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)

7
votes
3answers
60 views

Radon-Nikodym derivative of sum of two measures

Problem Statement: Suppose that $\mu$ and $\nu$ are two finite measures such that $\nu \ll \mu$, let $\rho = \mu + \nu$, and note that since $\mu(A) \le \rho(A)$, and $\nu(A) \le \rho(A)$, we have ...
3
votes
0answers
43 views

Doob's inequalities for not necessarily right-continuous martingales

In Revuz and Yor, they denote $\mathbb{H}^2$ the space of $L^2$-bounded martingales, and $H^2$ the space of continuous $L^2$-bounded martingales. They state "... by Doob's inequality ... ...
0
votes
1answer
38 views

Expected value of a piecewise function in two variables

Let X and Y to be two independent random variables with equal pdfs (pdfs are known). Does anyone know how to estimate the expected value of the following function??? \begin{equation} g(X,Y) = ...
2
votes
1answer
48 views

Central limit theorem: where is the martingale in this proof?

Yet another question from the depths of Durrett. Again in the proof of Theorem 8.8.3, the author notes that "by the orthogonality of martingale increments," $$ E \left( \sum_{m=1}^{[nt]} t_{n,m} - ...
1
vote
1answer
31 views

How to prove that $\Bbb{P}(X_{4l} = 0) \leq c_l (2d)^{-2l}$ for some constant $c_l$?

Let $(X_n)$ be a simple random walk on $\Bbb{Z}^d$ starting at $0$. (The dimension $d$ will vary, but I will suppress the dependence on $d$ for brevity.) I encountered a statement which claims that ...
2
votes
0answers
49 views

Good book for developing intuition for probability

I recently began to study "probability theory" in the sense of a rigorous mathematical treatment of probability in terms of measures,and etc. But, my background in probability is really elementary, ...
1
vote
1answer
33 views

Martingale CLT: “without loss of generality”?

(Hopefully last in a long series of posts from the "I don't have Rick Durrett's brain" department... apologies.) In Durrett's proof of a a simple martingale CLT (Theorem 8.8.3, p. 341), he loses me ...
1
vote
0answers
42 views

Stopped Brownian motion proof

I'm trying to work through a proof in Durrett's textbook of a martingale convergence theorem via an embedding of the martingale in Brownian motion, and am stuck verifying a detail as usual. I'm ...
2
votes
2answers
36 views

Show that $X_tY_t=X_0Y_0+\int_0^tX_sdYs+\int_0^tY_sdX_s+[X,Y](t)$, where $X_t,Y_t$ are Ito processes

So I have done this exercise and the proof holds, but I really don't believe it can be correct because the proof is worth twice as much as other exercises. I am also not 100% sure if $d_s ...
1
vote
2answers
39 views

Show that $\mathbb{E}\left(\int_0^1X_n(t)dW(t)-\int_0^1X(t)dW(t)\right)^2\to 0$

I tried to shove this question in another one of my posts as a follow-up question, but I deleted my comment and will post it instead. I would really appreciate if someone could help me spot out and ...
0
votes
1answer
111 views

Assignment of initial probability values

Suppose a coin is tossed until a head is observed for the first time. It is given that the coin lands heads with probability $p$ and tails with probability $1-p$. Based on only this information, can ...
1
vote
1answer
26 views

Joint distribution of Brownian motion and its running maximum

$B$ being standard Brownian motion, its running maximum is defined as $M_t = \sup_{0\leq s\leq t} B_s$. I am trying to follow the proof of the following result but I don't understand some of the steps ...
1
vote
1answer
11 views

Existence of a localizing sequence of stopping times for a continuous local martingale

I have a a question about continuous local martingales: the definition of continuous local martingale says that a continuous process $X_s$ is continuous local martingale if there is non decreasing ...
1
vote
1answer
36 views

Finding standrad deviation $\sigma$

Carton of milk can be reserved fresh for $20$ days in average, $\frac13$ from the milk cartons can reserved fresh for $22$ days or more. Let's assume that the period of fresh is exponential ...
1
vote
3answers
49 views

Finding maximum of two variables

Given $X$ is uniform on $[0, 10]$. Let $$Y = \max(5, X).$$ Determine Var(Y). I'm familiar with how to find the variance of a uniform random variable, as well as the max of two random variables. ...
2
votes
1answer
23 views

Lévy-Khintchine formula and Taylor expansion

I have trouble finding out why this condition $\int_{\mathbb{R}\backslash\{0\}} \min(1, x^2 ) \nu(dx) < \infty$ in the Lévy-Khintchine formula is necessary. The Lévy-Khintchine formula is ...
1
vote
1answer
28 views

Conditional expectation of insurance payment

I'm trying to solve the following problem: An insurance policy is written to cover a loss, X, where X has a uniform distribution on (0, 1000). At what level must a deductible be set in order for the ...
0
votes
2answers
44 views

