Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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Moment generating function vs Laplace transform

The moment generating function of a random variable $X$ is defined as $M_X(u)=E[e^{uX}]$ for $u\in\mathbb{R}$. On the other hand the (two sided)Laplace transform for the density $f_X(x)$ is ...
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1answer
20 views

Probability with qualifications and gender

Qualification Female Male Degree 5 1 None 5 4 School 8 12 Vocation 8 7 I've been going through some ...
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3answers
148 views

Probability of Distribution of Apples Question.

I have encountered this question which was actually assigned to a Biology class (the deadline has passed). It seemed simple at first but as more time passes by I realise how difficult it is. This is ...
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1answer
40 views

Find the conditional expectation $E[X|X\wedge t]$ [closed]

Let X be a random variable with exponential distribution with rate 1, $$f_X(x)=e^{-x}, x>0$$ Given $t>0$, denote $X\wedge t$= min{X,t}, find $E[X|X\wedge t]$ explicitly. Since $X\wedge t$ is ...
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1answer
49 views

Find an example of a sequence of random variables

Find an example of a sequence of integrable random variables $\lbrace X_n,n\geq1 \rbrace$ that has the following properties, $E[X_{n+1}\mid X_n]=X_n$ for every $n\geq 1$ but $E[X_{n+1}\mid ...
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22 views

ProbabilityTheory, precise definition of r.vs with same distribution

Studying probability theory, I wonder the definition of $X \overset{d}{=}Y$. However, it took me long time to search it to find nothing. Considering in the extension of limit law, i.e $X_n \to_d ...
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1answer
32 views

Student's distibution

If $X_i$ are independent equally distributed random variables, $S_n=X_1+...+X_n$, then $$\frac{S_n-n\mathbb E(X)}{\sqrt{n \sigma^2(X)}}$$ tends by distribution to $N(0,1)$ for $n \to \infty$. It is ...
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1answer
32 views

Let $E(X) = 1, E(X^2) = 3, E(XY) = -4$ and $E(Y) = 2$. Find $Cov(X, 2X + Y)$. [closed]

QUESTION: Let $E(X) = 1, E(X^2) = 3, E(XY) = -4$ and $E(Y) = 2$. Find $Cov(X, 2X + Y)$. I believe I have to use bilinearity to solve this, but I'm unsure as to how to go about doing that. Any help ...
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118 views

proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
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0answers
23 views

When is the joint density differentiable

My question is the following: given a real random vector $X = (X_1,...,X_k)$ with differentiable marginal densities $f_1,...,f_k$, what extra conditions on the marginals are needed to ensure that the ...
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1answer
40 views

CLT for bounded and dependent sequence

Let $\displaystyle X_1,X_2,...X_n$ be identically distributed such that $\displaystyle Pr\{a \leq X_i\leq b\}=1$ for bounded constants $\displaystyle a,b$. Further Let $\displaystyle ...
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1answer
18 views

Sufficient condition for u.c.p. convergence of processes

Assume that all processes are cadlag. I am trying to prove the following claim: Let a sequence of processes $X_n$ be given. Assume that for all $s$ in a dense subset of $\mathbb R^+$ $ X_n(s) ...
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0answers
23 views

Pollaczek–Khinchine formula for claims with expotential distribution - derivation

I am trying to understand ruin probability formula using Pollaczek–Khinchine formula described here: http://en.wikipedia.org/wiki/Ruin_theory $$\psi(x)=\left(1-\frac{\lambda ...
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1answer
50 views

Is $P(\bigcup_{k=1}^\infty A_k)=\lim_{n\rightarrow\infty}P(\bigcup_{k=1}^n A_k)$ equivalent to countable additivity?

The third axiom of probability theory, known as countable additivity, states that $$P(\bigcup_{k=1}^\infty A_k)=\sum_{k=1}^\infty P(A_k)$$ holds for any countable sequence of disjoint events ...
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36 views

Invariant Probability of Discrete Time MC from Continuous Time Markov Chain

Given rates α of an irreducible continuous-time MC on finite state space and told that π is the invariant probability measure of this chain, we define a discrete time MC as having transition ...
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1answer
29 views

Simple Random Walk and $n$th zero hitting time

I am reading an example in Durrett's book regarding the $n$th time the random walk hits 0. Consider a simple random walk, $X_i=1$ or $X_i = -1$ with equal probability. Let $S_n = X_1 + \dots + X_n$. ...
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1answer
16 views

Question about sum of random variables

Let's say we have two independent discrete random variables, $X=\left\{ 0,1\right\}$ and $Y=\left\{ 2,3\right\}$. What does it mean to sum these two variable? In other words: $S=Y+X=?$
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51 views

Show $E(X)= \int_{[0, \infty)} P[X>t] dt$, without using density argument [duplicate]

For $X$, a positive random variable, use Fubini's theorem applied to $\sigma$-finite measure to prove \begin{align*} E(X)= \int_{[0, \infty)} P[X>t] dt \end{align*} I found several references on ...
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0answers
7 views

How should I go about this insurance risk problem?

Number of claims a policyholder makes in one year has Poisson distribution with mean lambda. Over population of policyholders, lambda has a gamma distribution with mean k and r. P(number of ...
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0answers
23 views

What exactly do you mean by random variable converge almost surely?

