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

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0
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14 views

Convergence in mean square of function of Bernoulli

I am wondering if $Y_n$ converge in mean square: $Y_n=2^nX_1X_2\cdots X_n$ where $X_k, k=1,2,\ldots,n$ is equi-probably i.i.d. Bernoulli random variable. Here is what I have tried: ...
0
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1answer
40 views

Calculating average distance of the dart from a bullseye, with two different Gaussian random variables.

Im a master student and this course is about probability and random processes. I had troubles with some of the material and therefore I started reading from the start again. The exercise im stuck with ...
1
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0answers
52 views

Mutual information of independent fair binary random variables

Let random variables $X,Y$ independent fair random variables that take the values 0 and 1 with equal probability and $Z=X+Y$. So, $I(X;Y)=0$ and I am trying to find their conditional mutual ...
1
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0answers
16 views

The random variables and some definition

What do we mean when we have a random variable $X$, and we say $X$ is $O (1)$?or more general $O (n)$?
2
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1answer
74 views

Does weak convergence in $L^2$ imply almost everywhere convergence of Cesaro averages?

Consider a bounded sequence $f_n\in L^2(X,\mu)$, $\|f_n\|_{L^2}\leq C$. Is the following true: if $f_n \to f$ weakly (that is $\langle f_n,g \rangle \to \langle f,g \rangle$ for every $g\in ...
2
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0answers
29 views

Derivative of cumulative generating function at zero equals expectation value

Let $X$ be a random variable with values in $\mathbb{N_0}$. Then we can define the cumulative generating function of $X$ via $$ F_{X}: (-\infty, 0] \rightarrow \mathbb{R} \quad \quad t \mapsto ...
0
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1answer
76 views

There are 5 balls in an urn: 3 blacks and 2 reds. We draw 2 balls from the urn. Let X = number of blacks balls drawn.

There are 5 balls in an urn: 3 blacks and 2 reds. We draw 2 balls from the urn. Let X = number of blacks balls drawn. (a) Find the distribution of X. (b) Compute the expected value of X and the ...
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0answers
25 views

Joint convergence in distribution and independence

I am having some trouble getting my head round independence of random variables when it comes to convergence in distribution. Suppose $\{X_n\}_{n=1}^\infty$ are i.i.d. standard normal, and ...
3
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0answers
54 views

Limiting sequence of exponential random variables

Let $\eta_k$ be i.i.d. random variables having an exponential distribution, $$F_\lambda(x) = P(\eta_k \leq x) = 1-e^{-\lambda x}$$ for $x \geq 0$. Consider a sequence $\xi_k = ...
0
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1answer
22 views

Maximum of two Independent Random Variables with Erlang distribution

While I am deriving the maximum of two Erlang Indepenent random variables $$ z = \begin{cases} l, & \text{if $l$>$h$ and $l$>0} \\ h, & \text{if $h$>$l$ and $l$>0} \\ h, ...
0
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1answer
24 views

Distribution of the difference between two random variables

I have two independent random variables of Erlang distribution or you can consider them Gamma distributions but they are positive. Z = Y-X The difference between them should be a distribution that ...
2
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0answers
13 views

given the power spectral density, how to get $\|\|_{\infty}$ norm

Given the power spectral density of a random signal $P(\omega)$ It is possible to estimate or compute the upper bound of the signal, i.e., the $\|\|_\infty$ norm?
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3answers
33 views

5 distinct number distributed to 5 persons

$5$ distinct numbers are randomly distributed to players numbered $1$ to $5$. Whenever two players compare their numbers, the one with the higher one is the winner. Initially, players 1 and 2 compare ...
0
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0answers
40 views

Uniform distribution

Let $X_{1} ..., X_{n}$ be a sample from $U([0,\theta])$ for $\theta>0$, $X_{n:n}$ denotes the maximum observation, $\bar{X}=\frac{1}{n}\sum_{0 \leq k \leq n}X_{k:n} $ compute probability $$ ...
0
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2answers
56 views

PDF of several draws from an uniform distribution?

Suppose I draw several times from an uniform distribution, $X\sim\mathcal{U}(0, 1]$. (I'll use $\mathrm{R}()$ to denote an independent drawing.) What is then the PDF of several draws, added and/or ...
0
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0answers
6 views

Bounding the spread of a random variable

I have a random variable $y$ with a finite distribution of values $(y_1,y_2,\dots,y_d)$, with associated probability $(p_1,p_2,\dots ,p_d)$ and average $\langle y \rangle = \sum\limits_{i=1}^d x_i ...
2
votes
2answers
43 views

How to prove or disprove the following statement?

