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

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31 views

Problem that may use Borel Cantelli Lemma

So, there is a sequence of identically distributed independent random variables taking values on the integers, and they have a positive expectation. The problem is to prove that with probability 1 the ...
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2answers
51 views

Probability of rectangles area being less than 0.5 w/ total length of sides = 2

Question: A random point splits the interval [0,2] in two parts. Those two parts make up a rectagle. Calculate the probability of that rectangle having an area less than 0.5. So, this is as far as ...
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1answer
22 views

Distribution of the product of two lognormal random variables

Let $X_1$ and $X_2$ be two normal random variables. Write $X_1\sim N(\mu_1, \sigma^2_1)$ and $X_2\sim N(\mu_2, \sigma^2_2)$, to fix ideas. Consider the corresponding log-normal random variables: ...
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0answers
14 views

Continuous variable calculate conditional expectation

I have this math question that I am very stuck on. Let $X$ and $Y$ be jointly continuous with density $f_{X,Y}=x+y$ if $x,y\in [0, 1]$ and zero otherwise. Let $Z=1$ if $X>Y$ and zero ...
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1answer
22 views

Expected sum of the draws by Mr. B?

Ms. A selects a number X randomly from the uniform distri- bution on [0,1]. Then Mr. B repeatedly, and independently, draws numbers Y1,Y2 ,.... from the uniform distribution on [0,1], until he gets a ...
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1answer
44 views

Let $(X_i)$ be i.i.d. exponential, is the set $\{X_1,X_2,\ldots\}$ almost surely dense in $(0,\infty)$?

To be clear, I'm asking if the range of the random sequence $(X_i)$ is dense in $(0,\infty)$ a.s. I thinks the answer is yes, because for any $0<a<b<\infty$, we have $$P(X_1 \notin (a,b), ...
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45 views

Integrating a Random Variable and establishing the maximum of a related function

Frequency Regulation of a Power Grid I have a battery that is used to regulate the frequency of a power grid. That is, as the grid frequency varies about it’s ideal value, $f_{nom}$, the battery ...
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1answer
19 views

Characteristic function of discret random variable

I try to show the following: Suppose $(X_n),n\geq1$ is a sequence of random variables with uniform distribution on $\{1/n,\dots,n/n \}$. Show that $(X_n)$ converges in distribution to a random ...
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0answers
15 views

Simple and partial correlation

(i) The partial correlation coefficient $r_{12.3}=r_{12}-r_{13}r_{23}/(\sqrt{(1-r_{13}^2)(1-r_{23}^2)}$ and the simple correlation coefficient ...
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1answer
40 views

A probability transformation

Let $X,Y,Z$ be continuous random variables; $Z$ are independent of $X,Y$. Is the following transformation right ? \begin{align} P(X,Y \in (a,b),Y+Z \notin (a,b))&=\int_a^bP(X \in (a,b),y+Z \notin ...
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0answers
117 views

Convergence of an implicitly defined sequence of random variables

Let $\{X_n\}_{n\ge 1}$ be a sequence of independent identically distributed Poisson random variables with mean $\lambda^*$. Consider a sequence of random variables $\{\hat{\lambda}_{n}\}_{n\ge 1}$ ...
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1answer
32 views

Use the delta method to find the distribution of $Z_n$

Let $\overline{X}_n=\overline{X}$ the sample mean such that $\sqrt{n}\overline{X}_n\rightarrow^D N(0,1)$ where $\rightarrow^D$ means converge in distribution. Use the delta method to find the ...
2
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1answer
51 views

Conditional probability when conditioning on continuous-discrete variables

I am confused on the notion of conditional probability when the conditioning variable is continuous. Consider the random variables $X,Y$ on the probability space $(\Omega, \mathcal{F}, P)$ with ...
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0answers
19 views

Arbitrary vs. random subsets: computing probabilities

Let $G=([n],E)$ be a graph having minimum degree $\delta(G) \geq (1-\delta) n$. For some $q=q(n)$, let $G_q=([n], E_q)$ be the random subgraph of $G$ obtained by deleting each edge independently with ...
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0answers
32 views

Is a “deterministic” subset of a random subset random?

Let $S$ be some set and consider $X \subseteq S$ of size $|X|=x$ u.a.r. (among all the subsets having this size). Now, use some properties of this set $X$ to find some subset $Y\subseteq X$ of some ...
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0answers
22 views

Compute probability that a random subset has a certain property (when we know probability for an arbitrary subset)

Suppose we have a ground set $[n]:=\{1, \dotsc, n\}$. Now, we pick a random subset $S \subseteq [n]$ u.a.r. among all the subsets of $[n]$ having size equal to $s$. In general, if we know that for ...
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1answer
18 views

What is the probability distribution of the following random variable?

