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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62 views

Markov Chains and Return Times

Let $(X_n)_{n≥0}$ be a Markov chain with transition kernel $p$ on a countable state space $S$, starting at $x∈S$ $T^{(1)}=\inf\{n≥1:X_n=x\} \quad \quad$ first return time to $x$ ...
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1answer
50 views

Convergence of random variable

I've been facing the following problem: Let $(X_k, Y_k)_k$ be a sequence of $2$-dimensional, independent random variables, each with uniform distribution over $ B(0,k) $ Verify if the following ...
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0answers
28 views

Determine the Law of $F^{-1}(U)$

If, $F^{-1}(u)=\inf\{x\in\mathbb R:F(x)>u\}$ and $U$ is uniformly distributed in $[0,1]$, what is the law of $F^{-1}(U)$ ? ($F$ is a distribution function of some random variable $X$) How can ...
1
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1answer
79 views

SDE and Stochastic calculus

$W_t$ is 1 dimension Brownian morion. $X_t=(cosW_t,sinW_t)$ Write SDE about $X_t$ I thought that $f(t,x)=(cosx,sinx)$, but I can't how "$t$" is expressed. I heard that the hint of this question is ...
3
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1answer
141 views

Question on Gauss map - application of Birkhoff's ergodic theorem

Take a Gauss map $G: [0,1] \longrightarrow [0,1]$ which is $$G(x) = \frac{1}{x} \mod 1, 0<x<1$$ and $0$ if $x=0$. Let $\mu$ be the Gauss measure. For $x \in [0,1]$ let $[a_{1}(x), ...
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0answers
58 views

expectation by integral

I am trying to compute the expectation of $\mathbb{E}[XY]$ where $X$ and $Y$ are dependent on a third non-negative random variable $Z$. I can now compute the expectation as follows ...
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1answer
58 views

How to find the expectation value?

Suppose that an insurer has an exponential utility function $u(x)=−2e^{-2x}$. What is the minimum premium $P^{-}$ to be asked for a risk X? After solving this we reached the following, So,only ...
2
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0answers
45 views

Reconstruction of state covariance from output covariance

Let us be given an LTI system $$ \frac{d}{dt} x (t) = A x(t), \;\; x(0)=x_0 \\ y(t) = Cx(t) $$ where $x_0$ is a random vector (e.g. uncertainty). Then it is known that the expectation $\mathbb ...
2
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2answers
393 views

probability in cards that 4 people each get queen and king?

A 52 card deck is shuffled and then dealt out to 4 people (each person gets 13 cards). What is the chance that each person gets an a Queen AND King? My attempt: I know theres $\binom{52}{13}$ ...
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0answers
48 views

conditional expectation of the Brownian motion [duplicate]

$(B_t)$ is a Brownian motion and i assume that $s<t<u$ we have $$E[B_t |\sigma(B_s,B_u)] = G(B_s,B_u)$$ Does anyone knows the explicit expression of $G$ ? (the calculus is easy but ...
2
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1answer
457 views

Given two uniformly distributed independent random variables what should the PDF of multiplication of them? [duplicate]

I am a probability noob and I was solving the following problem, but my answer doesn't match the book's. The length and width of panels used for interior doors(in inches) are denoted as Xand Y, ...
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1answer
58 views

Expectation on product space

Suppose that $\mathbb{X}=\mathbb{X}_1\times \mathbb{X}_2$ and suppose that $ P$ is a probability measure on $\mathbb{X}$ with marginals $ P_i$ on $\mathbb{X}_i, i=1,2$. Is it then true that every ...
3
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0answers
96 views

Sum of sequence of random variables infinitely often positive

Let $X_1,X_2,\ldots$ be an infinite sequence of independent (but not necessarily identically distributed) random variables with $E(X_i)=0$ for all $i$. Set $S_n=\sum_{i=1}^n X_i$. I want to show that ...
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1answer
46 views

tightness of sequence of degenerate probabilities

If $\delta_x$ denotes for $x\in \mathscr{R} $, the degenerate distribution at $x$, prove that the sequence $\delta_{x_n}$ of probabilities on $(\mathscr{R,B})$ is tight iff $x_n$ is bounded. This is ...
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1answer
125 views

Does method of moments give consistent estimator?

