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

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0
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0answers
8 views

Show equalities of random variables

In the text we showed that a geometrically distributed random variable W has the lack of memory property. Now assume that the range of W is {1,2,3...} and that P(W = j + 1|W>j} = p for j = ...
-1
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0answers
7 views

Need help to solve transformation of random variable [on hold]

can anyone help me to solve this problem..it will be really helpful. Thank u
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2answers
22 views

Does $X_n \xrightarrow{\text{in distr.}} X$ and $|X_n|\leq Y$ imply $|X|\leq Y$?

We know that $$X_n \xrightarrow{\mathbb P} X \text{ and } |X_n|\leq Y \implies |X|\leq Y \text{ a.s.}$$ I was wondering if the same holds in case of convergence in distribution. So far, I've shown ...
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0answers
22 views

Why this is a martingale?

Setup: $W$ probability space $Z_i : W \to L_i $ random variables ($L_i$ finite, for example $\{0,1\}$) $f: Z_1 \times \ldots \times Z_n \to \mathbb{R}$ $X_i := \mathbb{E}[f \mid Z_1,..,Z_i]$ Why ...
-2
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0answers
28 views

Borel function and random variable [on hold]

Positive part of a function X+ is borel function and will be a random variable if X is random variable. I know this things, but how should i prove it mathematically ? It can also be put up as, "Borel ...
0
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1answer
19 views

When is a random variable is said to be well-defined?

In the paper On the Bootstrap of the Sample Mean in the Infinite Variance Case by Keith Knight, on page 1170 at the bottom of the page before the theorem, the author mentions that the random variable ...
3
votes
1answer
24 views

How can I find the distribution of a stochastic variable X^2 if X is normal standard distributed? [duplicate]

I am considering a stochastic variable X that is standard normal distributed i.e. $$ F_X(x) = \int_{-\infty}^x\frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}}dt $$ How do I find out the distribution of $X^2$? ...
0
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0answers
27 views

Discretization of the standard uniform dist.

I need some help. Sorry for the poor use of LaTEX... a) $U_{n} = \lfloor{nU}\rfloor/n$ prove that $\lim_{n \to +\infty} {U_{n}} = U$ where $U \sim unif[0,1]$ and thus that $lim_{n \to +\infty} ...
1
vote
1answer
26 views

What is the variance of this random variable: number of items

Let us assume that we have a capacity $n$ which tends to infinity. We have an infinite number of random variables $X_1, X_2, \dotsc$, where each $X_i$ is independent and identically distributed with ...
0
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1answer
26 views

Expected value and Variance calculation

Suppose $f$ is an uniformly distributed random variable with parameters $-1,1$ and $g$ is a Poisson-distributed random variable with parameter $\lambda >0$. We assume that $f$ and $g$ are ...
-2
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0answers
10 views

Relation between Bernoulli RV, binomial RV, geometric RV and Poisson RV [on hold]

what is the relation between Bernoulli RV, binomial RV, geometric RV and Poisson RV? And how we represent them?
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0answers
24 views

Number of Points in a circle [on hold]

Let $D$ be the density of points forming a random distribution within a specified area. We draw a circle of a particular radius in this region. How to calculate the expected number of points in this ...
0
votes
1answer
28 views

Corollary of Kolmogorov zero-one law

Here is another corollary of the theorem: Kolmogorov zero-one law given in my textbook (Probability path). How can I apply the said theorem given that if $X_n$ are independent random variables, ...
0
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0answers
9 views

Z transform of a random variable

Let's say we have: $\tau(k) = k\times r$ where $r \sim N(0,\sigma ^2)$. Therefore, in Z domain we have: $\tau(z) = -z\frac{dR(z)}{dz}$. But what is the Z transform of a Gaussian random variable ...
0
votes
1answer
43 views

Difficult probability problem [on hold]

I am stuck in this question $33$ miners are trapped in a mine. There´s an elevator that takes $10$ minutes to go down and another $10$ minutes to pick up a miner and return to the surface. One of ...
0
votes
1answer
33 views

Z~U[0,1] and X=f(Z) and f is:

I have found the f(z): Now, I need to find pdf of X. And I can see that 0< f(Z)=X<1, I don't know how I am going to get f(X), I just can see that f(X)=0 when X<0 and x>1, but I can see a ...
3
votes
1answer
43 views

Construct independent $X_n$ such that $\sum_{n=1}^\infty Var(X_n)=\infty$

How to construct an independent random variable $(X_n)_{n\in\mathbb{N}}$ such that $\sum_{n=1}^\infty X_n$ converges and $\mathrm{Var}(X_n)$ is uniformly bounded by some constant C, but ...
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0answers
27 views