Heads and tails Probability question

I have two coins in my pocket. One of them is normal. One side heads one side tails. The others both sides are tails. So if I pick a random coin from my pocket and the side I see is "tails" what is ...
1
vote
1answer
30 views

Probability distribution of data and likelihood in Bayesian and Frequentive Statistics

I have recently been studying Bayesian as well as Frequentive Statistics (mostly null hypothesis significance testing) and am confused as to the meaning of the distribution of the likelihood and ...
1
vote
2answers
41 views

Expected value definitions

Let $X : \Omega \to \mathbb{R}$ be a discrete random variable in a discrete probability space with countable sample space $\Omega$. Let $P(\omega)$ be the probability of an outcome $\omega \in ...
0
votes
0answers
20 views

Obtaining the PMF of the ratio of two discrete random variables.

I have the pmf of "a" discrete random variable and also of another discrete random variable "b". Now, I want to obtain the resulting pmf of "a/b". How to do this? I want the PMF and not the PDF.
3
votes
0answers
38 views

PDF of $X = \max\{X_1,X_2\}$, being $X_1$ and $X_2$ independent Normal distributed random variables

Let $X_1 \sim \mathcal{N}(\mu_1,\sigma_1^2)$ and $X_2 \sim \mathcal{N}(\mu_2,\sigma_2^2)$, what's the CDF of $X = \max\{X_1,X_2\}$? Both variables are assumed to be independent. I tried the ...
0
votes
0answers
17 views

Hoeffding's inequality, number of samples required

I was deriving the number of samples required to qualify certain confidence bounds, at the end I am getting slightly different results from what is stated in my lecture notes. can anyone explain why ...
2
votes
1answer
26 views

Conditional probability integral transform

It is well known that for a random variable $X$ with a continuous distribution function $F(x)$, $F(X)$ is uniformly distributed on $[0,1]$. Can we now prove the same for a conditional distribution ...
1
vote
2answers
32 views

Why are these two sigma algebras independent?

Given a probability space, let $E$ be an event and let $\{E_n\}_{n=1}^\infty $ be a sequence of events. Claim: If $\sigma(E)$ is independent of $\sigma(E_n)$ for each $n$, then $\sigma(E)$ is ...
2
votes
1answer
80 views

What exactly are random variables in probability theory?

What I understood about random variables is : We need to define a function that maps the set of possible outcomes of a random experiment to the 1-D real space. The notion of random variable is to ...
2
votes
1answer
39 views

Why does $[X\in A]=[(X, Y)\in A\times \mathbb R]$

$X$ and $Y$ are both random variables, $X,Y:(\Omega,\mathcal A)\to\mathbb (R,\mathcal B(\mathbb R) )$, $A\in \mathcal B(\mathbb R) )$ On an intuitive level, this is perfectly clear. But I am unsure ...
3
votes
1answer
37 views

On a problem of weak convergence for a particular convolution of pr. measures

Assume that $\{P_n: n \in N\} $ and $\{Q_n: n \in N\} $ are sequences of probability measures. Assume that $P_n \stackrel{w}{\to} P. $ Also, assume that $Q_n = \delta_{b_n}, $ the Dirac measure and ...
1
vote
1answer
34 views

Independence of sigma algebras

Let $(\Omega,\Sigma,P)$ be a probability space. What I know is that if $\{g_n: n \geq 1\}$ is a sequence of $\pi$ systems where $g_n \subset \Sigma$, then $\{g_n: n \geq 1\} $ is independent if and ...
4
votes
1answer
66 views

PMF estimation: concentration inequalities for the $l_1$ and $l_\infty$ errors

Assume that you are given $n$ i.i.d samples $X_1, ..., X_n$ drawn from a discrete distribution $p = (p_1, ..., p_k)$. We would like to estimate $p$ using the empirical estimator \begin{equation} ...
1
vote
0answers
16 views

Central Limit Theorem for asymptotically i.i.d. random variables

I have a ergodic stationary sample of independent column random vectors $\{\mathbf x_1, \ldots,\mathbf x_n\} \equiv \mathbf X_n$, $\mathbf x_i \in \mathbb R^k$, with finite moments and cross-moments. ...
0
votes
2answers
46 views

Calculating Expectation and Covariance

$X$ is a random variable which is uniformly distributed on the interval $[1,4]$. Define the random variable $Y = g(x)$ with $g(x) = x^2$ How can I calculate $E(g(X))$, $g(E(X))$ and the covariance ...
0
votes
1answer
16 views

Application of Ito's isometry in deduction of Wiener Ito Chaos expansion

I am trying to learn about the Wiener Ito Chaos expansion and starting reading Oksendal's notes on Malliavin calculus where it is treated in Chapter 1. For a link to the notes, please see ...
1
vote
0answers
35 views

Please validate the proof: a theorem about conditional expectations

I have encountered a theorem saying that: $X$ and $Y$ are random variables (real valued, integrable) on $(\Omega, \mathcal A, P)$, $G(X,Y)$ is a random variable (also integrable and real valued), ...
1
vote
2answers
37 views

Uniqueness of moments for probability distributions with infinite moments.

