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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3
votes
1answer
71 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
44 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 ...
-1
votes
0answers
13 views

Data set of variables with probabilities associated: standard notation [closed]

Let $X$ be a data set of $n$ independent observations and $m$ discrete variables $x_1,\cdots,x_m$. For each variable, there exists a vector $p$ containing the theoretical probabilities of the values ...
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 ...
0
votes
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
vote
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 ...
-5
votes
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.
0
votes
1answer
42 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
21 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
25 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
46 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 $P(X<Y)\neq 1/2$ ...
0
votes
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 ...
1
vote
0answers
28 views

Measurability of integrals with respect to different measures [closed]

Let $Y$ be a locally compact Hausdorff topological space (further assumptions like metrizability, separability, etc., may be added if necessary) and let $\mathscr Y$ denote the Borel $\sigma$-algebra ...
1
vote
0answers
29 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
vote
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
votes
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 ($(Z_t)$ ...
0
votes
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
votes
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
30 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?
0
votes
0answers
12 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
40 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 $\mathbb{E}(M_tM_sM_r)$?
1
vote
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
48 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: ...
0
votes
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
votes
0answers
8 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
votes
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 ...
0
votes
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
vote
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
35 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
votes
0answers
27 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
35 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
16 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
votes
0answers
29 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 ...
3
votes
1answer
19 views

Poisson Process independent Wiener Process using singular measures

I was reading some stochastic calculus of Jump processes and saw the result that if $W_t$ is Brownian and $N_t$ is Poisson both adapted to the $W_t$'s natural filtration then these processes are ...
3
votes
1answer
59 views

Question regarding Brownian motion

Hello I have two questions regarding the construction of Ito's integral in Øksendals book from here: http://th.if.uj.edu.pl/~gudowska/dydaktyka/Oksendal.pdf On page 25 he lists these 3 properties ...
0
votes
1answer
27 views

Proving an inequality involving conditional probability

Let $(X_t)_{t\ge0}$ be a stochastic process on a probability space $(\Omega,\mathcal F, \mathbb P)$ and let $\mathcal F_t=\sigma(X_s:0\le s\le t)$. Let $\Lambda\in \mathcal F_t$ with $\Lambda\subset ...