Let $A^n$ and $B^n$ be independent random variables taking values in $\{0, 1\}^n$. Let $Y^n = A^n + B^n$ (Hence, taking values in $\{0, 1, 2\}^n$). How can we express the distribution of $Y^n$ in the ...
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1answer
36 views

How to show $P(|X-E(X)|\leq x)=1\implies V(X)\leq x^2$

Let $X$ be a random variable with finite variance. I am trying to show if $P(|X-E(X)|\leq x)=1$ then $V(X)\leq x^2$. Could somebody please help me correct my working? $|X-E(X)|\leq ...
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1answer
34 views

Variance of a random variable [closed]

How do you get the variance of a random variable $X$ where $X = \frac{1}{6}(A \cdot B)$ and where $A$ and $B$ are two independent random variables with variances $\sigma_A^2$ and $\sigma_B^2$, ...
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1answer
17 views

A random variable is mapping from sample space to real numbers. How about random process?

A random variable is mapping from sample space to real numbers. How about random process? Can we think of the simplest random process as again a mapping from sample space to real numbers, with the ...
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2answers
93 views

Suppose that $E[X^n] = 3n$. Find $E[e^X]$…

Suppose that $E[X^n] = 3n$. Find $E[e^X]$. Hint from my professor: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} +···$ Not quite sure how to solve this problem, wouldn't $e^x$ go on exponentially. ...
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0answers
31 views

Could someone tell me what's wrong with my understanding about $var(Y)$?

Suppose $X$ is equally likely to take three values: $−1, 0, +1$. Let $Y = X^2$.The probability mass function for random variable Y is $P(Y=0)=\frac 13$ and $P(Y=1)=\frac 23$ Here is the thing, ...
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1answer
37 views

Let X be any random variable. Find $\displaystyle\lim_{b\to-\infty} P[X \le b]$…

Let $X$ be any random variable. Find $\displaystyle\lim_{b\to-\infty} P[X \le b]$ I would think $b$ is zero, making this an infinite sum but really not sure. Any help/direction with this problem is ...
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0answers
21 views

Show that if X is a continuous r.v. and it takes only positive values then $E(X) =\int_{0}^{∞}Z P[X ≥ t] dt $ [duplicate]

Show that if X is a continuous r.v. and it takes only positive values then: $$E(X) =\int_{0}^{∞} P[X ≥ t] dt$$ I am not really sure how to begin this proof. Any help or insight would be ...
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0answers
37 views

Sequence of random variables, mean zero, convergence to -infinity

What would be an example of a sequence $(X_k)$ of independent random variables with zero mean such that $$\frac{1}{n} \sum_{i=1}^{n} X_{i} \xrightarrow[\mbox{almost surely}]{n \to \infty}-\infty\ ...
2
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1answer
22 views

Are these facts about the Poisson process correct?

Before studying theorems one by one, I want to check whether it is right what I know about Poisson process. Let $\left\{N(t)\right\}$ be a Poisson process. Then the number of the event occur during ...
3
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0answers
16 views

Finding a random variable

Let $X_{1}, X_{2}, \dots$ be i.i.d Uniform[0,1] random variables. Find a random variable $X$ such that: $$(X_{1}\times\cdots\times X_{n})^{\frac{1}{\sqrt{n}}}e^{\sqrt{n}} \overset{d}{\to} X$$ The ...
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0answers
20 views

Show that an event has strictly positive probability

Consider the random variables $W_i,W_j, X_i, X_j$ with $X_i\sim X_j$, $X_i\perp X_j$ and $W_i\sim W_j, W_i\perp W_j$, where $\sim$ denotes equal probability distribution and $\perp$ denotes ...
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1answer
41 views

Discrete random variable decomposition over sample space

If $X$ is a discrete random variable taking values in $\{1,\dots,N\}$, does the sample space decomposition identity $X=\sum_{k=1}^N k1_{\{X=k\}}$ always hold, or are there instances where it might not ...
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1answer
20 views

Consistency of estimators

Let $(X_i)_{i\in\mathbb{N}}$ be normally distributed with parameters $\mu, \sigma^2$. Let $cov(X_i, X_j)=p_{j-i}$ for $i<j$. A sequence of estimators for $\mu$ is given by $\hat{\mu}_N = ...
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1answer
36 views

Probability Mass Function of having both loaded & fair coins [closed]

Suppose a box contains many coins that are either biased (loaded) or balanced. A loaded coin has probability of landing on its head as p ∈ (0.5, 1.0), and a balanced coin, of course has probability ...
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1answer
27 views

Expected sum value of permutaion

We have a set(A) of N elements. Let's assume elements are e1,e2,e3..etc. Value of each element can be 0 or 1. Another set of N elements(set B) are given, ...
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2answers
43 views

