Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use ...

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1answer
35 views

Expected value: proof of one property

I was asked to prove that the expected value of a continuous random variable $\xi$ having distrubition function F(x) and probability density function p(x) is equal to: $M\xi=\int_{-\infty}^{+\infty} ...
2
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0answers
26 views

help with a proof on Doob's Submartingale inequality - application of chebychev's inequality

I am stuck on a final step of the proof, we have that $(X_n)$ are non negative submartingale, and $c>0$. We let $T = \inf \{n: X_n > c \} \wedge N$ which is a stopping time. Let $E \{ ...
1
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2answers
36 views

Almost sure convergence of Bernoulli r.v.

Assume we have an independent sequence of Bernoulli r.v. $(X_n)_{n=1}^\infty$ each $X_n$ gets 1 w.p. $\frac{1}{\sqrt{n}}$ and otherwise $0$. How can we show that almost surely the following holds? ...
2
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0answers
31 views

Comparing Patrick Billingsley's Aniversary Edition to previous editions, and to Robert B. Ash's book.

I'm reading some of the reviews at amazon to the Anniversary edition of Billingsley's 'Probability and Measure', and several users state that the book is riddled with new typos, and plain errors, ...
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0answers
25 views

how can I find the PDF of g(x) when the Characteristic function is known?

Suppose that the characteristic function of X is given ($sin(\alpha\omega/2 \pi)$ ) how can I find the PDF of the $y=x^2$ ?I think we should find the PDF of the function X (using the related ...
3
votes
1answer
72 views

Median of a multinomial variable

Let $k\in\mathbb N^+$ be a positive integer. Consider a set of i.i.d. random variables $X_1,X_2,\ldots, X_n$, each of which is distributed uniformly over $\{1,2,\ldots,2k+1\}$. For $i\in ...
0
votes
1answer
25 views

Can become Probability mass function?

There are my question and my answer By definition, when $f(x) \geq 0$ for all $x$ which is element of real number Range =$\{x:f(x)>0\}$ is finite or countable sum of $f(a) =1$ $a$ is ...
0
votes
1answer
45 views

Computing the expected value of a random variable

I have to compute the following expectation \begin{equation} \mathbb{E}\int_0^{t \land \tau_n} \varepsilon ds \end{equation} where $\varepsilon$ is a positive constant and $\tau_n$ is a stopping time. ...
0
votes
1answer
27 views

memoryless property of exponential distributions with random variables

It is true that $P(X>t+s|X>t)=P(X>s)$ for certain values $t$ and $s$. However, how can I show that this still holds if: $T$ is a continuous random variable. That is ...
2
votes
2answers
53 views

Convergence of r.v. sequence

Say we have a sequence of independent r.v. $(X_n)^\infty_{n=1}$, we are given that $E[X_n]={\sqrt{n}}$ Is it true that the following holds a.s.? $$\lim_{M \to \infty }\frac{1}{M}\sum^M_{n=1} X_n = 0 ...
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0answers
42 views

How to evaluate the conditional probability P(X | Y∩Z) given P(X), P(Y), P(Z), P(X|Y) and P(X|Z)?

Is it possible to express P(X | Y∩Z) in terms of P(X), P(Y), P(Z), P(X|Y) and P(X|Z)? I tried using Bayes' Theorem, which says P(A|B)=(P(B|A)*P(A))/P(B), which ...
2
votes
0answers
48 views

When is this matrix positive semidefinite?

Let us fix dimension $n$. Consider the $n \times n$ matrix \begin{equation} S_n=\begin{bmatrix} 1 & z & z & \cdots & z \\ \bar{z} & 1 & z & \cdots & z\\ \bar{z} & ...
2
votes
2answers
56 views

Showing two random variables independent despite seemingly looking dependent

I just met this in probability and it got me completely stumped: We define an i.i.d sequence of normally distributed random variables $ \{ X_n \}_{n=1}^{\infty} $ such that $ X_n \sim ...
1
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0answers
23 views

Brownian Motion hitting time is finite yet has infinite expectation?

I've read that a hitting time of a Brownian motion (defined as $T_a = \inf\{t\ge0:W_t=a\}$ where $W_t$ is a standard Brownian Motion, i.e. a Wiener process), has the following two properties, which I ...
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3answers
23 views

Probability: 52 cards in a deck [closed]

If you are dealt two cards successfully (with replacement of the first) from a standard 52-card deck, find the probability of getting a heart on the first card and a diamond on the second.
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1answer
44 views

Almost everywhere convergence of random variables

This is a question that my teacher is having us do for homework but I think there might be a typo in it. I was hoping if someone clear this up for me. The sequence $\{X_n\}$ of random variables ...
0
votes
1answer
22 views

Why is chi-square distribution with 2 degrees of freedom an exponential distribution?