How to prove or disprove the following statement: "If random variables X and Y are independent, AND if random variables Y and Z are independent, then X and Z are independent." I tried to use Bayes ...
1
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2answers
26 views

Continuous probability; find the probability density function

Let $X$ and $Y$ be random variables and let $Z = \frac{(X+Y)}{2}$. Suppose you randomly choose two numbers $X,Y \in [1,3]$. Find the probability density function of $Z$ and the expected value $E(Z)$. ...
1
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2answers
28 views

Prove that the random variables $X$ and $Y$, $EX^2< \infty$ and $EY^2<\infty$ applies: $DX=E(D(X|Y))+D(E(X|Y))$

Prove that the random variables $X$ and $Y$, $EX^2< \infty$ and $EY^2<\infty$ applies: $$DX=E(D(X|Y))+D(E(X|Y))$$ $(D - \text{ variance }E- \text{ expectation })$ This semestar, we have been ...
0
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1answer
85 views

Average number of students who will get hw back. [duplicate]

The homeworks of $20$ students is collected, randomly shuffled and returned to the students. What is the average number of students who will get back their own homework? PS: I tried doing this using ...
1
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1answer
36 views

Expected value of sequence obtained by multiplying independent RVs by square-integrable RV

the problem in question is: Let $\{X_n\mid n\ge 1\}$ and $Y$ be random variables on probability space $(\Omega, \mathcal F, \mathbb P)$. Suppose that the $X_n$ are independent, $E[X_n] = 0$ and ...
0
votes
1answer
20 views

Probability of counter being 23

This is a probability question. Suppose we start our journey from step number 0. We now toss a coin. If the result is a head, we add 3 to a counter. If the result is a tail, we add 2 to the counter. ...
0
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0answers
37 views

Expectation of choosing one object?

There are $b$ balls in an urn, exactly one of which is black. You choose $n$ balls from the urn at a time and throw away all non-black balls. If the black ball is chosen, it is replaced back into the ...
0
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1answer
18 views

What's the probability that hitting shuffle on an album of 5 tracks 10 times will allow you to see the original order at least once?

I wanted to validate my approach, since I'm off from a solutions script I have on hand. The probability of getting the original tracklist is $p_s = \frac{1}{5!} = \frac{1}{120}$ Then using the ...
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0answers
27 views

Convergence in probability to zero of a random variable

For all $n\in\mathbb{N}$ let $B_{j,n}$ with $j=1,...,n$ be a triangular array of independent Bernoulli variables such that $$ \mathbb{P}\left[B_{j,n}=1\right] = ...
0
votes
1answer
21 views

$X$ RV with cdf $F$, $W \sim U[0,1]$ independent $\Rightarrow$ $V:=WF(X)+(1-W)F_{-}(X) \sim U[0,1]$

I try to prove: Let $X$ be a discrete random variable with cdf $F$, $F_{-}(x):=P(X<x)$, $W \sim U[0,1]$ a random variable and $X, W$ independent. Then $$V:=WF(X)+(1-W)F_{-}(X) \sim U[0,1].$$ My ...
0
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0answers
12 views

Using properties of the dirac delta to understand the probability density of a function of a random variable

I have a random variable $\eta$ defined as a function of a random variable $\xi$: \begin{align} \eta=g(\xi). \end{align} The probability density of $\eta$ is \begin{align} ...
5
votes
0answers
77 views

Random sphere-valued fields

I would like to generate random functions from an $m$-sphere $S^m$ to an $n$-sphere $S^n$ that are not too wild, some kind of generalization of random Gaussian fields. More precisely, I want $f(x)$, ...
2
votes
2answers
42 views

How to calculate the expected value of a discrete random variable?

A private investor has capital of £16,000. He divides this into eight units of £2,000, each of which he invests in a separate one-year investment. Each of these investments has three possible ...
-1
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1answer
40 views

Show that we always have $Y + Z = X + 4$.

Let $X$ be a Geometric random variable with parameter $p =\frac{1}{2}$. We define another random variable $Y$ in terms of $X$ as follows. $Y = \min\{X,4\}$ Here $\min\{X,4\}$ is the minimum between ...
0
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0answers
39 views

$X,Y$ independent from sigma algebra $\Sigma$ then also $X+Y$?

Let $X,Y$ be independent from a sigma algebra $\Sigma$. Does this mean that $X+Y$ is independent from $\Sigma$? I just don't know how to show it, but maybe a yes/no answer and a good hint could help ...
2
votes
3answers
112 views

Let X be a Geometric random variable with parameter p = 1/2. We define another random variable Y in terms of X as follows. Y = min{X,4}

Let X be a Geometric random variable with parameter p = 1/2. We define another random variable Y in terms of X as follows. Y = min{X,4} Here min{X,4} is the minimum between the value of X and 4. ...
0
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0answers
20 views

Check independence using the relative distance test for generating random numbers

I generated $500000$ random numbers of uniform distribution $[0,1]$ with $10$ Intervals, and I obtained the table as below. I compute chi-squared test is 3.037 with acceptable with the probability of ...
0
votes
1answer
46 views