Independence and expected value

I have a theorem that says If two random variables $X,Y$ are independent, then for any non-negative measurable functions $f:E\to\mathbb{R}$ and $g:E\to\mathbb{R}$ the following holds ...
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2answers
58 views

Find the mathematical expectation [closed]

Find the expectation of $$f(x) = a(1+x)^{-(1+a)}, \quad x>0.$$ The answer given is $\frac{1}{a-1}$. I am not getting the answer. Please help.
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1answer
49 views

Sum of random variables goes to infinity

I'm trying to show the following: Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d random variables with $\mathbb{E}[|X_1|]<\infty$ and $\mathbb{E}[X_1]=\mu$. Consider ...
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0answers
19 views

Distribution function of Sum of IID Exponentiation Variables of Variable amount

So I'm trying to determine the distribution function of a random variable, S, give: $N \sim Geo(\frac{1}{1+\lambda}) $ $S_i \sim Exp(\mu), \forall i\in [0,N]$ $S = \Sigma^{N}_{i=0}S_i$ $S = ...
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0answers
33 views

Let $X$ and $Y$ be iid real-valued random variables. Show $P[|X-Y| \le 2] \le 3P[|X-Y| \le 1]$. [duplicate]

Found this question in The Probabilistic Method and tried for hours to prove it, but I'm not getting anywhere. Can anyone walk me through it? I see that if we can show $P[1 \le X - Y \le 2] \le P[|X ...
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1answer
34 views

How do can i solve the integral, finding cdf [closed]

Let $X$ be an exponential random variable with mean 1 and Y a uniform random variable between $0$ and $1$. Assume X and Y are independent and let $Z =e^{X/2}$ Find the joint cumulative ...
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34 views

When convergence a.s. implies convergence in mean?

Can someone help me with proving the following: Assume that $X_n$ converges almost surely to $X$, where $X_n$ is a sequence of non-negative random variables. Furthermore, assume that the sequence ...
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1answer
26 views

Is a subsequence of an exchangeable sequence exchangeable?

Consider a finite sequence of random variables $X_1,...,X_n$ (1) SUFF COND: Suppose $X_1,...,X_n$ are exchangeable, meaning that the joint probability distribution of $X_1,...,X_n$ is equivalent to ...
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0answers
22 views

Linear independent in random variable and observations

I am confused with some fundamental concepts. Here for $n$ random variable $X_1,\cdots,X_n$, i.i.d and follow standard normal distribution, the probability that there exists a set of constant ...
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0answers
6 views

Deriving spectral norm or similar quantity for structured random matrix

I have a problem where I have no idea to start. Suppose a simple Least Squares system with $M$ unknowns $c$ and $N$ observations $y$ which is given through the linear mapping $X$: $$y = X c$$ It is ...
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1answer
96 views

Exchangeability and independence of random variables

I have a question on the relation between exchangeability and independence between random variables. Consider the random vectors $$u_1:= \begin{pmatrix} \epsilon_{1}\\ \epsilon_2\\ \epsilon_3 ...
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1answer
56 views

Questions on probability law

I'm trying to prove/disprove the following true or false statements, and I want to know if they are correct For every measurable function $g:\mathbb{R}\to \mathbb{R}$, $\mathbb{E}[g(X)]$ is ...
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2answers
20 views

Conditional Expectation and Variance Question

So I have a question I'm absolutely stumped with: Given two random variables, $X$ and $Y$ , with common variance, $\sigma ^2$, where $\mathbb{E}(Y|X) = X + 1$, find $\rho (X,Y)$. So I obviously need ...
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1answer
27 views

Prove $A_{\infty} < \infty$?

From Williams' Probability with Martingales How do we know that $A_{\infty} < \infty$? If $T = \infty$, then $$E[A_{T \wedge n}] \le (K+c)^2$$ $$\to E[A_{n}] \le (K+c)^2$$ $$\to ...
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0answers
21 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 ...
2
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2answers
33 views

Simple Probability - Enumeration and Geometric Distributions

I am not sure as to why this particular practice problem does not use a geometric distribution. A prize is randomly placed in one of ten boxes, numbered from 1 to 10. You search for the prize asking ...
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0answers
43 views

Is the notation $\mathbb{P}(X \cap Y) = \mathbb{P}(X,Y)$ common?

For two random variables $X,Y$, is the notation $\mathbb{P}(X \cap Y) = \mathbb{P}(X,Y)$ common? In a probability class last year we had always used $\mathbb{P}(X \cap Y)$. This year in a stochastic ...
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1answer
98 views

Is sharing the same support a necessary condition for exchangeability?

I am confused on the meaning of exchangeable random variables. The question is: consider the random variables $X_1,X_2,X_3$ defined one the same probability space $(\omega, \mathcal{F}, P)$; is ...