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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Why are these two conclusions equivalent?

Here is a theorem: Let $X$ and $Y$ be independent random variables with law $L_X$ and $L_Y$, respectively. (e.g. $L_X(B) = P(X \in B) \text{ for every Borel set } B$). Then, $X+Y$ has law $L_X ...
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17 views

Confusion concerning joint probability function

In the course notes I am studying, the author comments that the joint distribution function $F_{X,Y}$ completely determines the stochastic vector $(X,Y)$ and then proceeds to calculate the following ...
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67 views

Probability that the nunber of ties in $2n$ coinflips is $k$

A fair coin is flipped $2n$ times. If the number of "heads" and the number of "tails" coincide, a tie is reached. What is the probability $p_k$, that the number of ties occuring is exactly $k$, ...
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29 views

Law of large numbers for the function of sum of random variables

Consider a large number of Bernoulli trials $X_1,\ldots,X_n$, where trial $X_k$ has a probability of success $\sigma_k$. So with $\sigma_k$ probability $X_k = 1$ and otherwise $X_k=0$. Further, denote ...
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60 views

iff $E\Bigl(|X_1|\log ({1+|X_1|)}\Bigr)<\infty$

Question: $X_n$'s are i.i.d then $$E\Bigl(\sup_{n\geq 1} \frac{|X_n|}{n}\Bigr)<\infty \iff E\Bigl(|X_1|\log ({1+|X_1|)}\Bigr)<\infty$$ My attempt: for $\Rightarrow$ part, because $\limsup ...
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2answers
28 views

existence of probability density function

Let $X$ be a random variable on a probability space $(\Omega,\Sigma,\mathbb{P})$. Let $F_X$ denote the probability distribution function of $X$ given by: $$F_X(y) = P(X\leq y) \text{ for } y \in ...
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230 views

Is “random variable” really random?

This is a concept question. The fundamental of modern probability theory is measure theory. A probability space is just a finite measure space and a random variable is just a measurable function. We ...
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33 views

Distance between two p.m.fs

I am stuck with the following problem from research. Is there any existing distance measure which can compare two probability mass functions with different support? For eg. for pmfs $p_1$ and $p_2$ ...
2
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1answer
38 views

Existence of the law of a random variable

Here is the definition of the law of a random variable. Let $X$ be a random variable on $(\Omega,\mathcal{F}, \mathbb{P})$. Then, the law of $X$, denoted by $L_{X}$, is a probability measure on ...
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204 views

The theory in probability

Consider a real-life experiment (perhaps written as a problem in a textbook): A coin is continually tossed until two consecutive heads are observed. Assume that the results of the tosses are mutually ...
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45 views

Computing expectation of: $\small E\left[f(Z,U)e^{\frac{V^2-(V+W)^2}{2}} \right] $

Suppose we have three mutually independent random variables $U,V,W$ where $W \sim \mathcal{N}(0,1)$, $V \sim \mathcal{N}(0,c)$ and $E[U]=0$, $E[U^2]=1$. Lets define $Z=U+V+W$. Can we compute (or ...
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1answer
31 views

Characterization of uniform integrability of random variables

Let $\{X_n \}$ be a sequence of random variables on a probability space $(\Omega,\mathcal{F},P)$. Then, $\{X_n\}$ is uniformly integrable if $$\lim_{M \to \infty} \sup_n \int_{|X_n| > M } |X_n| = ...
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25 views

Klenke's definition on exchangeable families of random variables

In the beginning of Ch12 (page 231) of Klenke's book "Probability theory", I find his definition on exchangeable families of random variables confusing. Let $I$ be an arbitrary index set and let ...
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23 views

Rate of convergence in sum of random variables

So I know that if $X_n+Y_n\xrightarrow{p} 0$ and $Y_n\xrightarrow{d} N(0,1)$ then by Slutsky's theorem we get $X_n\xrightarrow{d}N(0,1)$, but I was interested in knowing the rate of convergence. ...
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1answer
40 views

Is $X_n=O_p(1)$ defined on same probability space?

"$X_n$'s are bounded in probability" means "for every $\epsilon >0$ $\exists M_{\epsilon}$ such that $P(|X_n|>M_{\epsilon})<\epsilon$ $\forall n\geq1$ i.e. a single $M_{\epsilon}$ will ...
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1answer
38 views

Probability that a random bridge board does not contain a sequence?

