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 ...

learn more… | top users | synonyms (1)

0
votes
0answers
33 views

how to derive exponential growth equation from stochastic growth?

consider an exponential growth process of a population starting that has initial size $N_0$ and grows at rate $r$: $$\frac{dN}{dt} = rN$$ assuming deterministic and constant growth, the population ...
1
vote
0answers
31 views

Under what circumstances can two intersecting events be independent?

I'm toying with the concept of independence, which I understand to be the idea that "one event's occurrence has no bearing on the other element's occurrence." I know it is wrong to think that two ...
0
votes
0answers
19 views

Calculating the variance of an average of $N$ iid random variables

I'm having some problems proving that the variance of an average of $N$ iid random variables is equal to $\frac{1}{N}\text{Var}[X_1]$, where $X_1$ is one of the considered random variables. Formally, ...
3
votes
0answers
25 views

Prove convergence in measure (i.e., in probability) “distributes” over addition and respects nonnegativity.

Suppose $X_{n}$, $Y_{n}$, and $Z_{n}$ are random variables, with $Z_{n} \geq 0$ a.s. and $X_{n} \xrightarrow{p} X$, $Y_{n} \xrightarrow{p} Y$, and $Z_{n} \xrightarrow{p} Z$. Prove the following ...
0
votes
0answers
20 views

For which p is the Markov chain recurrent (almost random walk)

We have a Markov chain on Z with matrix: $p_{ii+1}=p=1-p_{ii-1}$ for $i\leqslant-1$, $p_{ii-1}=p=1-p_{ii+1}$ for $i\geqslant1$, and $p_{00}=p_{01}=p_{-10}=\frac{1}{3}$. For which values of p ...
0
votes
0answers
27 views

Convergence of a Sum

I need to show that the $\lim_{n\rightarrow \infty } \sum_{i=1}^{n} (\frac{p_{i}(1 - p_{i})}{n^{2}} )$ converges to $0$. Where the $p_{i}$'s are just constants. In particular, they are probabilities ...
1
vote
1answer
25 views

Proving the set of all finite or countable unions of intervals is not a Sigma Algebra

I would like to extend on a question I asked here Consider a set $J$ of all (open, closed, half-open, singleton, empty) intervals on $[0,1]$ Now consider further a set $B$ which is the set of all ...
1
vote
1answer
18 views

Equality of probability of finite hitting time for irreducible states in Markov Chain

Suppose I have a finite state Markov Chain with state space $S=\{1,2,3,4,5,6\}$. Suppose I further have that $\{1,2\}$,$\{3,4\}$ and $\{5,6\}$ are irreducible classes where $\{1,2\}$ and $\{3,4\}$ are ...
0
votes
2answers
43 views

Why is the set of all intervals on [0,1] not a sigma algebra

I am working my way through "A First Look at Rigorous Probability Theory" by Rosenthal and I'm a bit confused by something. In chapter 2 p.9 he mentions the set $J$ as being all intervals ...
1
vote
0answers
25 views

Bernoulli-Shift is a stationary process

Let be $(\Omega, \mathcal{F}, \mathbb{P})=([0,1), \mathcal{B}([0,1)), \lambda)$. We define $Y_n : \Omega \rightarrow \Omega$ by $Y_n := 2Y_{n-1} \mod 1$. Many sources claim that this is a stationary ...
3
votes
1answer
30 views

Minimum of a function that is an expectation

Let $X$ and $Y$ be two random variables. Define a function $f$ by $$f(t)=E[(X+tY)^2]$$ Find the value $t$ that minimizes $f(t)$ in terms of $E(X^2),E(Y^2),$ and $E(XY)$. Evaluate the ...
2
votes
1answer
29 views

Distribution of a random variable waiting for a consecutive sequence of bits?

Suppose we're trying to transmit a message comprised of $n$ bits. Assume each bit has a probability $p$ of being correct. Success means we succeed at consecutively transmitting all $n$ bits. As soon ...
3
votes
1answer
51 views

The partial derivative of a characteristic function (exercise).