Discrete Random Variable and Its Probability Distribution

EZ Language Center offers a 2-month summer course on three of the most popular and romantic languages aroun the world. French, spanish, and italian. Their database shows that .27, .40 and .33 of their ...
3
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0answers
34 views

Distribution of $\frac{X}{|Y|}$, where X and Y are standard normal r.v.'s

Let X and Y be independent standard normal random variables. What is the distribution of $\large \frac{X}{|Y|}$? Attempt: Let $\large U = \frac{X}{|Y|}$ and $ V = |Y|$. This transformation is not ...
1
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1answer
16 views

Exponential random variable

Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked is an exponential r.v. with parameter 1/20. Smith has a used car that he claims ...
0
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1answer
44 views

Proving Negative of Standard Normal is Standard Normal

Let X be standard normal random variable $N(1, 0)$ prove that $-X$ is also standard normal. I think I am stuck on a technicality but here is my attempt: Let $Y = -X$ P(Y $\leq$ u) = P($-X$ $\leq$ ...
0
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0answers
11 views

Question regarding piecewisely defined random variable

If I have two independent random variables X and Y with known PDF. And another random variable defined piecewisely as $Z=0.9\times(X-Y)$ if $X-Y<c1$ $Z=0.4\times(X-Y)$ if $c1<=X-Y<=c2$ ...
2
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2answers
53 views

Would Evaluating a polynomial at uniformly random points outputs random values?

I`m wondering if we evaluate a polynomial on many points picked uniformly at random. Can we say the output values Y's are uniformly at random?
2
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2answers
30 views

Definition of multivariate random variable

Let $(\Omega,\mathcal E,P)$ be a probability space and $X_1,\dots,X_n$ random variables on $(\Omega,\mathcal E, P)$. Then I can define the vector $X=(X_1,\dots,X_n)\colon \Omega \to \mathbb R ^n$ and ...
0
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2answers
46 views

Uniform Random Number

Two uniform random numbers are chosen one after the other. what is the probability of second number second random number greater than first number? I tried this way Please correct me if I am wrong. ...
-3
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0answers
36 views

Expectation value and Variance [closed]

If $E[X]=1$ and $Var(X)=5$, find a) $E [(2+X)^2]$ b) $Var (4+3X)$ Please all maths experts kindly help!!!
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votes
1answer
24 views

Discrete RV problem

A flight control system uses four independent computers working in parallel. At each critical step, the computers “vote” to determine the appropriate step. The probability that a computer will ask for ...
0
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0answers
23 views

Convergence of random saddle point

Let $y_n^*$ be the solution of $$ y = g_n(y) $$ where $g_n(\cdot)$ is a random function. Suppose that for fixed $y$ $$ g_n(y)\to h(y) $$ almost surely and pointwise as $n\to\infty$. Is there any ...
1
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0answers
39 views

Find the type of noise a random variable should be contaminated with, such that certain properties are satisfied.

Let $\mathbf{x}$ be a random column vector in $\Bbb{R}^n$, given as follows $\mathbf{x}=(x_1,\ldots,x_n)^\top$. We require $\mathbf{x}$ to follow a multivariate Gaussian distribution with mean vector ...
0
votes
2answers
14 views

Marginalization question $\Pr[a] = E_X[\Pr[\ a|X\ ]\ ]$

I'm reading explanation of a theorem, and there's one step that I can't understand. I know it should be simple enough, but I just can't think of the reasoning atm. The step says, According to ...
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2answers
14 views

$L^1(P)$ random variable limit.

for random variable $X$ in $L^1(P)$, What is $\lim_{a\to \infty}a * P( |X| > a )$ ? I think its value equals 0. But I can't solve this problem. I used markov's inequality But I can not ...
1
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1answer
14 views

Product of sequence of uniform integrable random variables are uniformly integrable?

If $\{X_i\}$ for $i \in I $ and $\{Y_j\}$ for $j \in J$ are uniformly integrable.Then prove that, $\{X_i+Y_j\}$ for $(i,j) \in I \times J$ is uniformly integrable.What about $ \{X_iY_j\} $ for $(i,j) ...
0
votes
1answer
49 views

Chebyshev Inequality

I am reading a research paper, and the author claims to get to a desired result by making use of the Chebyshev Inequality. I can get to the desired result also with some reasoning, but I fail to ...
2
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0answers
25 views

how to prove independence in this case

The question is : $X_1,...,X_n$ are i.i.d.$Uniform(0,\theta)$. Let $X_{(n)}$ denote the maximum of these $n$ random variables. Prove that $\frac{X_1}{X_{(n)}}$ and $X_{(n)}$ are independent. What I ...
0
votes
1answer
19 views

