Tagged Questions

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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2answers
25 views

Stopping Time Subset Proof

My probability textbook has a really crappy proof for the following result. Suppose $S$ and $T$ are stopping times, with $S(\omega) \le T(\omega)$ for all $\omega$. Prove that $\mathcal{F}_S \subset ...
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1answer
18 views

Convergence in Distribution

Suppose $X_n$ are $\mathbb{R}$-valued random variables. Then how would I be able to show that $X_n\rightarrow^D X$ where $X$ is also a $\mathbb{R}$-valued random variable iff $F_n(x)\rightarrow F(x)$ ...
2
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0answers
11 views

laplace transform and fourier transform of kernel

Suppose $p_t(x)=\frac1{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}$ is the Gaussian density. $p_t(x)$ is also the Green kernel of the heat equation in 1D: $(\partial_t-\frac12\Delta)u=0$. The Fourier transform ...
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0answers
4 views

On a problem of convergence of measure for Levy measures

I have a question that pertains to the Levy representation of infinitely divisible distributions. However, the technical item that is relevant to me right now is one that relates to weak or vague ...
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0answers
19 views

Comparing probabilities [on hold]

Two assumptions: The probability of being born is 1 in 400 trillion The probability of winning the lottery is 1 in 175 million How much greater (% wise) is the probability of winning the lottery as ...
1
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1answer
22 views

Existence of strictly positive probability measures

Let $X$ be a Hausdorff space (or let's even assume it is metrizable). A strictly positive measure on $X$ is the that gives positive measure to any non-empty open subset of $X$. Under which condition ...
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3answers
31 views

Formal proof that X and X squared random variables are dependent.

Intuitively I know that any $X$ and $Y = X^2$ random variables are not independent, but I can't come up with a formal proof. In the case I'm most interested in, $X$ is uniformly distributed on ...
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0answers
15 views

Questions regarding a Gamma distributed Random Variable ( first moment and square density)

Consider the following Gamma distributed RV $$\operatorname{Gamma }(m_S,\theta_S)$$ with the following shape and scale parameters $$m_S = \frac{(\theta_1+\theta_2)^2}{\theta_1^2+\theta_2^2}$$ ...
2
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1answer
27 views

Showing that $X_n$ ~ $N(0, a_n)$ converge to $0$ when $a_n \to 0$ sufficiently fast

If $X_n$ have distribution $N(0, a_n)$ with $\sum_{n=1}^\infty a_n^b < \infty$ for some $b > 0$, then $X_n$ converge almost surely to $0$. I was able to show (for a previous part of the ...
0
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0answers
16 views

If the difference of two i.i.d. random variables is normal, must the variables themselves be normal?

I previously asked a similar question about the sum of two i.i.d. random variables, thinking the two cases to be equivalent. But I can't see how to apply the proof of that case to this one. It is ...
0
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1answer
8 views

Calculate edge probability of a graph

I wonder what is the edge probability $p$ for which a random graph with $n = 5000$ nodes has the largest expected diameter? How can I calculate that? Is there someone who can help me? This would be ...
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1answer
15 views

Probability of edges in Graph

I have given a random graph G(n, p) with n = 5000 vertices and an edge probability of p = 0.004. I calculated the expected number of edges which is (0.004 * maximum number of possible edges) $pE = ...
2
votes
3answers
60 views

Poisson Process: Finding the sum of interarrivial time

One hundred items are simultaneously put on a life test. Suppose the lifetimes of the individual items are independent exponential random variables with mean $200$ hours. The test will end when ...
0
votes
1answer
9 views

what will be the PDF of the magnitude of this random variable x+j y?

if we have a complex random variable [x+j*y] where (j :sqrt(-1)) and x,y both have Gaussian distribution and statistically dependent , so what will be the distribution (PDF) of the magnitude of this ...
1
vote
3answers
36 views

Proving two random variables differ with positive probability

Suppose that conditional on $x$, $y$ is normal with mean $x'\beta_0$ and variance $\sigma_0^2$. The log of the conditional density is then $$ ...
0
votes
1answer
11 views

Asymmetric simple random walk?

It comes from the book Probability: Theory and Example. I don't understand the part marked with red line. Why it cannot converge to an interior point of $(a,b)$? Can anyone help? Thanks so much!
1
vote
1answer
14 views

Prove existence/non-existence of a pdf given mean, std, range

Given: Mean = 100, Range = [4, 10000], std = 3000 Is it possible to prove whether a pdf exists or not that satisfies these values? If it does exist, what would be approximate shape of the ...
2
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1answer
32 views

If the sum of two i.i.d. random variables is normal, must the variables themselves be normal?

It is well known that if two i.i.d. random variables are normally distributed, their sum is also normally distributed. Is the converse also true? That is, suppose $X$ and $Y$ are two i.i.d. random ...
0
votes
1answer
44 views

E(XY) = E(X).E(Y|X) . Is this true for mean = zero.