I was taught the collection of a distribution's moments uniquely defined the distribution. Recently, I have been studying Pareto distributions, which have infinite means for shape parameters less than ...
0
votes
1answer
68 views

Infinite heads from Infinite coin tosses?

If I toss a coin an infinite amount of times, can I be sure to get an infinite amount of heads? Is it possible for it to be tails every flip meaning I get no heads at all?
1
vote
0answers
59 views

Strong law of large numbers

Suppose $X_i\in\mathcal{L}^2$ with expectation $0$ such that $\sum_{i=1}^\infty \mathbb{E}[X_i^2]/i^2<\infty$ and suppose they are pairwise non correlated. Does then the SLLN still hold?
0
votes
0answers
32 views

Counter example of non continuity

I present in the following a variation of the problem described in Continuity of a deterministic function generated from a probability function. There, it has been proved that $g(x)$ is not ...
1
vote
2answers
54 views

Finding the number of days that should be written on carton milk

Carton of milk can be reserved fresh for $20$ days in average, $\frac13$ from the milk cartons can reserved fresh for $22$ days or more. standard deviation $\sigma=4.651$ Let's assume that the ...
0
votes
2answers
36 views

Finding standard deviation $\sigma$

Carton of mlik can be reserved fresh for $20$ days in average, $\frac13$ from the milk cartons can reserved fresh for $22$ days or more. Let's assume that the period of fresh is normaly ...
0
votes
1answer
33 views

Finding k using the Central limit theorem

In a swamp there are two regular frogs and three princes frogs, the queen takes out $k$ frogs with replacement. Let $R$ be the number of times that a regular frog is selected. Using the Central ...
2
votes
1answer
29 views

stochastic dominance definition

I was wondering if, for positive random variables $X$ and $Y$, $\Pr(X\geq Y)\geq 1/2$ implies $\Pr(X\geq x)\geq \Pr(Y\geq x)$. Intuitively it "makes sense", since $X$ tends to be more often bigger ...
1
vote
1answer
42 views

Conditional Expectation

I am learning conditional expectation for fun. I cannot solve the following problem, I think it is just my vocabulary is limited. This problem came from a book on Stochastic Processes: Let $\Omega = ...
0
votes
0answers
36 views

Are these sigma algebras?

Consider the experiment of tossing a fair coin infinitely many times. Let $w$ be an outcome of this experiment written as $w = (w_1,w_2,.....)$ where $w_i = H \text{ or } T$ Let $E_n = \{w : w_n = ...
2
votes
1answer
25 views

Obtaining a $3$-dimensional simple random walk from a $d$-dimensional simple random walk with $d>3$.

Suppose $S_n$ is a $d$-dimensional random walk with $d>3$. Let $T_n=(S_n^{(1)},S_n^{(2)},S_n^{(3)})$, that is, we obtain $T_n$ by looking only at the first three coordinates of $S_n$. It is clear ...
1
vote
1answer
33 views

$\frac{\chi^2_n}{n}$ Stochastically increasing in $n$?

I was wondering whether $\frac{\chi^2_n}{n}$ is stochastically increasing in $n$. My main problem: Suppose $\hspace{5pt}\frac{(n-p)\hat{\sigma}^2}{\sigma^2} \sim \chi^2_{n-p}$. Then the expected ...
0
votes
1answer
11 views

Expectations of stopping times in general

I have a very basic question: So for a stopping time $\tau$ with $E(\tau)<\infty$ we have $E(\tau)=\sum_{n=0}^\infty P(\tau>n)$, right? Why is that? Thanks!
1
vote
1answer
26 views

$EY_n^\alpha \to 1$ and $E Y_n^\beta \to 1$ implies $Y_n \to 1$ in probabiltiy

Studying for quals... This is from Durrett, problem 2.13. If $Y_n \geq 0$, $EY_n^\alpha \to 1$ and $E Y_n^\beta \to 1$, for some $0 <\alpha < \beta$, then $Y_n \to 1$ in probabiltiy. I note ...
2
votes
0answers
30 views

On different versions of the Levy-Khinchin formula

I have a question that is not so much technical in nature, but more about the historical development of the topic of infinitely divisible distributions. In older but highly reputable probability ...
3
votes
2answers
106 views

Probability in heterogeneous sample space

Consider tossing a fair coin once followed by rolling a fair die, however the die is rolled once only when we get a head in the preceding toss. Need to find out probability of getting a six or a ...