Notation in sequences of random variables

I read in statistics books that a sequence of random variables are often written as ${X_n}$. But in all the theorems it just says $X_n$. Why is that? And does ${X_n}$ symbolize the WHOLE sequence ...
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0answers
4 views

A derivation of expected values of difference of random processes

Hope to ask a calculation step from a paper: Let $\mathcal{S}$ be a subset in the Euclidean space $\mathbb{R}^n$. Let $x \in \mathcal{S}$. Let $y \in\mathcal{S}_2^{k-1} = \{y \in \mathbb{R}^k ...
0
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0answers
21 views

How to understand a complicated random process

I read a paper and it defines a r.p as following: $x \in \mathbb{R}^n$, $y \in \mathbb{R}^k$ $\{h_i\}_{i=1}^n$ and $\{g_j\}_{j=1}^k$ are two indep. sets of orthonormal r.v.'s Define a r.p: ...
3
votes
2answers
56 views

Finding $f_Y$ such that $Z=Y\cos(X)\sim\mathcal{N}_{0,\sigma}$ for $X\sim\mathcal{U}[0,2\pi]$

I need to choose the probability distribution $f_Y(y)$ of a random variable $Y$ such that the variable $Z=Y\cos(X)$ is normally distributed with zero mean, i.e. ...
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0answers
19 views

Bounding the Correlation Coefficient

Let us assume we have two random variables $X$ and $Y$ where $X = f(A, B, C)$ and $Y = g(A, B, C)$. $A, B, C$ are 3 independent random variables and the functions $f, g$ are known but rather expensive ...
0
votes
1answer
16 views

Central Limit Theorem & Delta method problem

Let $U_1$,...,$U_n$ be a random sample from the U(0,1) a. Let $X$=-log($U$). Find the distribution of X b. Let $Y$=$1/{\prod_{i=1}^n U_i^{1/n}}$, where $U_1$,...,$U_n$ be a random sample from the ...
0
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0answers
38 views

Expectation and variance of a comboflip

Here's the question, a comboflip is the simultaneous flip of a fair coin and toss of a fair die. Comboflips are done until at least one head has been seen and at least one 6 has been seen(they do not ...
0
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2answers
40 views

How can I do a constructive proof of this:

Say Z is a non-negative R.V, and P(Z>0)>0. Then exists a a>0 and an b>0 such P(Z>a)>b. I am not sure how to start with the proof, I've been assigning numbers than can qualify for some CDFs but I don´t ...
0
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0answers
35 views

Continuous random variable - probability density function question

A continuous random variable $X$ has a single parameter $a$. The probability density function of $X$ is $f(x)=c(1-x^2)$ for $-a<x<a$ and some constant $c>0$. a) What are the allowable values ...
1
vote
1answer
12 views

Notation for image of a discrete random variable?

Suppose we have a discrete probability space $(\Omega,\Sigma,\mathbb{R})$ and a discrete random variable $X:\Omega \to \mathbb{R}$. A usual way to denote the set of values that $X$ takes is simply ...
0
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1answer
11 views

Proving something is strict stationary

Let $W$ be a uniform distribution on $(0,\pi)$. Let $Z_t=\cos(tw)$. I know that $Z_t$ is a strict stationary but I have no idea how to prove this. Can someone give me some methods?
1
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2answers
30 views

Expectation of a function of pairs of random variables

For positive random variables $(X_1, Y_1)$ and $(X_2, Y_2)$, suppose that $(X_1, Y_1)$ and $(X_2, Y_2)$ have the same distribution and (the two pairs) are independent. Also suppose that $E[Y_1|X_1] = ...
0
votes
1answer
32 views

Pdf of the product of an exponential r.v. and a beta r.v.

Let $X$ and $Y$ are 2 independent random variables, where $X$ has an exponential distribution with parameter $1$ and $Y$ is $\beta(a,b)$ distributed. What is the Pdf of $W=XY$ ? Thanks !
0
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1answer
26 views

Uniformity of the difference between two random variables

What can I say about the distribution of two random variables $A$ and $B$ such that $A-B$ is uniformly distributed?
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2answers
68 views

Clarification of Identification [closed]

This is more of an observation question. When you see $x$, In $f(x) = x^2$ And when you see $g(x) = x^3$ You automatically identify $x = x$ Wouldn't the $x$'s be off by a little bit? But ...
0
votes
1answer
34 views

Explicit CDF associated to Gamma PDF [closed]

Thanks in advance for the help with this! I'm struggling to follow the solution in the book for this problem. Any help is greatly appreciated. Let the distribution function of X for x>0 be: $$F(x) = ...