Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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30 views

Which of the following is always true for A and B

Given that: $ P(A) = 0.5$ $P(B) = 0.7$ $P(A \cap B) = 0.3$ I have to choose one option that is true... However they all seem to be false which means I am possibly making a mistake.. The only option ...
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0answers
18 views

$(X,Y)$ is distributed uniformly over the square with vertices $(1,1)(1,-1)(-1,1)(-1,-1)$. Compute $\mathbb{P}(|X+Y|<1)$

$(X,Y)$ is distributed uniformly over the square with vertices $(1,1)(1,-1)(-1,1)(-1,-1)$. Compute $\mathbb{P}(|X+Y|<1)$ My ...
0
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1answer
23 views

Is it possible to obtain the following inequality?

Let $X,Y$ be two random variables such that $E\|X\|^{2n}\leq c_1^n,n\in\mathbb{N}$ and $E\|Y\|^{2n}\leq c_2^n,n\in\mathbb{N}$. Clearly, we have then $$E\|X+Y\|^2 \leq 2E\|X\|^2 + 2E\|Y\|^2 \leq ...
1
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2answers
46 views

Probability involving bread and jam!

SO, I drop a piece of bread and jam repeatedly. It lands either jam face-up or jam face-down and I know that jam side down probability is $P(Down)=p$ I continue to drop the bread until it falls jam ...
1
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0answers
40 views

What is a convolution kernel?

What is a convolution kernel? (in measure theory, probability theory) In which book can I read about kernels on measurable spaces and convolution kernels? Thank you!
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27 views

Urn problem- distribution after all balls of x randomly selected colours are removed

Apologies for any notational problems or lack of clarity: I'm a linguist not a mathematician. Anyway, here goes: There is an urn with $n$ balls divided into $k$ colours, where the number of each ...
2
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2answers
48 views

Probability of guessing the colors of a deck of cards correctly

10 years ago when I was about 15 I sat down with a deck of shuffled cards and tried to guess if the next card in the deck would be red or black. In sequence I guessed 36 cards correctly as red or ...
0
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0answers
11 views

Notation for Markov Model

I have a statement that says: "Show that conditional on $X_m = i$, $(X_{m+n})_{n \in \mathbb{N}}$ is Markov($\delta_i, p$) independent of $X_1, \ldots X_m$". What does the notation Markov($\delta_i, ...
0
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1answer
36 views

probability generating function of a sum and its expected value

The questions don't seem to be that hard but I think that I'm missing something... Question: N assumes values in the nonnegative integers. (a) Show that ...
0
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0answers
11 views

Distribution of AYB in terms of distribution of Y

Let $A$ and $B$ be two random orthogonal matrices and let $Y$ be a random diagonal matrix. The distribution of $Y$ is known to be $p_y$. How can we express the probability distribution of $X = AYB$ in ...
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0answers
78 views

A problem with kernels on measurable spaces

Let $(E, \mathcal{B}(E)), (F, \mathcal{B}(F))$ be two measurable spaces. A $kernel$ from $(E, \mathcal{B}(E))$ to $(F, \mathcal{B}(F))$ is an application $N : p \mathcal{B} (E) \rightarrow p ...
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0answers
20 views

Necessary Properties of a General Random Variable

Are there any other properties that a random variable X must satisfy besides (1) $X: \Omega \to R $ and (2) the cdf of x be defined for all real x ? Is there anything else that a random variable MUST ...
0
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1answer
26 views

Prove $f(x) = \frac 1 {\sqrt{2\pi}} \int_{\mathbb R} \hat f(t) e^{itx} \ \lambda(dt)$ for every $x \in \mathbb R$.

Let $\mathcal L_{\mathbb C}^1(\lambda)$ such that $\hat f \in L_{\mathbb C}^1(\lambda)$ (Fourier transformation). I've proven that $f(x) = \frac 1 {\sqrt{2\pi}} \int_{\mathbb R} \hat f(t) e^{itx} \ ...
0
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1answer
16 views

Combinatorics: Sampling with Replacement (Door Key Question)

You have a key ring with N keys, exactly one of which is your house key. You randomly try one key at a time until you get the correct key. However, you mix the keys that you have already tried with ...
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0answers
16 views

If $\mu'$ denotes the pushforward measure, then $\int f\circ X\;d\mu=\int f\;d\mu'$

Let $(\Omega,\mathcal{A})$ and $(\Omega',\mathcal{A}')$ be measure spaces $\mu$ be a measure on $(\Omega,\mathcal{A})$ and $\mu':=\mu\circ X^{-1}$ be the pushforward measure of $\mu$ under $X$ ...
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0answers
37 views

Show $\int_{\mathbb R} f * g \ d \lambda = \int_{\mathbb R} f \ d \lambda \cdot \int_{\mathbb R} g \ d \lambda$.

