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
21 views

Not getting the answer as given in Feller

Find the probability that the equation $x^2-2ax+b=0$ has complex roots, if $a,b$ are random variables following the Uniform $(0,h)$ distribution individually and independently. So we effectively ...
1
vote
1answer
127 views

Markov's Inequality for Negative Binomial distribution

Given that $Y$ follows Negative Binomial distribution (counts y successes before $k$th failure), using Markov's inequality show that for any $q \in[p,1]$, there exists constant $C$, such that ...
-1
votes
0answers
22 views

How to prove stochastic dominance? [on hold]

Consider the set of constant vectors $p_i$ and $\tilde{p}_i$, such that $p_i \succeq \tilde{p}_i \succ \mathbb{0}\; \forall i$ (component wise inequality) and define: $M \triangleq ...
1
vote
1answer
15 views

Martingale convergence without $L^p$ boundedness

I read the following example in the book titled "COUNTEREXAMPLES IN PROBABILITY AND REAL ANALYSIS" (by GARY L. WISE, and ERIC B. HALL): Does anyone know simpler examples? I do have one! I would be ...
1
vote
1answer
351 views

Simulation of a Gaussian process on $R^2$ with a stationary kernel using the Karhunen-Loève expansion

Assume $X(\omega, t) \sim \mathcal{N}(0, K(\cdot, \cdot))$ is a real-valued, centered Gaussian process on $R^2$, i.e., $X: \Omega \times R^2 \to R$. Let the covariance function of the process be ...
2
votes
2answers
23 views

Convergence in law of sample means of random variable

Let $\{X_n | n \in \mathbb{N} \}$ be a sequence of independent identically distributed random variables with density function: $$f_X(x) = e^{\theta - x}I_{(\theta, \infty)}(x)$$ with $\theta > ...
0
votes
1answer
13 views

Question about “integrable” random variable

I was reading the definition of Markov's Inequality on Wikipedia and it says If $X$ is any nonnegative integrable random variable and $a > 0$, then $\mathbb{P}(X \geq a) \leq ...
9
votes
3answers
257 views

Expected Value of R squared

Let $n$ be a fixed positive integer. Generate $n$ numbers $x_1, x_2, ..., x_n$ from the set $[0,1]$, with the probability distribution being the uniform one and the $x_i$ all being independent of each ...
1
vote
1answer
43 views

Probability space defined by function from $X$ to $[0,1]$

Let $X$ be a non-empty countable set. If there is a function $f:X\rightarrow [0,1]$ such that $p(S)=\sum_{x\in S} f(x)$ for all $S \in 2^X$, then prove that $(X,2^X, p)$ is a probability space. My ...
5
votes
1answer
93 views

Showing a Borel-Cantelli-esque result for not necessarily independent random variables

$A_1, A_2,\dots$ are events that are not necessarily independent. Prove that $$P(A_n \text{ i.o}) = 1 \iff \sum_n P(A_n \cap B) = \infty \text{ for all $B$ with $P(B)>0$}.$$ $(\Rightarrow)$ ...
-4
votes
0answers
33 views

Probability of an unbalanced coin [on hold]

Let's say i find a coin on the ground and i flip it 100 times getting 99 heads, what is the probability that the coin is unbalanced?(in particular that the probability of getting head is higher than ...
4
votes
1answer
35 views

Martingale with respect to a decreasing filtration

I am trying to solve problem 2.16 from the book "Continuous Martingales and Brownian Motion" by Revuz and Yor. There are two things that confuse me from the exercise so hopefully someone can shed some ...
0
votes
0answers
26 views

How to demonstrate the pdf of $P_{\sigma} (t)=\lambda_c e^{- \lambda_c t} / (1 - e^{- \lambda_c T})$

In $t_c$, there are $n$ expirations of $T$ and the remnant $\sigma$ seen from the above figure. Let the time $t_c$ forms the exponential distribution with parameter $\lambda_c$. How to demonstrate ...
0
votes
1answer
22 views

Doob-Kolmogorov Inequality

Denote by $(X(t),t\ge 0)$ a standard Brownian motion, i.e random variables with the following properties: $X(0)=0$. With probability 1, the function $t\mapsto X(t)$ is continuous on $[0,\infty)$. ...
0
votes
0answers
21 views

Limit definition of Sets.

