this one, why? Expectation, mean of random variable.

Characteristic function of random variable $X$ let us denote as $f_X(t)$ and $EX$ it's mean or expectation. Does the following hold in all cases, because it keeps coming up and I don't know why it is ...
0
votes
1answer
31 views

Solving inequality of two independent exponentially distributed RVs

I have huge problems solving following excersice: There are two molecules. The decay of the molecules is exponentially distributed with $\alpha_1 = 1$ (for molecule 1) and $\alpha_2 = 2$ (for ...
0
votes
1answer
29 views

Proof that normalized vector of Gaussian variables is uniformly distributed on the sphere

I have seen in various places the following claim: Let $X_1$, $X_2$, $\cdots$, $X_n \sim \mathcal{N}(0, 1)$ and be independent. Then, the vector $$ X = (\frac{X_1}{Z}, \frac{X_2}{Z}, \cdots, \frac{...
1
vote
0answers
12 views

Calculating the probability distribution function of a realization of a random process

I have a realization of a continuous random process, $y = f(t)$, a function of time ($t$). I am trying to calculate $P(y = Y_0)$, the probability distribution function of $f(t)$. Am I right in saying ...
2
votes
0answers
18 views

The expected distortion of a linear transformation (continued)

Let $A: \mathbb{R}^n \to \mathbb{R}^n$ be a linear transformation. I am interested in the "average distortion" caused by the action of $A$ on vectors. Consider the uniform distribution on $\mathbb{S}...
1
vote
1answer
16 views

Is $\sigma(X,Y) = \sigma(X, X \cdot Y)$ for two Random Variables $X$ and $Y$?

Suppose we have two real random variables $X,Y$. Then clearly \begin{equation} \sigma(X, X \cdot Y) \subset \sigma(X,Y) \end{equation} since both $X$ and $X \cdot Y$ are $\sigma(X,Y)$-measurable ...
-1
votes
0answers
30 views

Find a function of a Gaussian RV whose variance is the inverse of the Gaussian variance [closed]

For a Gaussian random variable $X\sim\mathcal{N}(0,\sigma)$ where $\sigma$ is unknown, is it possible to find a function $f(X)$ of $X$ (independent of $\sigma$) such that its variance $Var[f(X)]=1/\...
0
votes
1answer
24 views

White noise - in terms of associated Legendre polynomials

I am trying to draw random numbers $$Z_{l,m} = \int_{-1}^1 dx \, P_l^m(x)W(x)$$ Here $P_l^m(x)$ are the associated Legendre polynomials with integers $l\geq0$ and $-l\leq m \leq l$. The variable $W(...
0
votes
1answer
40 views

Expectation of absolute random variables with mean 1 and standard deviation 1

For a random variable $\gamma \sim \mathcal{N}(\mu,\sigma)$ , were is $ \mathcal{N}$ is the normal distribution. What is the way to calculate the following: $ \mathbb{E}[|\gamma|] = ? $ And ...
1
vote
1answer
36 views

Expectation of concave transformation of random variable

Suppose I have two different discrete random variables $y>0$ and $x>0$. Now I want to compare two expected values involving these and a nonlinear transformation: When is one larger than the ...
0
votes
0answers
9 views

Relationship between complex normal and bivariate normal distributions

Suppose I have a complex random variable $X$ which follows a complex normal distribution (with $0$ mean). I've been trying to represent the complex normal in a simpler way, but I'm not sure how. Is ...
4
votes
1answer
19 views

Distribution of sums of random variables over finite field

Let $q$ be an odd prime, $X_1, \ldots, X_n$ be i.i.d. random variables over $\mathbb Z_q$, and $0 < p < 1$ be some constant. Let $X_i$ take on the value $0$ with probability $p$, and the ...
0
votes
2answers
29 views

Fine E(4X^2+4X+1)

So I have the following tables $$ \left[ \begin{array}{c|ccc} x&-3&6&9\\ f(x)&\frac{1}{6}&\frac{1}{2}&\frac{1}{3} \end{array} \right] $$ I am tasked to ...
2
votes
1answer
14 views

Proof that the sum of two independent exponential random variables is gamma with $\alpha=2$