Suppose $f,g \in \mathcal L_{\mathbb C}^1(\lambda)$. Show $\int_{\mathbb R} f * g \ d \lambda = \int_{\mathbb R} f \ d \lambda \cdot \int_{\mathbb R} g \ d \lambda$. I see that $\int_{\mathbb R} ...
0
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1answer
21 views

Probability of breaking the enigma cipher

I assume that most of you are already familiar with how the ENIGMA machine works, that the germans used during WWII. We now that the enigma machine has 3 scramblers with each 26 setting each. That ...
11
votes
1answer
229 views

Probability of a zero product given one previous zero product

Consider two random vectors $v=(v_1,\dots, v_n)$ and $w=(w_1,\dots, w_{n+1})$. Each element of $v$ is independently $\pm1$ with prob $1/2$. Each element of $w$ is independently $1$ with probability ...
1
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1answer
27 views

Value of E(XY), dependant variables?

Let $A_1, A_2, A_3$ be independent variables. Let $X=2A_2-A_1$ and $Y=A_3+3A_1$ How can I calculate $E(XY)=E[(2A_2-A_1)(A_3+3A_1)]$? NOTE: I can calculate each $EA_n$ separately: $EA_1=0.5, EA_2=0, ...
0
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1answer
20 views

How to use Chernoff bound on binomial trails

(Not fair coin!) Head with probability $0.9$ and tail with probability $0.1$ use Chernoff bound for the probability of more than 70% head in $n$ trails that tested. I think its binomial distribution ...
1
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1answer
31 views

Criterion of two measures are absolute continuous or singular

I was reading the Durrett's book: probability theory and example and stuck at some stages about the radon-nikodym derivatives related topic: Here is the setting: Let $\mu$ and $\nu$ be measures on ...
1
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1answer
53 views

A probability problem with maximum and summation

Let $X_n$ be iid nonnegative r.v.s, suppose there exists positive sequence $a_n$ such that $S_n/a_n\xrightarrow[]{P}1$, then show $$\max_{1\le i\le n} X_i/S_n\xrightarrow[]{P}0.$$ I have shown that ...
1
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1answer
35 views

How to give a good upper bound on tail probability for $P\{|\frac{R_n}{\sqrt{n}}-1| \ge \varepsilon\}$?

Suppose $X_1,X_2,\ldots$ is a sequence of i.i.d. standard normal random variables. $R_n=\sqrt{X_1^2+\ldots+X_n^2)}$. How could I prove $P\{|\frac{R_n}{\sqrt{n}}-1| \ge ...
0
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1answer
56 views

Unifying the treatment of discrete and continuous random variable

I have been working on the reconciliation of the treatment of discrete and continuous random variable in a measure theoretic sense. But I found myself blocked on fundamental results. We know that If ...
0
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1answer
16 views

Convergence of Remainder from Taylor Expansion

For a distribution function $F$ and its variance functional $T(F)$, it can be shown that the Taylor expansion of $T(F)$ at $F$ in the direction of the empirical distribution function $F_n$ gives the ...
0
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1answer
17 views

Generalised probability density functions using Dirac deltas

I just cannot understand how a discrete random variable $X$ may be represented using a generalised probability density function (p.d.f.) $f_X$ as, \begin{equation}\tag{1} f_X(x) = \sum_{x_k \in R_X} ...
0
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0answers
100 views
+50

Condition for a process to be a supermartingale

I am struggling in this question: Let $W$ denote a Brownian motion. Given that $ X_t = e^{- \lambda t} X_0 + \int_0^t \sigma e^{- \lambda (t-s)} \,dW_s$ solves the SDE \begin{equation} dX_t = - ...
0
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2answers
23 views

Can a continuous random variable be jointly continuous with itself?

Furthermore, is it always the case? I don't see why not, but my issue is that $f_{XX}(x, x)$ looks weird. Does it accept two parameters, or only one?
2
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2answers
35 views

Formally proving that $E(|X|) < \infty \iff \sum_{n=1}^{\infty}P(|X|\geq n) < \infty$?

In other words, I have to prove that $\int_{-\infty}^{\infty}|x|f(x)dx < \infty \iff\sum_{n=1}^{\infty}P(|X|\geq n) < \infty$ where $f(x)$ is the density function. I know that the summation ...
0
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1answer
25 views

Proof of Transformed Binomial Converging to $N(0,1)$

I'm trying to (efficiently) prove that a transformation of the sum of a sequence of $n$ Bernoulli$(p)$ trials ($X_1,\dots,X_n$) converges in distribution to $N(0,1)$. Specifically, if we denote $B = ...
1
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1answer
13 views

Application of Borel-Cantelli for sequence of two parameters

Let $(A_{m,\ell})_{\ell \geq 0, m \geq 0}$ be a sequence of events in some probability space. How to show by using Borel Cantelli that, if $$\sum_{\ell \geq 0, m \geq 0} P(A_{m,\ell}) < \infty,$$ ...
0
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1answer
18 views

Bernoulli Distribution (PMF) of random variables X,Y

A fair coin is tossed three times, let X be the number of cases in which the HEAD is obtained, and Y be the absolute value of difference between the number of HEAD and the number of TAIL. Seek the ...
0
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3answers
44 views

Simply normal sequence of $-1$'s and $1$'s as coefficients of harmonic series

Suppose $s_{n}$ is either $1$ or $-1$ for $n=1,2, 3,\ldots$ and that half the $s_{n}$'s are 1; i.e. $$ \lim_{n\to \infty} \frac{\#\{i\leq n: s_{i}=1\}}{n}=\frac{1}{2}. $$ Does then the series ...
2
votes
1answer
40 views

What does it mean u(dx) in the Fourier transform of a probability measure u?