I know that Joint Probability density function for two random functions $X$ and $Y$ $$P(XY) = P(X)\cdot P(Y|X)\tag{1}$$ But I just read in a set of lecture notes that for E(X)=E(Y)=0 $$E(XY) = ...
-1
votes
2answers
21 views

Property of submartingale and supermartingle?

Is it true that for a submartingale, $$E(X_n) \le E(X_m)$$ for $n \le m$. And for a supermartingale, $$E(X_n) \ge E(X_m)$$ for $n \le m$. If it is true, then why? I feel confused because the ...
1
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2answers
35 views

Understanding a proof on almost sure convergence

I'm having trouble with the following proof: $\color{red}{\text{This is not the end of the proof.}}$ I'm not understanding the definition of $A_k$. For instance $A_5\nsubseteq A_6$ because $A_6$ ...
3
votes
1answer
32 views

$\phi(v)/\Phi(v)$ is decreasing for $\phi$ and $\Phi$ being the PDF and CDF of $N(0,1)$

Let $\phi(v)$ and $\Phi(v)$ denote, respectively, the PDF and CDF of the standard normal distribution. How would one show that $$ \frac{\phi(v)}{\Phi(v)} $$ is decreasing? I tried the quotient rule ...
2
votes
1answer
25 views

Tail probability of a max of iid

If $X_{i}$ are iid random variables with $X_{i}>0$ and $\mathbb{P}(X_{i}>t)\sim t^{-\alpha}$ as $t\to \infty$. Then my question is: Is it also true that $\mathbb{P}(\max_{1,\dots n} ...
2
votes
1answer
25 views

Problem similar to Kolmogorov's inequality using martingale.

Suppose that $X_k$ is a sequence of independent random variables with mean zero and variance $1$. Let $S_k=X_1+\cdots+X_k$ and let $$ h(\lambda)=\limsup_{n \rightarrow \infty}P\left(\max_{1\leq k\leq ...
1
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0answers
36 views

fractional Brownian motion is not a semimartingale. How to apply Ergodic theorem in the proof of this theorem?

Here is the proof of the theorem. I couldn't understand how to apply Ergodic theorem in this proof. Let $X=(X_t)_{t\geq0}$ be a fractional Brownian motion with self-similar parameter $H\in(0,1)$. We ...
0
votes
1answer
56 views

How to deal with probability problems in proper class sample spaces?

Here my focus is mainly on $Ord$ and questions such as: An ordinal is chosen by random. What is the probability of the event that it is a cardinal number? A coin is tossed proper class many ...
2
votes
1answer
28 views

Martingales and variance

For a martingale $(Z_n)_{n\in \mathbb N}$ define $X_i=Z_i-Z_{i-1}$ with $Z_0=0$ Show: $$Var(Z_n)=\sum_{i=1}^nVar(X_i)$$ My attempt: We can write $Z_n=\sum_{i=1}^nX_i$, so we actually just have to ...
0
votes
0answers
17 views

what will be the PDF of magnitude and phase of this random variable?

i have a random variable as shown in the figure and i tried to find the PDF of the magnitude and phase of this random variable using central limit theory as i mentioned , i know that if we have ...
-1
votes
0answers
41 views

how likely is it that 2 strangers in a city of 8 million will cross paths? [on hold]

I am looking for stats regarding likelihood of 2 people from very different social economic stratus cross paths in a city of 8 million? even if its just riding on the same subway train. or more ...
-1
votes
1answer
25 views

What is the probability of this question?

On a single draw from a deck of playing cards the probability of selecting heart is 1/4 the probability of selecting a black card is 1/2. what is the probability of selecting both a heart and a black ...
2
votes
1answer
18 views

Exponential distribution: waiting time at post office

Consider a post office with two clerks. Three people, A, B and C, enter simultaneously. A and B go directly to the clerks, and C wait until either A or B leaves before he begins service. What ...
2
votes
2answers
20 views

Markov Chain: Moving on a circle

A particle moves on 12 points situated on a circle. At each step it is equally likely to move one step in the clockwise or in the counterclockwise direction. Find the mean number of steps for ...
1
vote
1answer
16 views

Notation with random variable $\overline{X}_{n^2}$ in Strong Law of Large Numbers proof.

I'm reading the proof for the strong law of large numbers. It says: Let $X_1,X_2,\ldots$ be a sequence of independent and i.i.d. random variables with finite mean $\mu$ and finite variance ...
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0answers
33 views

what is the PDF of this random variable? [on hold]

please i want to find the PDF of this random variable :according to the center limit theorem i have random variable as expressed below which is summation of exponential of many random phases and i ...
1
vote
0answers
19 views

The distribution of a function of uniform and a complex Gaussian random variable

Hope this question is clear and straight to the point, if not than I can edit it accordingly. Given the following independent random variables with distribution $$X_i\sim \mathcal{CN}(0,1) $$ ...
0
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0answers
12 views

Posterior tail probability is absolutely continuous?