Assume that you have a probability space $(\Omega, \mathcal{F},P)$ and a random varaible $X: (\Omega, \mathcal{F})\rightarrow(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$. Define the characteristic ...
0
votes
0answers
8 views

What does Karhunen-Loève expansion have to do with cosine-sine basis expansion?

According to my research, Karhunen-Loève(KL) expansion is a version of Fourier series for stochastic processes and states that under some conditions, a stochastic process $X\left(\omega, t\right)$ can ...
1
vote
0answers
42 views

Integral Inequality (CDFs and PDFs)

Suppose I have a function $g \geq 0$ defined by $$g(x) = \int_{-\infty}^{x}f(t)\text{ d}t \geq 0\text{, }x \in \mathbb{R}\text{. }$$ I know for a fact that $g$ is continuous and nondecreasing. Is ...
3
votes
1answer
51 views

Structure of the $L_1$ space of measurable subsets of $[0,1]$

Let $\mathcal A$ be a Borel $\sigma$-algebra on $[0, 1]$, and let's introduce a metric on it by $$ d(A, B) = \lambda(A\mathbin\Delta B) \qquad \forall A,B\in \mathcal A $$ where $\lambda$ is the ...
0
votes
0answers
40 views

If $X$ does not have a density, what is $\int x \ d X(P)$?

Let $X$ have distribution function $$F(x) = 1_{(-\infty,0)} e^x + 1_{[0,\infty)} (1 - e^{-x}/3).$$ My questions are: If a general distribution function is not a nice continuous function, does X ...
1
vote
1answer
44 views

RVs independently and uniformly distributed on interval $[0,1]$, prove every order is equally likely

Finitely many random variables $p_1,p_2,...,p_n$ are independently and uniformly distributed on interval $[0,1]$. They form an ascending sequence $p_{i_1} \le p_{i_2} \le ... \le p_{i_n}$. For ...
3
votes
0answers
24 views

Good book on quantum probability

Who can suggest me a book on quantum probability? I'm mostly interested in its geometric aspects (complex projective space, Fubini-Study...). I have a background in quantum mechanics and differential ...
6
votes
0answers
50 views

A random variable is symmetric if and only if its characteristic function is real-valued

Quick summary: I am stuck on the implication: $\phi_X$ real-valued $\rightarrow$ $X$ symmetric. Assume you have a probability space $(\Omega, \mathcal{F},P)$, and a random varaiable $X: \Omega ...
2
votes
2answers
76 views

Conditional probability and almost sure equality confusion

I initially began writing a lenghty post asking for clarification of my ideas, but decided to delete it and stick to one question that I hope might help my confusion: We're on $(\Omega, \mathcal F, ...
0
votes
0answers
36 views

Why is $E(X_t|B_t)=\frac{E(X_tB_t)}{E(B_t^2)}B_t$?

Why is $E(X_t|B_t)=\frac{E(X_tB_t)}{E(B_t^2)}B_t$ ? Does this always hold In an exercise I have to show that $E(X_t|B_t)\neq X_t$, where $X_t=\int_0^t B_s ds$, I think the definition of $X_t$ ...
0
votes
2answers
54 views

Finding marginal distribution, unit sphere

I'm asked to find the marginal distribution of $(X,Y)$ as $(X,Y,Z)$ is a point chosen uniformly on the unit sphere. I've worked out that the joint density function $f_{XYZ}(x,y,z) = \frac{3}{4\pi}$ ...
-1
votes
1answer
28 views

Questions on symmetric difference of events

From a comment on my math overflow question: No, $P(A\bigtriangleup B)=0$ means $A$ and $B$ are essentially the same except in situations that almost surely do not happen. $P(A)=P(B)$ says much ...
2
votes
2answers
33 views

Irreducible Markov chain. Pakes Lemma.

I've got problem with that task: Consider $\{Z_n\}_{n>0}$ is iid with integer values with expected value $\mathbb EZ_1<0$ and $\{X_n\}_{n\ge0}$ is homogeneous Markov chain defined by $$ ...
1
vote
1answer
40 views

Probability of lim sup, lim inf for sequence of random variables.