Let $\{x_i\}$ be identically continuously distributed variables (not independent in general, let's say it can be a stationary AR(1) model). Define function $f_b$ depending on parameter $b\geq 0.$ ...
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2answers
62 views

Tricky Probability (Looks like a Quiz)

Suppose that we have a computer server. The probability that server works normally is p. Also we use n copies of the server in order to increase its reliability. What is the chance the whole system ...
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1answer
32 views

Existence of expected value with complex power

Suppose that $X$ be a random variable taking values on $(0,+\infty)$ with density function $f(x)$ and we have $\mathbb{E}(X^2)<\infty$. Can we conclude that $$\mathbb{E}(X^{-2-it}) ...
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2answers
59 views

Iterating functions of expectations

We all know that $E[E[X]]=E[X]$. I was wondering, does it also hold that $E[g(E[X])]=E[g(X)]$ for "any" function $g$?
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1answer
67 views

Lottery probability of winning

On a certain day, $N$ lottery ticket are sold and $M$ win. To have a probability of at least $\alpha$ of winning on that day, approximately how many ticket should he purchased? The answer is the ...
7
votes
1answer
189 views

Compute a probability in Random Walk by Martingales

Let $X_n$ be the state at time $n$ of a Markov chain with these transition probabilities : $$p_{i,i+1}=p_i\qquad,\qquad p_{i,i-1}=q_i=1-p_i$$ $(a)$ Show that $Z_n=g(X_n)\,;\,n\geq0$, is a ...
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1answer
43 views

Rescaling function for probability of $k$ adjacent ones in a binary string

Call $\xi$ a random variable taking values in $\{0, 1\}^{\{0, 1, 2, \ldots, n\}}$, where each character of the string has vaalue $1$ with probability $p$ and $0$ with probability $1-p$ independently. ...
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1answer
125 views

Variance of sum of random variables independent when not consecutive

So I have 20 different 'weeks' that are being considered as part of a weight loss exercise. The weight lost each week is distributed normally around mean $\mu =0.8kg$ and std. deviation $\sigma^2= ...
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1answer
264 views

Distribution of Standard Brownian Motion squared [closed]

If the standard Brownian Motion $B_t$ has pdf $$f(x)=(2\pi*t)^{-1/2}*\exp (-\frac{x^2}{2t})$$ how would one compute $B_t^2$? I normally would show a lot of different things that I try myself ...
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0answers
78 views

proving Lyapunov's condition for a sum of random variables

Let $X_1,X_2,...$ be independent random variables such that $P(X_n = \pm1) = \frac {1-2^{-n}}{2}$ and $P(X_n = 2^k) = 2^{-k} $ for $k = n+1,n+2,...$ Define a new sequence of random variables by $Y_n ...
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1answer
36 views

Prove or disprove that $\frac{\partial}{\partial u} P(T\geq u, U \geq t) =\frac{\partial}{\partial u} P(T \geq u, U \geq T)$ at $u=t$

Let $U$ and $T$ be two non-negative random variables, $T$ has density. Prove or disprove this: $$\bigg (\frac{\partial}{\partial u} P(T\geq u, U \geq t) \bigg )_{u=t}= \bigg( \frac{\partial}{\partial ...
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1answer
28 views

Which probability should use to approximate Binomail distribution?

As I know, we can appoximate binomial distribution by normal distribution and poisson distribution. But i don't clear it right and which better? Please help me if you know Thks.
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1answer
41 views

Which kind of probability should use in here?

I don't know how to compute the probability in the problem below. $10$ people take an exam, $6$ male and $4$ female. The probability for a male to pass is $0.5$, while the probability for a female ...
1
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1answer
59 views

Characterization of conditional independence

Definition: Let $\mathcal{G},\mathcal{K},\mathcal{H}$ be $\sigma $-subalgebras of $\mathcal{F}$, where $\left( {\Omega ,\mathcal{F},\mathbb{P}} \right)$ is a given probability space. We say that ...
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0answers
74 views

How a 1-point set can have positive probability measure?

Suppose I have a program: x := bernoulli() if (x == true): return 0.5 else: return uniform-continuous(0,1) If I am not mistaken, the distribution out output ...
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2answers
82 views

What does it mean to sample, in measure theoretic terms?

Suppose I have some random variable $X$ defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$. What does it mean, in measure theoretic terms, to draw a sample from $X$? When $\Omega$ ...
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1answer
133 views

Copulas and their properties

I am working with the following copula, and have a few questions about it: $C(x,y) = xy + \theta (1-x)(1-y)xy$ Here $\theta \in [-1,1]$ and $x,y \in [0,1]$ First, I am trying to show this copula is ...
2
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1answer
196 views

Is a sigma algebra (up to null sets, as in conditional expectation) always generated by a random variable?

Motivation Let $(\Omega, \mathcal F, \mathbb P)$ be a standard probability space. For two $\mathcal H, \mathcal G \subset \mathcal F$, we say $\mathcal H = \mathcal G$ mod 0, if they are same up to ...
2
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1answer
60 views

class of slowly varying functions

The (seemingly) universal example of a slowly varying function in extreme value theory is a power of the natural log, e.g. $\lambda (\ln (x))^{a}$ for $x > 0$, $\lambda \ge 0$, and $a \ge 0$. In ...
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0answers
26 views

Expression for $B_1$

I think that it is indeed the case that $$ B_1 = \int_0^1 \frac{B_1 - B_t}{1-t} dt, $$ where $B$ is a standard one-dimensional Brownian motion. Am I right? If so, how you we prove it? Thanks a lot ...
3
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1answer
61 views