Proposition 1.32 $X_{n}\xrightarrow{a.s.} X$ if and only if for any $\epsilon>0$ $P( | X_{n}- X |<\epsilon, \; \forall n\geq m )\rightarrow1$ $as$ $ m\rightarrow\infty$ Proof. Suppose first ...
0
votes
1answer
24 views

Sigma-fields and probability

I'm unsure what this question asks of me. For (i) I have given a power set with 16 elements in terms of a,b,c and d. I don't understand what I need to do for (ii). I believe (iii) is fairly ...
3
votes
5answers
128 views

Conditions for uniqueness of the median

A median of a random variable is defined as any $m \in \mathbb{R}$ such that $P(X \le m) \ge 1/2$ and $P(X \ge m) \ge 1/2$. Alternatively, in terms of the CDF $F$ of $X$ defined by $F(x) := P(X ...
-1
votes
0answers
58 views
+50

analytic solution of a definite integral

Integrate the following $$\int_0^\infty \alpha\,\beta\, c\, k\, x^s\, x^{c-1} (1+x^c)^{k-1} \left[(1+x^c)^k-1\right]^{-\beta-1} \left[1+\gamma ...
0
votes
1answer
21 views

ways to choose 4 people including two or more male from N people with 1/3 male 2/3 female

A sample of four people is randomly drawn from a population of N > 4 people.Assume that 1/3 of the total population is male, and 2/3 is female. (To simplify things, let’s assume that N is always ...
0
votes
0answers
31 views

ODE for the normal distribution [on hold]

The normal density function $\phi(x)=\tfrac{1}{2\pi}e^{-\frac{x^2}2}$ can be described via the ODE $$\phi^\prime(x) = -x \phi(x)$$ under the condition $\int_{-\infty}^\infty \phi(x) = 1$. Is there ...
0
votes
0answers
19 views

Dominated Convergence Theorem.

Dominated Convergence Theorem "Suppose $X_{n}\rightarrow X$ a.s., and there is a random variable $Y$ with $E[Y]<\infty$ such that $|X_{n}|<Y$ for all $n$. Then $E[lim_{n \to ...
15
votes
2answers
311 views

Problem on EU commission

Consider the following problem. A collection of $n$ countries $C_1, \dots, C_n$ sit on an EU commission. Each country $C_i$ is assigned a voting weight $c_i$. A resolution passes if it has the ...
1
vote
1answer
35 views

A Seemingly Trivial but Computationally Complicated Probability Problem

Suppose $X,Y$ are independent $Uniform(-1,1)$ random variables. Determine the distribution of $Z=X-Y$. I do not really think I should add my work here because whatever I have tried until now, has ...
0
votes
0answers
28 views

How to obtain a certain expression as an expectation

I have a probability space $(\Omega, M, \mathbb{P})$, where each $\omega \in \Omega$ is a random subset of natural numbers (i.e. This is a probability space of sequence of natural numbers sometimes ...
0
votes
1answer
14 views

Determining density involving scaled beta distribution

Suppose $Y \sim \mathrm{Beta}(2,1)$. If $X = \theta{Y}$ (for some $\theta > 0$) how do I determine the joint density $f(x, \theta)$? Edit: the density for $Z$ is $2z$. Would it be correct to say, ...
2
votes
1answer
13 views

Showing that the union of two algebras will give us a disjoint finite union of sets.

Suppose we have a space $X$ that could be anything and $\mathcal{A}_{1}$, $\mathcal{A}_{2}$ are algebras. How would I show that that algebra generated by $\mathcal{A}_{1} \cup \mathcal{A}_{2}$ ...
0
votes
1answer
28 views

Law of total probability explanation

What is the intuition behind the law of total probability? http://en.m.wikipedia.org/wiki/Law_of_total_probability
0
votes
2answers
416 views

Mean and covariance of Wiener process

Let $x(t), x(0)=0$ be a Wiener process with the parameters $a$ and $\sigma.$ Prove that its mean equals $a \cdot t$ and its covariance $R(t,s)$ is equal $R(t,s)=\sigma \min(t,s)$
0
votes
1answer
23 views

Independent events and Kolmogorov

Suppose we have a probability space $(\Omega, \mathfrak{F}, P)$, and independent events $(E_n)_n$. Consider $$M_n = \sum_{k=1}^n I_{E_k}$$ Is it correct to say that by the Kolmogorov $01$ law $M_n$ ...
1
vote
1answer
14 views

Semimartingale jumps question

I am reading a statement which contains $\Delta X \cdot Y$ where $X$ is a semimartingale and $Y$ is a finite variation process and the notation means the lebesgue stieltjes integral. My problem is ...
1
vote
2answers
25 views

Need a Probability Theory book that also focusses on Analysis

I am in search for a Probability Theory book which also contains elements and proofs from Analysis. A non-Measure Theoretic approach is most desirable. I have gone through great books like Ross but I ...
1
vote
2answers
30 views

In Markov chains a limit distribution is invariant

Suppose we have a Markov chain $(X_n)_{n \geq 0}$ with state space $S$. Suppose that $(\pi_i)_{i \in S}$ is a limit distribution. Then is $(\pi_i)_{i \in S}$ an invariant distribution ? I know the ...
3
votes
2answers
49 views

Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$

Question: Let X be a nonnegative random variable and $0 < \lambda \leq EX$. Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$ At first glance I thought I could use some ...
0
votes
0answers
7 views

Approximation of objective based on statistical distance

I am a computer science researcher (mostly theoretical) currently in midst of statistics and not able to figure out how to proceed. At an abstract level, I have a hypothesis for an unknown ...
2
votes
2answers
37 views