Is there any explanation on why these two distributions are equivalent? How can the sum of two square of Gaussians represents the limit of a geometric distribution? I found an answer here, which ...
3
votes
1answer
26 views

A Property of Martingale of Sum of i.i.d. Random Variables

I am trying to solve the following problem: Let $\{Y_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. random variables with finite mean. Let $F_n =\sigma(Y_1,...,Y_n)$. Let $\tau$ be a stopping time ...
0
votes
1answer
47 views

Probability of two IID random variables

Let $X$ and $Y$ be independent and identically distributed. Show that if $X$ and $Y$ are continous, then $P(X<Y) = 1/2$ Give an example of two IID RVs $X$ and $Y$ such that $P(X<Y)\neq 1/2$ ...
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0answers
19 views

What are the two levels of probability theory?

What I mean is what is the Probability theory of using integrals called? (typically undergraduate course, Probability Theory I) Then what is the probability using measure theory? (typically graduate ...
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0answers
30 views

Generalizing lemma 1 in Tao's notes on operator norms of random matrices

My question concerns the proof of Lemma 1 in this blog post of Terence Tao. In the first paragraph, he says: Applying standard concentration of measure results (e.g. Exercise 4, Exercise 5, or ...
1
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1answer
25 views

Verification an inequality

This problem arose when I was reading the proof of Lemma $A.1$ of Koenker and Portnoy ($1988$)'s paper L-Estimation for Linear Models. For better exposition, I reformulated the original statement as ...
0
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0answers
12 views

Spectral measure of a stationary time series

Let $(Z_t)$ be white noise with $E[Z_t^2]=1$ and $A$ and $B$ random variables such that $E[A] = E[B] = 0$, $E[A^2] = E[B^2] = 1$, $A$, $B$ and the infinite sequence $(Z_t)$ are independent ($(Z_t)$ ...
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0answers
8 views

Showing that a process is autoregressive

Consider a sequence $(Z_t)$ of i.i.d. standard normal variables and real numbers $\alpha > 0$ and $\theta \in (0,\frac{1}{3\sqrt{3}})$. Let $X_t = \sigma_tZ_t$ for $\sigma_t^2$ defined by ...
0
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0answers
13 views

How to recover a measure from the product

Let $(X,\mathcal{F}_X), (Y,\mathcal{F}_Y)$ be measurable spaces and $\mu :\mathcal{F}\rightarrow [0,\infty]$ be a measure (assume that $\mathcal{F}\supseteq \mathcal{F}_X\otimes\mathcal{F}_Y$). I do ...
0
votes
1answer
31 views

We randomly chose a number on the interval $[0,1]$. Find the probability that the first $10$ decimals of the chosen number are all equal to $1$

This question was part of a test on an advanced probability class, centered around Lebesgue Measure. Let $\Omega=[0,1]$ be a Borel $\sigma-$Algebra and $P([a,b])=b-a$, $0\le a\le b\lt1$ the ...
0
votes
1answer
22 views

Intuitive explanation of double expectation

This has been bugging me for some time. The famous result in probability is like $E[Y] = E[E[Y|X]]$ Can someone write an intuitive explanation of the above?
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0answers
13 views

Analytical expression for expected the n-th and (n-1)th order statistic of minimum of two uniform random variables

I have two independent random variables $X_1 \sim Uniform(b,1)$ and $X_2 \sim Uniform(0,1)$, where $0<=b<1$. Their CDF's are: $$ F_{X_1}(z) = \begin{cases} 0 & z<b \\ \frac{z-b}{1-b} ...
0
votes
1answer
41 views

Prove a.s. convergence of random variables.

I need to prove this: Assume that you have a probability space $(\Omega, \mathcal{F},P)$, $X_t$ is a stochastic process which is jointly measurable with respect to $\mathcal{B}(\mathbb{R})\times ...
0
votes
1answer
22 views

Encryption - show probability for obtaining specific bit

Assume a person A encrypts a message which consist of the bits m1, ..., mn. The person is using the one-time pad algorithm. Another person B intercepts the ciphertext and we suppose he knows that mi ...
2
votes
1answer
26 views

Expectation of martingales [closed]

If I know that $\{M_t\}_t$ is a martingale, we know that $$\mathbb{E}({M_tM_s})=\mathbb{E}(M_{\min({t,s})}^2)$$ Is there something I can say about $\mathbb{E}(M_tM_sM_r)$?
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0answers
13 views

Tight upper bound for expectation of function of a positive and bounded random variable

This problem popped up in my research. Let $X$ be a positive and bounded ($X\in (0,B) \ a.s.$)random variable with degrees of freedom $d$ and noncentrality parameter $\lambda$. My goal is to find ...
0
votes
1answer
49 views

Conditional probability question for n trials`

E and F are mutually exclusive events, trials are independent and performed repeatedly. I have already calculated the probability E or F occurs for the first time on the nth trial. The question is: ...
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0answers
19 views

Find pmf for binomial distribution with prior

Let $X$~$Bin(n,P)$ where $P$~$Beta(\alpha,\beta)$. How do I find the pmf for $X$? I have a vague idea that I have to condition on $P\leq \tilde{p}$ to find $Pr(X=x|P\leq\tilde{p})$ but I'm not ...
0
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0answers
9 views