Prove $\lim_{n \to \infty} \frac{\Gamma(n+1/2)}{\Gamma(n)~n^{1/2}}=1$

Prove $$\lim_{x \to \infty} \frac{\Gamma(x+1/2)}{\Gamma(x)~x^{1/2}}=1.$$ I got this problem from Probability and Statistics by Degroot & Schervish. There is a hint to use Stirling's formula ...
2
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0answers
49 views

problem on almost sure convergence

Let {$X_i$} be iid with finite second moment. Let $Y_n = \frac {2} {n(n+1)} \sum_{i=0}^n i*X_i $, n$\ge$1 Show that $Y_n \to E(X_1) $ I tried to define $Z_i = \frac {2} {(n+1)} i*X_i $ Then $Y_n = ...
2
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0answers
121 views

Conditional measure with respect to a sigma-algebra generated by the level sets of a function has full measure on its level set.

Let $(X,\mathscr{B},\mu,T)$ be a measure-preserving system, where $X$ is a compact metric space, $\mathscr{B}$ its Borel $\sigma$-algebra, $\mu$ a Borel probability measure and $T$ continuous. Let ...
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0answers
30 views

Modeling Gaussian Error

Context I am designing a simulation of a robot receiving input from a sensor which has gaussian error. The robot will start from a known position and move at a constant speed; the sensor will ...
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2answers
51 views

Problem on Convergence of random series

Suppose that $\{X_n\}$ is an independent sequence and $E[X_n]=0$. If $\sum \operatorname{Var}[X_n] < \infty$, then $\sum X_n$ converges with probability $1$. Is independence necessary condition ...
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2answers
62 views

When do normal distributions not occur?

I know that in many cases one can assume a normal distributed probability density. But what the situations when the distribution in non-normal. Some examples would be nice. For example, suppose ...
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1answer
120 views

Law of total expectation and conditional expectation

We know from law of total expectation that $$ \mathbb{E}[\mathbb{E}[Y|X]]=\mathbb{E}[Y] $$ Does that still work if there is a further condition, i.e. does this equation hold? $$ ...
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1answer
38 views

Equality involving a sequence of independent exponentially distributed variables

I'm trying to prove the following statement: Let $\left( {{T_n}:n \geqslant 1} \right)$ be a sequence of independent, exponentially distributed random variables with ${T_n} \sim Exp\left( {{q_n}} ...
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1answer
98 views

convergence in probability of function of random variables

Suppose that $X_1, X_2, \ldots, X_n$ be a sequence of i.i.d random variables. If we have $E(|X_1|^k) <\infty$ for some $k>0$ and $f(x)$ is a bounded continuous function on $\mathbb{R}$. Is the ...
3
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1answer
135 views

Property (ii) of increasing functions in Chung's “A Course in Probability Theory”

I am a bit confused by the line of reasoning on page 2 of Kai Lai Chung's "A Course in Probability Theory". In particular, he is considering a real-finite valued function $f$ which is defined and ...
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1answer
25 views

Markov Processes: How to show $\int P(X_t\in B\mid X_0)dPX_0^{-1}=P(X_t\in B)$?

Let $\left\{X_t \right\}_{t\in T}$ be a time homogeneous Markov process with state space $S$. How do I formally demonstrate$$P(X_t\in B)=\int_S P(X_t\in B\mid X_0)dPX_0^{-1}$$(here $PX_0^{-1}$ is the ...
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3answers
2k views

Joint density problem. Two uniform distributions

This is the problem: An insurer estimates that Smith's time until death is uniformly distributed on the interval [0,5], and Jone's time until death also uniformly distributed on the interval [0,10]. ...
0
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2answers
29 views

random variables with $\sigma(A,B)=1$ and gamma distributed

We've discussed about the following (without solution) in class: Let $(X_k)_{k=1}^\infty$ be a sequence of indipendent and uniformly distributed random variables in $[0,1]$. Are there random ...
3
votes
1answer
121 views

Is there a proof for the Central Limit Theorem via some fixed point theorem?

This question arose in my mind when I learned that the Gaussian is a fixed point for the Fourier transform. On the other hand, in e.g. the Banach fixed point theorem we have convergence to a fixed ...
1
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1answer
110 views

Convolution of two Uniformly distributed r.v. ove

Assume a continuous random variable $X$ that is uniformly distributed $\underline{\text{on}}$ a $k$-sphere. For simplicity, lets assume a simple circle with radius $R$ in 2 dimension. Therefore ...
2
votes
1answer
50 views

What is the distribution of $x^TAx$ when $x$ is gaussian ($A$ may be not symmetric)

Suppose that $A \in \Re^{d \times d}$, $x \in \Re^d$ and each component of $x$ is independently sampled from $N(0,1)$. I wonder what is the distribution of $x^TAx$. To be more concrete, how will the ...