I'm trying to prove that the sum of two exponential random variables is gamma. This proof is straightforward using the uniqueness of moment generating functions however I'm asked to find the density ...
1
vote
1answer
31 views

Transformation of density and $W=(X+Y+Z)^2$

I want to solve this exercise with the transformation formula, what did I do wrong in my solution?: Let $X,Y,Z$ be independent random variables with uniform distribution on [0,1]. What's the ...
1
vote
0answers
14 views

Partition Probability Proof

I'm tasked with proving the following Lemma: Let X be a set of $n^1$ < n elements, and B $\subset$ X, |B| = k. Suppose $P_1, P_2,...P_r$ are random partitions of X, where each $P_i$ partitions X ...
1
vote
2answers
22 views

Cnditional expectation of exponential on UDF

Let $X$ be an exponential random variable with $\lambda = 5$ and $Y$ a uniformly distributed random variable on $(-3,X)$. Find $\mathbb{E}[Y]$. I tried to solve it by just integrating $f(y)$ from $(-...
4
votes
0answers
26 views

Transformation of random variables exercise

I want to know if my solution to the following exercise is correct: Let $X$ be a gamma distributed random variable with parameter 2, meaning with distribution $$P_X(\mathrm{d}x)=\mathbb{1}_{\{x>...
2
votes
0answers
47 views

Probability of sum of 10 dice throws [duplicate]

If a die is rolled 10 times. What is the probability that the sum of the results is less than or equal to 20? I was trying to solve this using something like $P(X_1 + X_2 + ....+X_{10} \le 20)$ but ...
2
votes
1answer
38 views

Convergence of $V_n=\prod\limits_{i=1}^n U_i$

I struggle to do this exercise: Let $U_1,U_2,\dots$ be a sequence of i.i.d. random variables. We define $$V_n=\prod\limits_{i=1}^n U_i$$ Show that $V_n^{1/n}$ converges almost sure and calculate the ...
1
vote
1answer
40 views

Stochastics exam Exercise

The professor uploaded an exam to practice, but unfortunately I have no solutions. Let U be a unifomly distributed random variable on $[0,1]$. 1) Let $X=-ln(U)$. Show that $X$ is distributed ...
2
votes
1answer
63 views

The variance of the expected distortion of a linear transformation

Let $A: \mathbb{R}^n \to \mathbb{R}^n$ be a linear transformation. I am interested in the "average distortion" caused by the action of $A$ on vectors. (i.e stretching or contraction of the norm). ...
0
votes
2answers
36 views

Transformation of the uniform distribution

I struggle to understand the transformation of a random variable with uniform distribution. For example: Let $X\sim \text{Uniform}(0,1)$ and $T=-2\ln(X)$ and I want to find the CDF of $T$, then I ...
1
vote
1answer
26 views

What is the expected distortion of a linear transformation?

Let $A: \mathbb{R}^n \to \mathbb{R}^n$. I am interested in the "average distortion" caused by the action of $A$ on vectors. (i.e stretching or contraction of the norm). Consider for instance the ...
2
votes
1answer
47 views

Expectation of $|X-Y|$ when a coin is thrown six times

If a fair coin is thrown six times. Let $X =$ number of heads and $Y = 6-X =$ number of tails. What is $E|X-Y|?$ I was able to come up with this table, but I am not sure if this is correct or not and ...
1
vote
1answer
38 views

Distribution of Expectation function into a $|X-Y|$

We know that $E(X+Y) = E(X) + E(Y)$. But why is $E|X-Y|$ $\ne$ $E|X| - E|Y|?$
3
votes
0answers
39 views

Random variable for storing cost to get the target.

There is a simple game for a single player. Player's initial level is $n$ and player want to get level $m$. If player's level became the target level $m$, the game terminates. Player should pay $c_i$...
0
votes
1answer
22 views

convergence in probability of division and their expected values

Let $\frac{X_n}{Y_n} \rightarrow 1$ in probability. Then does $\frac{\mathbb{E}[X_n]}{\mathbb{E}[Y_n]} \rightarrow 1$? If not, what are the conditions required for this to hold?
1
vote
0answers
9 views

Probability density transformation for non-invertible mapping

I am looking for a generalization of the result which states that the density of the sum of two random variables is the convolution of their densities. Specifically, if I have $Z=f(X,Y)$, where $p_{X,...
-1
votes
1answer
54 views