Suppose that the distribution of $X$ given $\theta$ is absolutely continuous with respect to Lebesgue measure on $\mathbb{R}$, for each value of $\theta$. Denote the conditional density with ...
0
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2answers
38 views

Formula the conditional probability of mables

I have a interesting question that need your help. I have two sets A and B. Set A have 10 marbles that numbered from 1 to 10. Set B have 6 marbles that numbered from 1 to 6. Randomly choose $g$ ...
1
vote
2answers
34 views

Relating a Gamma Distribution to an Exponential one?

Question related to Gamma and Exponential random variables. Suppose I have a Gamma random variables with shape and scale parameter $m$ and $\theta$ i.e $$X\sim\Gamma(m,\theta)$$ respectively. Can I ...
2
votes
1answer
25 views

On Schwarz Zippel Lemma

Theorem (Schwartz, Zippel). Let $P\in F[x_1,x_2,\ldots,x_n]$ be a non-zero polynomial of total degree $d≥0$ over a Field $F$. Let $S$ be a finite subset of $F$ and let $r_1,r_2,...,r_n$ be selected at ...
2
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0answers
27 views
+150

Why do two points never 'arrive at once' in a Poisson point process

In the following, all the measure spaces are endowed with the Borel $\sigma$-algebra corresponding to their topology (we take the usual topology on $[0,\infty)$). Let $E$ be a Polish space and let ...
0
votes
1answer
8 views

Expected number of visits to a state of a Markov chain up to a certain time

Let $P=\{p_{ij}\}$ be a stochastic matrix (with rows and columns indexed by a countable set) and let $p^{(k)}_{ij}$ be the entries of $P^k$. I'm trying to prove that, if the associated Markov chain is ...
2
votes
0answers
20 views

(Billingsley, 2nd ed, 1968) D space, (12.33) inequality proof

Convergence of probability measures, Billingsley, 2nd ed, p132, Theorem 12.4 This is what I want to prove where $x \in D \equiv$ the set of cadlag functions defined on $[0,1]$ and $w_x^{''}(\delta) ...
2
votes
1answer
24 views

Help with Linear Transformation of a multivariate normal

Given X ~ $N_2$ (μ, Σ)$ Find the Distribution of $$ \begin{pmatrix} X+Y \\ X-Y \end{pmatrix} $$ Show independence if $Var(X) = Var(Y)$ Attempt: Given proper of ...
2
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0answers
34 views

Probability and Quantum mechanics

I don't quite understand how the probability language of sample spaces, $\sigma-$algebra, random variables, etc, fit into the quantum mechanics' formalism. To wit, we usually say that an observable ...
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votes
0answers
19 views

conditional expectation convergence in L1 [on hold]

Let $\mathcal{F}_n \uparrow \mathcal{F}_{\infty}$ and $Y_n \rightarrow Y$ in $L^1$. I'm stuck on how to show that $E(Y_n | \mathcal{F}_n ) \rightarrow E(Y | \mathcal{F}_{\infty})$ in $L^1$?
1
vote
0answers
20 views

martingale convergence proof

This is out of Durrett 5.5.7. Let $X_n \in [0,1]$ be adapted to $\mathcal{F_n}$. Let $\alpha,\beta > 0$ such that $\alpha + \beta = 1$. Suppose that $$ P(X_{n+1} = \alpha + \beta X_n | ...
4
votes
0answers
73 views

Find a symmetric random walk on $\mathbb{Z}$ that is transient.

I wanted to know if it is possible find a symmetric random walk on $Z$ that is not recurrent. Let $X$ have the following distribution, with a probability $1/2^{i+1}$, $X=\pm b_i$. Let ...
3
votes
1answer
41 views

Where do I go wrong?

Suppose $X,Y$ are independent Uniform$(0,1)$ random variables. Find the probability $P(Y\geq X\mid Y\geq\dfrac{1}{2})$. Please note that I know the correct answer and that I have arrived at the ...
0
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0answers
10 views

conditional dependence and sum of random variables

I know that $Y \perp\!\!\!\perp (X,Z)|A \Rightarrow Y \perp\!\!\!\perp X|A$, but is the following true? 1) $Y \perp\!\!\!\perp (X+Z)|A \Rightarrow Y \perp\!\!\!\perp X|A$ I feel like the addition ...
2
votes
0answers
44 views

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ has the distribution of $aX$, find all characteristic functions of $X$.

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ have the same distribtution as $aX$ for some real $a$, what are the possible characteristic functions of $X$. Let $\varphi_X(t)$ be ...