Two cards with adjacent values and the same suit produce a SEQUENCE. For example, the heart-ten and the heart-jack form a sequence. The order of the values in bridge is $$23456789TJQKA$$ -What is the ...
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56 views

Probability, that a sequence of $n$ coin-flips contains $k$ changes of the lead

A fair coin is flipped $n$ times. What is the probability $p_k$, that the lead between "heads" and "tails" changes exactly $k$ times ? For example, the sequence $$HHTTTHH$$ contains two changes ...
2
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1answer
35 views

Convergence of random variables

Let $X_n$ and $Y_n$ be random variables (both with mean zero and variance one). I read a paper that proves $\mathbb{E}[(X_n+Y_n)^2]\rightarrow 0$ and since it was known that $Y_n\rightarrow N(0,1)$ ...
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1answer
38 views

Computing expected value of exponential

Let $U$ and $V$ be two independent random variables where $E[U]=$ and $E[U^2]=1$ and where $V$ is standard Gaussian. We also let $W=U+V$. How to compute the following expectation or find an an upper ...
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64 views

Define the random process $U(t) = A$ where $A$ is uniform over $[-1,1]$

Define the random process $U(t) = A$ where $A$ is uniform over $[-1,1]$. How would one sketch a sample realization of this?? Can someone give me a simple idea so I could attempt my own definition? ...
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29 views

Confused if this is conditional or dependent probability.

John observers the following while driving to work. • 4 were driving a red car. • 3 were driving a blue car. • 3 were driving a black car. He also notices ...
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49 views

Why does the uniform probability function have the y value it does? (1/b-a)

It seems very clear in the discrete case. Rolling a die for example, you have a discrete valued function $f(x)= \frac{1}{6}$, $x \in \{1,2,3,4,5,6\}$. The sum of all the values is 1. In the ...
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Intuition behind almost sure limit of $\frac{|S_n|}{n^{1/p}}$

Suppose $X_i$'s are non-degenerate i.i.d. Then (1) If $E|X_1|^p=\infty$ we've $\limsup_{n\rightarrow \infty} \frac{|S_n|}{n^{1/p}}=\infty$. And this is true $\forall p>0$ (2) However for $p=2$ ...
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31 views

Help proving generalized Jensen's inequality $\mathbf{E}[f(\cdot,X(\cdot))\mid \mathscr{G}] \geq f(\cdot,\mathbf{E}[X\mid\mathscr{G}](\cdot))$

I'm reading Meyer's seminal work Probability and Potentials (1966), in which he states the following "borrowed" theorem from Dubins "Rises and Upcrossings of Nonnegative Martingales" (1961). ...
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1answer
20 views

Stationarity and Ergodicity vs. Memorylessness

A (discrete) memoryless information source is (usually) defined as a collection of random variables that are independent and identically distributed. My question is, does memorylessness imply ...
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1answer
76 views

Packing of discrete random variables with finite second moment

I am considering a discrete random variable $X \in\mathbb{R}$ with $N$ points (where each point has non-zero probability) and $E[X^2]=1$ and $E[X]=0$. Let $d_l$ be the the smallest distance between ...
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2answers
37 views

Is this set of event axioms complete?

In the notes' first chapter of the course "Discrete Stochastic Processes" presented by Prof. Robert Gallager, the "Axioms for events" defined as follows: 1.2.1 Axioms for events [Chapter 1: ...
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48 views

Breaking a stick randomly at two points: Expected value of the largest piece. [duplicate]

If a stick of unit length is broken randomly at two points, to make 3 pieces of stick. What is the expected value of the largest stick. Is there an elegant solution to this problem? Thanks.
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1answer
60 views

$\sum X_n$ converges a.s.

This one is from old qualifying exam. $\{X_n\}$ be non-negative, independent and $\{Y_n\}$ is another sequence (not necessarily independent) but $X_n \sim^{d} Y_n$. Then $\sum X_n$ converges a.s. ...
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2answers
48 views

Inequality involving $X_{(n)}$

I am trying to prove that for random variables (not necessarily iid) $X_1,...,X_n$ that $X_{(n)} \le x + \sum_{i=1}^n X_i \mathbb{I}_{X_i > x}$ for all $x \in \mathbb R$. I tried to show this by ...
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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 ...
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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 ... ...
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1answer
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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) = ...
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1answer
49 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} - ...
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1answer
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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 ...
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52 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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
113 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 ...
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1answer
29 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 ...
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1answer
12 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 ...
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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 ...
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3answers
51 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. ...
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1answer
24 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 ...
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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 ...
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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 ...
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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 ...
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2answers
42 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 ...