Density of a bimodal distribution

I'd like to make a density function which contains the item: $exp(-(x^2-c)^2)$ or any exp(f(x)), where f(x) is a polynomial of x with degree 4. I tried that in mathematica, but found it really ...
2
votes
1answer
34 views

determining if a tail event

I am to determine if $$\{\sup X_n < \infty \}$$ is a tail event, the solutions are as follows: I don't understand how they got the line of equalities, specifically the last one, and why it holds ...
1
vote
2answers
22 views

Independence of two transformed variables from a symmetric distribution

Let $X$ be a continuous random variable with a symmetric probability density function centered at zero. Let, $Y=|X|$ and $Z=\mathrm{sgn}(X)$, where $\mathrm{sgn}(X)=1$ if $X>0$ and ...
0
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0answers
7 views

Stationary solution of a autoregressive process

I have an autoregressive process given by the following equation $$X_t = X_{t-1} - 3X_{t-2} + 3X_{t-3} + Z_t \tag{1}$$ where $(Z_t)_t$ is a given white noise process. I am wondering if I can find a ...
1
vote
1answer
18 views

proving a random variable is a martingale

I am on the final part. I have shown all the properties of martingales, except for the fact that $E|N_n| < \infty$. The solutions state $|N_n|$ is bounded, but I don't see how it is as $S_n$ is ...
2
votes
1answer
23 views

Show that $|\phi_{X}(t)-1-t^2\log|t||\le3t^2$

Let $f(x)=|x|^{-3}1_{|x|\ge1}$ be the density function of a random variable $X$ and $\phi_{X}(t)=E[e^{itX}]$. Show that $\forall t\in[-1,1] $ $$|\phi_{X}(t)-1-t^2\log|t||\le3t^2$$ I couldn't think of ...
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0answers
16 views

Probability and expectation of three ordered random variables

I am really stuck on the following question during my exam preparation. Let $X_1$, $X_2$ and $X_3$ ...
1
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0answers
17 views

Expectation of the maximum of random variables

I'm trying to get $E(\max \{ a-X, b-X-Y, 0 \})$, where $X$ ~ $N(0,\sigma^2)$, and $Y$ ~ $N(\mu, \gamma^2)$, and $X,Y$ are independent. I've been trying to figure this out by doing, $E(\max \{ a-X, ...
2
votes
1answer
36 views

continuous local martingale brownian motion

$B$ is a one-dimensional Brownian motion and $X_t$ is defined as$\\$ $X_t:=f_{1-t}(B_t)$, $0\le t<1$ and $0$, $1\le t<\infty$ where $f_s(x)=\frac{1}{\sqrt{2\pi s}}e^{-\frac{x^2}{2s}}$. I have to ...
2
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0answers
28 views

Integral Representation of Brownian Motion [duplicate]

B is a Brownian motion with values in $\mathbb{R}$. I have to find a process $(F_t)_{t\in[0,T]}$ such that $X=E[X]+\int_0^T F_s dB_s$, for $X=B_T$, $X=\int_0^T B_tdt$, $X=B^2_T$, $X=B^3_T$ and find a ...
4
votes
1answer
36 views

Determining the values a random variable takes

Let $(X_n)$ be IID bernoulli random variables and set $$Y_n = \sum_{i=1}^n \frac{X_i}{2^i}$$ I am trying to show this converges weakly to the uniform distribution on $[0,1]$. I am given a hint that I ...
0
votes
2answers
33 views

Generalized prisoners' problem

I am trying to generalize the prisoner's problem. The problem reduces to this: find the probability that a random permutation of $1,...,n$ has no cycle of length $>L$. If the number of ...
0
votes
1answer
32 views

T-Distribution, Normal Distribution, and Confidence Intervals

In my probability class we were given the following problem: Suppose you take a sample of your friends and measure their heights. You calculate the sample mean to be 5 feet tall and the sample ...
0
votes
1answer
17 views

What metric is it in the definiton of converge in probability?

What metric or norm is it in the definiton of converge in probability ; $ \forall \epsilon, \lim_{n} \mathbb{P}(\mid X_{n}-X \mid > \epsilon) \rightarrow 0 $ as $n \rightarrow \infty$ noone seems ...
2
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0answers
30 views

How to prove convergence for the following?

Let $X_m$ be distributed as follows: $$\mathbb P(X_m=\pm m)=\frac{1}{2m^\beta}, \mathbb P(X_m=0)=\frac{m^\beta-1}{m^\beta}.$$ Let be $S_n=\sum_{m=1}^{n}X_m$, then I. If $\beta>1$, then ...
5
votes
1answer
63 views

$\aleph_1$ almost sure events that almost never all hold

This recent question sparked my curiosity. Is there a family of events $(E_k)_{k \in I}$ such that$\def\pp{\mathbb{P}}$ $\pp(E_k) = 1$ for any $k \in I$ but $\pp( \bigcap_{k \in I} E_k ) = 0$? Clearly ...