Maybe this is extremely simple, but i havent found a specific answer for this online. For a sequence of independent continuous random variables $X_n$ ,$n=1,2,3,...$ , all with the same probability ...
0
votes
1answer
28 views

Computation of a joint distribution function

Let $Y = X+W$ and suppose the joint PDF of $X$ and $Y$ is $$ f_{X,Y}(x,y) = \lambda^{2}e^{-\lambda\cdot y} \hspace{2mm}:\hspace{2mm} 0 < x < y < \infty$$ What is the density of $W$? I have ...
1
vote
0answers
29 views

Stopping Time Sum of Random Variables

Let $X_1,...,X_t$ be an i.i.d. sequence of random variables with support $\{a,-b\}$, where $a,b>0$, and measure $P(a)=p_1$, $P(-b)=p_2$. Assume $p_1a-p_2b<0$, so that $E[X_t]<0$. Let ...
1
vote
0answers
14 views

Problems about Farlie-Morgenstern family of bivariate CDFs

Hi I am trying to solve the following problem: Let $F_X:\mathbb{R}\to[0,1]$ and $F_Y:\mathbb{R}\to[0,1]$ be unnivariate Cumulative Distribution Functions (CDFs) and suppose $-1\le\alpha\le 1$. Define ...
0
votes
0answers
25 views

Coupon Collector with infinitely many coupons

There are plenty of coupon collector problems on here, but I couldn't find any with infinitely many coupons. Please let me know if this is a duplicate. The problem is taken from these notes: ...
0
votes
2answers
30 views

Random Walk Stopping Time

Let $(X_1,X_2,...)$ be i.i.d random variables, with $P(X_t=1)=P(X_t=-1)=1/2$. Then $S_t= \frac{1}{t}\sum_{i=1}^{t}X_i $ is a zero mean random walk. Let $\tau$ be the stopping time corresponding to ...
4
votes
1answer
32 views

Proving weak convergence of random probability measures

I don't understand the following as I read along a proof in a paper: We denote by $\mathcal{P}({M})$ the space of probability measures on a metric space $M$, equipped with the weak topology. ...
2
votes
1answer
26 views

Probability that a biased asymmetric random walk reaches the origin

I am working on the following problem for my probability class and I am a little stuck: A particle moves at each step two units to the right or one unit to the left, with corresponding ...
1
vote
0answers
30 views

Symmetric difference and approximation of measure [duplicate]

Let $\scr{A}$ be an algebra of subsets. Let $(\Omega, \sigma(\mathscr{A}), P)$ be a probability space. Then for each $B \in \sigma(\scr{A})$ and $\epsilon > 0$, there exists $A \in \scr{A}$ such ...
1
vote
1answer
66 views

Proof that process is martingale, exponential distribution

Let $X_1,X_2,\dots$ be i.i.d. random variables with exponential distribution with parameter $1$ and define $$Y_m= \sup{\{k\ge1:X_1+\dots+X_k\le m\}}$$ Prove that $Y_m-m$ is martingale and ...
0
votes
1answer
34 views

PDF of negative $\cos(X)$

Let $Y = - \cos(X)$, then what will be the pdf? Please share if you have any idea. If $Y = \cos(X)$, where $X$ is uniformly distributed in the interval $(0, 2 \pi]$, then the pdf is given by ...
2
votes
1answer
27 views

Show that the following limit is $AB$.

I am interested in showing the following: $$ \lim\limits_{p\rightarrow 1/2} \frac{B}{1-2p}-\frac{A+B}{1-2p}\frac{r^B-1}{r^{A+B}-1}=AB. $$ Here $A,B\in(0,\infty)$ and $r=\frac{1}{p}-1$. This is for a ...
1
vote
1answer
35 views

Does the law of large numbers hold when $\epsilon_n\rightarrow 0$?

Let $Y_n\sim \text{Pois}(n)$.I need to find for each $\alpha\in\Bbb R$ the limit $$\lim_{n\rightarrow\infty}P(Y_n\geq n+n^\alpha)$$ I got to: $$P(Y_n\geq n+n^\alpha)=P\left(\frac {Y_n} n -1 \geq ...
1
vote
0answers
16 views

Is the space of discrete probabilities measurable in the space of probabilities with weak topology?