Convergence a.e. of the series $\sum_{i=1}^{n^2} \frac{X_i}{n^2}$

Let $(X_n)_{n\geq 1}$ be independent random variables with expected value $m$ and $\sup_n Var(X_n)\leq K < \infty$, and they are uncorrelated. Then $1)$ $$\sum_{i=1}^n \frac{X_i}{n} $$ ...
0
votes
1answer
27 views

Equality of two conditional expectations

I would like to show that for any random variable $X$ and $Z$ such that $X$ and $Z$ are independent and for any measurable functions $f$ and $g$, $$ \mathbb E \left[ f(g(X),Z) | g(X) \right] = ...
1
vote
1answer
23 views

argument technique to prove convergence of random variable

I witness a lemma in my class note and I think the proof is not quite clear. Could anybody give me some ideas about argument technique to prove the lemma? The lemma 3 in the beginning of the text: ...
-1
votes
1answer
40 views

Probability of a nonnegative submartingale converging to zero [on hold]

Suppose that $\{X_k\}$ is a nonnegative submartingale, and $\Pr(X_1 = 0) = 0$. Then could we conclude that $\Pr(\liminf X_k=0) = 0$? What about $\Pr(\lim X_k=0) = 0$? Thanks a lot. Some background ...
-1
votes
0answers
27 views

Poisson Process Alterations

If we have a Poisson process of rate $\lambda$, do the following alterations still result in a Poisson process? 1) Deleting every alternate point 2) Inserting points at times $1, 2, 3, ...$ 3) ...
3
votes
2answers
214 views

Are Polish space and LCCB space related?

It seems these spaces are the most useful ones for doing probabilities. Are LCCB (locally compact with countable basis) somewhat more general spaces that when endowed with a metric become Polish? I ...
0
votes
0answers
28 views

The Derivation of the Ito-Wentzell Formula

Is there a good derivation of the Ito-Wentzell Formula which is a generalization of the Ito's Lemma? Here are some unsatisfactory references to the Ito-Wentzell Formula: ...
0
votes
1answer
4 views

Random Variable Modeling

I am trying to understand how to model a random variable. So using a biased coin with $P(Head) = q$. If I am to generate a random variable $Y$ that is equally likely to be either a or b depending on ...
1
vote
4answers
69 views

Difference between $E[X^2]$ and $E[X^3]$

Hope to ask a dumb question. $Y = aX$,with $a \in N_+$. Here, we know the correlation coefficient is 1. Now, suppose $X \sim N(0,1)$. Here, we know $X, Y$ are not independent. Cov($X,Y$) = ...
0
votes
1answer
45 views
+100

Conditional independence given elementary events implies conditional independence given $\sigma $-algebra

Proposition: Let $X$ be a continuous markov chain with discrete state space $S$. Let $Z$ be the corresponding jump chain and $\left\{ {{W_i},i \in \mathbb{N}} \right\}$ its holding times. Let ...
1
vote
1answer
35 views

Showing That Two Normal-Based Random Variables Have the Same Distribution

Above is my question. $\overline X$ has distribution $N(0,1/n)$ - that's fine to work out. Similarly, $X_n / \sqrt{n}$ has distribution $N(0,1/n)$. These follow from the general relation $$ ...
1
vote
0answers
18 views

Uniform integrability of specific sequence of RV

I am investigating the following limit $$ \lim_{n \to \infty} E \left[ n \ln^-\left(1 - 2 \frac{\sigma}{n} [{\cal N}]_1 + \frac{\sigma^2}{n^2} \underbrace{ \| {\cal N} \|^2}_{\chi^2 \mbox{ ...
0
votes
1answer
23 views

Dyadic expansion

I'm reading the appendix in Billingsley book "Probability and measures" and I can't understand the following. If $$\sum_{i=1}^n\frac{d_i(\omega)}{2^i} < \omega \leq ...
-3
votes
0answers
25 views

Billingsley 2.5a) [on hold]

The field $\mathfrak{F}(\mathcal{D})$ generated by a class $\mathcal{D}$ of subsets of $N$ is defined as the intersection of all fields over $N$ containing $\mathcal{D}$. (a) Show that ...
0
votes
1answer
9 views

Smallest (sub-) Sigma algebra of a null set

Given a probability space ($\Omega,\mathcal{A} ,P$) and $N \in \mathcal{A}, N \ne \emptyset$ with $P(N) = 0$ What is the smallest sub-sigma algebra of $\mathcal{A}$ containing $N$. I'm kind of ...
2
votes
2answers
50 views

$X_n \to X$ in $L_2$, show that $\lim_{n \to \infty}E[X_n^2]=E[X^2]$

$X,X_1,...$ are random variables, $X_n \to X$ in $L_2$. Show that $\lim_{n \to \infty}E[X_n^2]=E[X^2]$. My attempt: $X_n \to X$ in $L_2 \implies \lim_{n \to \infty} E[(X_n-X)^2]=0 \implies \lim_{n ...