Let $\mu$ be a probability on $\mathbb{R}^n$ and consider its Fourier transform $\overset{\wedge}{\mu} (u) = \int e^{i (u ,x)} \mu( dx)$, where $(u, x)$ is the scalar product of $u$ and $x$. What ...
0
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1answer
38 views

Identically Distributed Functions of Random Variables

How can you tell if two functions of random variables are identically distributed. For example, $Y_{1} = 2X_{1}$ and $Y_{2} = X_{1} + X_{2}$ How does one determine whether they are identically ...
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0answers
19 views

Independent Random Variables with Uniform Distribution

Let $X_1, X_2,...,X_n$ be independent random variables, each having a uniform distribution over $(0,1)$. Let $Z:=\min(X_1, X_2,...,X_n)$ and $Y:=(X_1, X_2,...,X_n)$. I need to find the cdf and pdf of ...
0
votes
1answer
13 views

Normal distribution problem; distribution of height

The problem is: the height of children in age from 3.5 to 4 years is described by normal distribution with parameters $\mu =103$ centimetres and $ \sigma=4.5$ centimetres. What is the percent of ...
1
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1answer
22 views

Quantifying over all random variables

I often encounter statements in the literature in probability theory of the form: "Let $(\Omega, \mathscr{A}, P)$ be a probability space, $S$ a state space and $X : \Omega \to S$ a random variable ...
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0answers
22 views

how to prove $\mathop {\lim }\limits_{n \to \infty } {\{\Phi [(1 - \varepsilon )\sqrt {2\log n} ]\}^n}=0$?

$\Phi (x)$ is the distribution function of standard normal distribution. $\varepsilon$ is some positive tiny number that is less than 1. How to prove this beautiful and important limitation: ...
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1answer
43 views

IId random variables from Exponential distribution

If $X_1$ and $X_2$ are iid random variables from the exponential distribution with parameter $\lambda$. I need to find the pdf of $X_1/(X_1 + X_2)$. As of now I have used $X_1=\lambda e^{-\lambda x}$ ...
-1
votes
1answer
42 views

Cdf and Pdf of independent random variables(iid)

Let $X_1, X_2,...,X_n$ be independent random variables, each having a uniform distribution over $(0,1)$. Let $Z:=\min(X_1, X_2,...,X_n)$ and $Y:=(X_1, X_2,...,X_n)$. I need to find the cdf and pdf of ...
1
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2answers
29 views

Probability regarding 26 letters and one event before another question

The 26 letters A, B, ... , Z are arrange in a random order. [Equivalently, the letters are selected sequentially at random without replacement.] a) What is the probability that A comes before B in ...
0
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1answer
39 views

Itô formula + SDE

I have a problem with solving the following problem: I.e. I want to show that $X_t$ is a solution to the SDE by employing the Itō formula. Now the problem is I don't get how I should set the ...
0
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1answer
34 views

Lebesgue-Stieltjes Integral (Several Variables)

Let $\mathcal F$ be a convex set of probability measures or distribution functions and $F, G$ be two elements in $\mathcal F$. Let $T$ be a functional on $\mathcal F$ defined as follows. Note that $h$ ...
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2answers
29 views

Proof of the law of large numbers for higher moments

Let us work on some probability space $<\Omega,\mathscr{A},\mathbb{P}>$: I'm looking for (independent) proofs of two proofs, of the generalised weak and strong law of large numbers ...
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1answer
51 views

Counterexample to conditional probability with dependent events

Let $X1,X2,X3$ be i.i.d. taking values in a finite set, and not constant. Is it necessarily true that $P(X3=X2|X2≠X1)≤P(X3=X2)$? Give a proof or a counterexample. Since the two events $A=\{X3=X2\}$ ...
3
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0answers
44 views

Is the distance induced by the topology of the set a metric

First of all excuse me if something with the question is wrong. I am not very knowladbable in this area, though I know what I am asking and my question should be infact a reasonable one. The topology ...
0
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0answers
16 views

Is it true that for every signed probability distribution `f`, there are positive distributions `g` and `h` st. `fg=h`?

While reading the article Half of a Coin: Negative Probabilities, I came across the following theorem: For every generalized g.f. f (of a signed probability distribution) there exist two ...
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1answer
17 views

Fnd a sequence to be convergence in distribution

Varablies $X_1,\ldots ,X_n$ are independent and $\forall {i\in\{1,\ldots n\}}: X_i \sim \exp(1)$. Find numeric string $a_n$ such that sequence of random variables $$Y_n= \max\{X_1,\ldots, X_n\} - ...
1
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0answers
26 views

How to find relation between expected value and P(X>0)

Let random variable Y ≥ 0 with E[Y]^2 < ∞. Prove that P(Y > 0) ≥ (E[Y])^2/E[Y^2] ? How will we approach this question