Let $X$ be a uniform random variable on $(0,1]$. Find the distribution function and the density function of $Y=-\frac{1}{\lambda} \ln(X)$

I'm doing this problem and I have some doubts: Let $X$ be a continuous uniform random variable on $(0,1]$. Find the pdf and the density function of $Y=-\frac{1}{\lambda} \ln(X)$, with $\lambda ...
0
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0answers
24 views

Convergence of sums of dependent Bernoulli random variables

Let $A_{j,n}$ and $B_{j,n}$ be two triangular arrays of independent Bernoulli variables such that $$ \begin{array}{l} \mathbb{P}\left[A_{j,n}=1\right] = p_{A,n},~~\mathbb{P}\left[A_{j,n}=0\right] = ...
1
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0answers
55 views

Does a sigma algebra specifies a random variable?

If we are given a random variable $X$, we call $\sigma(X)$ the sigma algebra generated by $X$ If I am given a sigma algebra $\mathcal F$, 1) when does exists a random variable $X$ such that ...
0
votes
1answer
17 views

How can I extract probabilities from my data?

Suppose I have a series of IID RVs $\theta_1, \theta_2, \dots , \theta_J$ all taking values between $0$ and $\pi$. Suppose further that the distribution is unknown. Is there anyway I could extract ...
4
votes
1answer
155 views

Can two rv-s $X,Y$ be dependent and $E(XY)=E(X)E(Y)$?

Can someone define independence of two random variables with this "product rule", or are there any counterexamples?
0
votes
1answer
23 views

Probability density function of a function of a uniform random variable

Suppose that a random variable $X$ has a uniform distribution over the interval (0,1) i.e. $X$~U(0,1). Let $Y$ be a random variable such that $Y$ = |$X$-1/3|. What is the probability density function ...
0
votes
1answer
37 views

how to prove that every positive r.v. X is the non-decreasing limit of a sequence of simple r.v.’s

Every positive r.v. X is the non-decreasing limit of a sequence of simple r.v.’s. That is, if $X \geq 0$, then there exist simple r.v.’s Xn, defined on the same probability triple, such that: ...
0
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0answers
27 views

distribution density and probability distribution function?

random variable: the number of tosses at least as large as $5$ obtained in $18$ rolls of a fair dice. $P(X=5)=1/6 P(X=6)=(1/6) (1/3)^{18}$ how do I compute the distribution density and probability ...
1
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0answers
51 views

The convergence of $\sum \pm a_n$ with random signs

Suppose that $\left(a_n\right)$ is a sequence of real numbers and that $\left(\varepsilon_n\right)$ is a sequence of IID RVs with $$P\left(\varepsilon_n = \pm 1\right) = \frac{1}{2}$$ According to ...
1
vote
1answer
40 views

Probability problem - Converse to SLLN

I'm having trouble with an exercise (E4.6 Converse to SLLN) in "Probability with Martingales" by David Williams. The problem is as follows: Let $Z$ be a non-negative RV. Let $Y$ be the integer part ...
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votes
3answers
58 views

Functions of random variables

I have two random variables $X$ and $Y$, specific elements that these random variables can take are $x$ and $y$. Now, say I define a random variable $f(x,Y)$ (a function of the random variable $Y$) ...
0
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0answers
14 views

proof that aX, XY are random variables

X and Y are random variables on the same probability triple. how should proofs of the below look like? a) $aX$ is a r.v. for $X \in R$. b) XY is a r.v. what should be the starting point and what ...
0
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1answer
22 views

Calculating the probabilities of a certain random variable

If I have an exponential random variable with a mean of $0.5$ and we consider a sample of length $10$; How can I calculate if its minimum value is bigger than a certain probability or how can I ...
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0answers
16 views

Discounted stochastic linear regulator problem with a transversality condition: I think I have a solution, need proof

Consider a sequence of i.i.d. random variables $ \left\{ {{\varepsilon _t}}\right\}_{t = 1}^\infty $ with $E\left( {{\varepsilon _t}} \right) = 0 $ and $ E \left( {\varepsilon _t^2} \right) = ...
1
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1answer
36 views

Meaning of probability density function - continuous random variables

Suppose we have a random variable X uniformly distributed over the interval (0,1). The probability density function of X is given by: $$f(x)=\left\{\begin{array}{l} 1 \space\space if \space\space ...
3
votes
1answer
56 views

If $X$ is a Random Variable, finding a $Y$ for which $\mathbb{P}(\vert X-Y\vert >\varepsilon)<\varepsilon$

Suppose we have a probability space $(\Omega ,F,\mathbb{P})$ where $F=\sigma(A)$ for some algebra $A$. For any bounded random variable $X$ and $\varepsilon>0$ I am trying to prove that there exists ...
0
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1answer
28 views

calculating expected values to earns

I am having trouble with this question with regards to calculating expected values: Peter and Maria players each rolling one die and earns \$1 the player with the best score (Each player rolls one ...