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

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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 ...
0
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1answer
20 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 ...
0
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1answer
41 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 x\iff(X-E(X))^2\...
0
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1answer
35 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 $\sigma_B^2$, ...
0
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1answer
22 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 ...
3
votes
2answers
95 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
32 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, ...
-1
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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 ...
1
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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 appreciated....
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0answers
43 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
votes
1answer
24 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
votes
1answer
28 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 ...
1
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0answers
22 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 ...
0
votes
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 ...
1
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1answer
21 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 = \frac{1}{N}...
-1
votes
1answer
37 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 0....
1
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1answer
31 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, ...
0
votes
2answers
44 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 $$\mathbb{E}...
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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.
0
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1answer
51 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 $$S_n:=X_1+\cdots+X_n,\...
0
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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 = ...
2
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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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votes
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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0answers
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 $...
3
votes
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 ...
0
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0answers
24 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 $a_1,\...
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0answers
8 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 ...
2
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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 \end{...
1
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1answer
58 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 ...
0
votes
2answers
23 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
47 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 \lim ...
0
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0answers
25 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
34 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 ...
0
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0answers
46 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 ...
0
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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 ...
0
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1answer
56 views

How to find the joint probability mass function of X and Y of a die toss?

Suppose that the die is tossed. Let X equal 1 if the result is an even number, and let it be 0 otherwise. Also, let Y equal 1 if the result is a number greater than three and let it be 0 otherwise. ...
1
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1answer
28 views

What is the approach to understand this algorithm?

Given $\{x_1, x_2,\ldots x_n\}$ where $x_i \in \{0, 1\}$ there is a binary equation $\varphi$ that is $x_{t_1}+x_{t_2}+\cdots+ x_{t_m}=0 \mod 2$ where $t_i \in \{1,2,\ldots,n\}$ for $x≥1$, $i=1,2,\...
0
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0answers
34 views

Expected Value of $P(|Y_n^{(K)}| > \epsilon)$ where $Y_n^{(K)}$ is the random sum of a sequence of RV converging to 0 in Probability

I have been struggling with this for countless hours, I would appreciate a hint to get me going in the right direction (no complete answer please) Problem: Assume that for all $k \in \mathbb{N}$ ...
1
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1answer
16 views

Probability of intersection of multiple 2-way universal events

If given a set of events that are known to be 2-way universal, is there a closed-form solution for the probability of their intersection? If so, how would you go about finding it? I know that for 2 ...
1
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1answer
30 views

Proving that $|P\{X=m\}-P\{Y=m\}| \leq P\{X\neq Y\}$

Let $X$, $Y$ be random variables on the same probability space. Show that for all $m$, $$|P\{X=m\}-P\{Y=m\}| \leq P\{X\neq Y\}$$ I'm actually not even sure how to start. I think it's going to rely ...
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0answers
7 views

representative sample

population is students taking a Chemistry class; Sample of 60 students as a random sample from this population. Select one variable on the survey and argue whether or not you think this sample is ...
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1answer
17 views

Difference between random number and random variant?

After generating random number we can get the random variant by using inverse transform or other techniques. What is the difference between random number and random variant. Can anyone explain it with ...
0
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1answer
35 views

Finding PDF of function of a random variable

Suppose $X$ has PDF: $f_X (x)= \lambda e^{-\lambda(x+2)}$ , for $x \ge-2$ $f_X(x)=0$ , for $x <-2$ Determine the PDF of $Y = X^2$. I am stuck because for $-2\le X \le 2$, $0\le Y \le 4$, and I ...
3
votes
0answers
52 views

$P(|X_1+X_2|<x)\le P(|X_1|<x)$ for every independent centered continuous $X_1$ and $X_2$?

Let $X_1$ and $X_2$ be zero mean independent continuous random variables. Then, is it true that $P(|X_1+X_2|<x)\le P(|X_1|<x)$. The intuition is that summing independent variables increase ...
1
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0answers
16 views

Multivariate normal distribution problem

Consider three Gaussian variables $X_1,X_2,X_3$ with $\mathbb{E}[X_i]=0$ and $\mathbb{E}[X_iX_j]=\rho_{ij}$ for $i,j=1,2,3$. Then, three new variables are defined: $$ \left\{ \begin{array}{l1} Y_1 ...
0
votes
1answer
33 views

What is $E[Z|Z\ge 0]$ where $Z$ is a continuous random variable with support in $[-1,1]$?

I have a random variable $Z$,I seek an expression for $E[Z|Z \geq 0]$. I assume this is easy to get hold of but I just can't seem to get it. As a further complication $Z=X-Y$, where $X$ and $Y$ are ...
5
votes
1answer
75 views

On the linear combination of $\pm 1$ random variables

Let $X_1,\dots, X_n$ be i.i.d symmetric $\pm 1$ random variables, i.e. $X_j$ takes values in $\{-1,1\}$ with $$\mathbb{P}(X_j=1)=\mathbb{P}(X_j=-1)=\frac{1}{2}.$$ Let $a_1,\dots,a_n\in\mathbb{Z}$, ...
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votes
2answers
33 views

Generating a random variable from a uniform random variable [closed]

I have no idea how to go about doing this. Any help would be much appreciated.
2
votes
1answer
59 views

How many students would have to take the exam to ensure with probability at least $.9$ that the class average would be within $5$ of $75$?

I'm having trouble solving this problem: From past experience, a professor knows that the test score of a student taking her final examination is a random variable with mean $75$. How many ...
0
votes
2answers
41 views

Find the density function from a joint density function

I try to solve the following task and I don't know what the correct way to do is. Let $p\in(0,1)$ and $(X,Y)$ be a pair of random variables with distribution density function $$f(x,y)=\frac{1}{2\...