Find value of P and E(x) [closed]

The random variable $X$ takes on the values $1$, $2$, or $3$ with probabilities $$\frac{2+5P}{5}, \frac{1+3P}{5} , \frac{1.5+2P}{5}$$, respectively. What is the value of $P$ and $E(x)$?
1
vote
2answers
43 views

Almost sure convergence implies convegence in distribution - proof using monotone convergence

I'm trying to understand the following proof of the statement : "Almost sure convergence implies convegence in distribution" The definition of convergence in distribution is given as follows : $X_n$...
0
votes
1answer
16 views

Central moments of a Bernoulli distribution

Consider a discrete random variable distributed as a Bernoulli: $$ Y=\begin{cases} 1 & \text{with probability } p\\ 0 & \text{with probability } 1-p \end{cases} $$ The $n$-th central moment ...
0
votes
1answer
24 views

Random walk - probability of first pass through zero.

In the proof that symmetric random walks end up regressing to the origin with probability $1$, I have found this didactic post on-line, where the power series of the probability mass function of the ...
0
votes
3answers
44 views

distribution of $X+X$

My question is about the interpretation of the expression of $X+X$, where $X$ is a random variable. For example, suppose $X$ is the number seen when we roll a regular six-sided dice. Is $X+X$ the ...
1
vote
1answer
29 views

Minimum of maximum of independent variables

I'm trying to find the probability distribution and expected value of the minimum of maximums of a combination of random variables. For example, say $$X_1 \sim \mathrm{Exp}(\text{rate}=\lambda_1), ...
0
votes
0answers
16 views

How do you create a Gaussian distribution on a polynomial ring?

In the specification for the YASHE homomorphic encryption algorithm, it says that for the parameter generation subroutine, you need to: Given the security parameter $λ$, fix ... distributions $\...
0
votes
2answers
78 views

Equivalent random variables and sigma algebras

Consider two random variables $X$ and $Y$ defined on the same probability space $(\Omega,\sigma,P)$. We know that they are equivalent in the sense that $P(\{X \ne Y\})=0$. Let $A_X$ and $A_Y$ be the ...
0
votes
3answers
36 views

What would be the graph of absolute value of $X - Y$

I am solving a question, where I need to find the expected value of the absolute value of $X-Y$ where $X$ and $Y$ are two independent uniformly distributed random variable. So I was looking for the ...
-2
votes
0answers
62 views

Probability that $\max[X,Y]$ is less than $\frac{1}{2}$ [duplicate]

Two inpendent random variables $X$ and $Y$ are uniformly distributed in the interval $[-1,1]$. Find the probability that $\max[X,Y]$ is less than $\frac{1}{2}$.
1
vote
0answers
88 views

Exact Probability of reducibility of Bivariate Polynomials

I am considering polynomials of the form $$P(x,y)= \sum_{k=0}^n\sum_{l=0}^n a_{k,l}x^{k}y^{l}$$ where $n \in \mathbb{N}$. The coefficients $a_{k,l}$ are considered to be randomly generated from the ...
0
votes
0answers
41 views

Is this integral analytically solvable?

I am trying to find the average of exponential of a random quantity which obeys log normal distribution. For that I need to evaluate $$\int_{-1}^{\infty } \frac{\mathrm{e}^{-\left(\log (\delta +1)+\...
0
votes
0answers
7 views

Convergence of Ritz polynomial in mean square

If i use a method of weighted-residual or Ritz method and obtain a numerical approximation as a polynomial ... How can i prove the convergence of this solution to the exact(in mean square sense)?
1
vote
1answer
22 views

Why doesn't the Variance of a Random Variable use $\frac{1}{n}$ instead of $P(X=x_i)$? Aren't the weights already expressed in the Mean?

I understand that the population variance of a set is just a special case of the variance of a random variable where $P(X=x) = \frac{1}{n}$ for $n$ elements in the set. Still I can't help but feel ...
0
votes
0answers
15 views

Very basic clarifications on sampling processes

Could you help me to clarify some basic notions from sampling theory? Please highlight if anything of what is written below is wrong because I am very confused on the order of logical steps. ...