Let $S$ be a polish space and $\mathcal{P}(S)$ be the space of all probability measures with weak topology. I am wondering whether the set of all discrete probability measures is measurable. Thanks ...
2
votes
1answer
39 views

What's the distributional derivative of a Banach space valued almost surely continuous stochastic process?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $\lambda$ be the Lebesgue measure on $[0,\infty)$ $(H,\left\|\;\cdot\;\right\|)$ be a Banach space over the field $\mathbb ...
3
votes
1answer
25 views

Finding a random variable $X$ such that $X_n$ (given) converges in distribution to $X$

For every $n\in\Bbb{N}$, let $X_n$ be a random variable which gets the values $\{-1, -\frac{n-1}n,...,-\frac 1 n, 0, \frac 1 n,...,\frac{n-1}n, 1\}$ with equal probability. Find a random ...
1
vote
1answer
39 views

$\sigma$-algebra generated by random variable : Show that if $\sigma(X)=\sigma(Y)$ then $\sigma(X+Y)\subseteq \sigma(X)$

Let $(\Omega,\mathcal{F},P)$ be a probability space and $X$ be a random variable. The $\sigma$-algebra generated by $X$ is defined as $$\sigma(X):=\{X^{-1}(B)\; | \; B\in B_{\mathbb{R}}\}$$ where ...
0
votes
1answer
56 views

how can I find E(XY) using the moment generating function?

Let $X,Y$ a random variable with density $f(x,y)=\frac{2}{5}(2x+3y),\quad 0<x<1, 0<y<1$. Find the joint moment generating function. I find that the answer is: ...
2
votes
1answer
25 views

Can you create non transitive dice for any finite graph?

Let's say you have a finite directed graph, with no two nodes that point at each other. Can we assign each node a dice, so that each node beats the node it is pointing at. This is easy for acyclic ...
2
votes
0answers
43 views

Supermartingale-like property : does convergence still obtains?

A super-martingale $\{X_n\}$ in discrete time is usually represented as having the defining property $$X_n \geq E[X_{n+1} \mid \mathcal F_n] ,\;\; \forall \,n \tag{1}$$ where $\{\mathcal F_n\}$ ...
1
vote
3answers
42 views

What can go wrong if we let sigma algebra to admit the union of uncountable union of elements?

By definition we only allow the union of countable infinite of elements to be also include the $\sigma$ field, why not uncountable many? Is there a historical view behind this?
3
votes
1answer
41 views

Prove a condition equivalent to the Lyapounov's one.

This is an exercise our professor gives to us during an exam. I did it partially, but now I would like to solve it completely. Show that for a sequence of random variable $\{X_n \}_{n \ge 1}$ and ...
3
votes
1answer
33 views

Which assumptions on $Ω\subseteq\mathbb R^d$ do we need in order to show density of $C_c^∞(Ω)$ in $(L^p(Ω),\left\|\;\cdot\;\right\|_{L^p(Ω)})$?

Let $\Omega\subseteq\mathbb R^d$, $u\in\mathcal L^1(\Omega)$ and $$u_\varepsilon(x):=\frac 1{\varepsilon^d}\int_\Omega\rho\left(\frac{x-y}\varepsilon\right)u(y)\;{\rm d}\lambda(y)\;\;\;\text{for ...
1
vote
0answers
32 views

If $D_1\subseteq D_2$ then the generated sigma-algebras are such that $\sigma(D_1)\subseteq \sigma(D_2)$

Let $A$ be a set of subsets of $\Omega$. Then, the $\sigma$-algebra generated by $A$ is defined as $$\sigma(A):= \cap \{\mathcal{F}\; |\; \mathcal{F}\text{ is a }\sigma\text{-algebra and }A\subseteq ...
0
votes
2answers
29 views

Conditional expectation with respect to $\tau$

Can somebody help me with the following expectation: $E[(\sum_{i=1}^{\tau} X_{i})^2 \mid \tau)$ where $X_{i}$, $i=1,2,..$ are independent random variables taking values $1$ and $-1$ with equal chance ...