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

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1
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0answers
9 views

Adjacent order statistic of transformed random variables

I have $n$ draws $X_1, X_2, \ldots,X_n$ from a random variable $X$ which is continuous and has values between zero and $\overline{x}$. Let $Y(k)$ be a linear transformation of $X$ such that $Y(k)=a(k)...
0
votes
1answer
29 views

For $X,Y $ random variables, $h $ a function, show that $E (Xh(Y)|Y)=h (Y)E (X|Y) $ almost surely

Question in the title: For $X,Y $ random variables, $h $ a function, show that $E (Xh(Y)|Y)=h (Y)E (X|Y) $ almost surely My main problem is that I don't even understand what $E (Xh(Y)|Y)$ means.....
4
votes
1answer
67 views

What is the difference between $\mathbb E[Z|\mathcal G]=Y$ and $\mathbb E[Z|\mathcal G]\stackrel{\text{a.s.}}{=}Y$?

I'm somewhat confused by the definition of martingale: Let $(\Omega, \mathcal F, \mathcal F_n, \mathbb P)$ be a filtered probability space. We call $(X_n)_{n\in\mathbb N}$ martingale if for all $n\...
1
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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 ...
2
votes
0answers
62 views

When to stop pumping up balloons?

Yesterday I acted as a volunteer in a psychology/neurology experiment where one of the trials consisted of playing a computer game in which you had to click the mouse to pump up a balloon. For each ...
0
votes
2answers
38 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 ...
2
votes
0answers
73 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 ...
0
votes
0answers
23 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
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2answers
24 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 ...
-1
votes
0answers
14 views

Sum of Independent Levy RVs is Levy RV [closed]

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
443 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 $\frac{\...
0
votes
1answer
32 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{...
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(...
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
1answer
64 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). ...
2
votes
0answers
19 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}...
2
votes
1answer
511 views

Variance stabilization for Poisson data

Intro Let $Z > 0$ be a random variable with the mean and variance defined as $\mathbb{E}\{ Z \}$ and $\operatorname{Var}\{ Z \}$, respectively. The variance stabilization transform (VST) $f(z)$ ...
5
votes
5answers
6k views

Expected Value of Flips Until HT Consecutively

Suppose you flip a fair coin repeatedly until you see a Heads followed by a Tails. What is the expected number of coin flips you have to flip? By manipulating an equation based on the result of the ...
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 ...
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
2answers
79 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
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
32 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
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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
39 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
38 views

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

We know that $E(X+Y) = E(X) + E(Y)$. But why is $E|X-Y|$ $\ne$ $E|X| - E|Y|?$
1
vote
1answer
41 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 ...
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
48 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 ...
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 $c_i$...
1
vote
1answer
857 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 ...
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
32 views

Prove $|M_{T \wedge n}| \le c + K$

From Williams' Probability with Martingales Is $\sigma_k^2$ random (and not constant)? How can that be? As far as I know unconditional variance and unconditional expectations are supposed ...
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
1answer
72 views

Show that $P(X_n \leq x_n) \rightarrow P(X\leq x)$

Suppose $X_n$ converges in distribution to $X$, $x_n\rightarrow x$ and the cumulative distribution function for $X$ is continuous at $x$. Show that $P(X_n \leq x_n) \rightarrow P(X\leq x)$. I tried ...
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 $E(x)$?
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 ...
5
votes
3answers
552 views

Negative Variance

I have two independent variables $X$ and $Y$. $W=X-Y$ when $X\sim \mbox{Bernoulli}\left(1/2\right)$ and $Y\sim N(0,1)$. This puts $\operatorname{Var}(x)=1/4$ and $\operatorname{Var}(Y)=1$, but I have ...