If somebody claim that $\sum_{k=1}^n X_k$ converges almost surely to a random variable , does it imply that $Y$ is finite? If so, first of all, is it the point of proving Kolmogorov convergence ...
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1answer
39 views

A stronger version of Kolmogorov Inequality

I came across this question which says that there is a stronger version of the Kolmogorov Inequality for symmetrically distributed random variables. The question is as follows Let $\xi_1, \ldots, ...
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1answer
18 views

Probability of winning in a die rolling game

A fаir die is rolled by 3 plаyers. The plаyer with the biggest score wins аnd the gаme stops, but if аt leаst two plаyers hаve the mаximаl score, then the gаme stаrts аgаin. Whаt is the probаbility of ...
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2answers
79 views

Show that $\int_{\mathbb{R}} F(x) F(dx)=\frac{1}{2}$ if $F$ is a continuous distribution function

If $F$ is a continuous distribution function, prove that \begin{align*} \int_{\mathbb{R}} F(x) F(dx)=\frac{1}{2} \end{align*} What I tried \begin{align*} \int_{\mathbb{R}} F(x) ...
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105 views

Inverse of $f(x) = xe^x-x$

I'm wondering if there is a way to obtain the inverse of the function $y=xe^x-x$. I am aware of the use of Lambert's W function in the inverse of $xe^x$ but as can be seen this is a different animal ...
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1answer
37 views

Show $\lim_{n \to \infty} n^{-1} E \left( \frac{1}{X}1_{[X>n^{-1}]} \right) =0$

Suppose $X$ is a non-negative random variable satisfying \begin{align*} P[0 \le X < \infty ]=1. \end{align*} Show a) \begin{align*} \lim_{n \to \infty} n E \left( \frac{1}{X}1_{[X>n]} ...
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1answer
32 views

Does convergence in distribution of discrete random variables with same finite support imply convergence in probability?

Let ${X_m}\mathop \to \limits^D Y$ and ${\text{supp}}\left( {{X_m}} \right) = {\text{supp}}\left( Y \right) = \left\{ {0,1, \ldots ,n} \right\},\forall m \in \mathbb{N}$. Does ${X_m}\mathop \to ...
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1answer
55 views

Find the probability that $B$ obtains a new kidney.

Look at this problem: Two individuals, $A$ and $B$, both require kidney transplants. If she does not receive a new kidney, then $A$ will die after an exponential time with rate $\mu_A$, and ...
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20 views

Which is a good book to read about convergence of posterior measure?

I am working on Bayesian statistics and would like to know about a good text book about convergence of posterior measure.
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39 views

Weak convergence to a constant implies convergence in probability

If on some probability space the random variables $X_1, \dots, X_n$ with distributions $\mu_n$ convergence weakly to the constant random variable $c \in \mathbb{R}$, i.e. $ \int f \, d \mu_n \to ...
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1answer
24 views

Is this formula a KL divergence?

As everyone knows KL divergence's formula is $KL(p||q) = \sum_{i=1}^{n}p(i)\log (p(i)/q(i))$. In the image, formula(9) is really calculate KL(X||($(UZ^TA^T)$)) , however i have no idea why there is ...
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2answers
387 views

Mean E(X - Y)^3

If X, Y are two independent and identically distributed random variables of mean 0, it is true that $\mathbb{E}(X - Y)^3 = 0$? Thank you!
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30 views

Applicability of Law of Large numbers for a given sequence of random variables

What are the general methods of showing that law of large numbers doesn't hold for a sequence of random variables?
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1answer
40 views

A confusion on conditional probability

I'm confused on two kinds of conditional probabilities: ${y=x+n}$, where ${x}=\pm1$ with equal probability(0.5). And $n$ is $\cal{N}(0,1)$. Then I know, the conditional probability of ${y}$ ...
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1answer
43 views

Let $X_1,X_2,X_3$ be independent $\mathrm{Exp}(λ)$-distributed random variables. Find the probability that $X_1 < X_2 < X_3$

I'm trying to figure out how to do this using exchangeability, but can't figure it out. Any help is appreciated.
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24 views

Generalized Form of Fano's Inequality

The Wikipedia article on Fano's Inequality presents a generalization as follows: Let $\mathbf{F}$ be a class of probability densities with a subclass of $r+1$ densities denoted $f_{\theta^{(i)}}$ ...
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1answer
44 views

Using Stirling's formula to uniformly bound Bernoulli success probabilities

In this paper, the authors say that for any $\gamma \in [1/2,1)$, there is a positive constant $B=B(\gamma)$ such that for any $n$, $$ \sum_{n\gamma\leq k \leq n} \binom{n}{k} \leq B n^{-1/2}2^{n ...
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0answers
37 views

Events in the tail $\sigma$-algebra

I am having a little trouble understanding what exactly is the tail $\sigma$-algebra. Just so we are all on the same page, my book defined the tail $\sigma$-algebra like this: Let $X_n$ be a ...
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1answer
20 views

Calculating Posterior Expected Utilities

I need help figuring out how to calculate the posterior expected utility. E(U(D = 1,X | Y = 1)) I have the following information/probabilities. There is more ...
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1answer
26 views

Computing Posterior Probability

I'm new to all of this, so please give explanations with answers. My goal is to learn it. I've given the start of a first attempt below. Suppose that X (Boolean true, false) influences another ...
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1answer
33 views

Finite limit involving characteristic function implies values of first and second moments

If $$\lim_{c \to 0} \frac{\phi_X(c) - 1}{c^2} = -\frac{\sigma^2}{2} < \infty$$ where $\phi_X(c)$ is the characteristic function of the random variable $X$, then $$E[X] = 0,\qquad E[X^2] = ...
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1answer
16 views

On convergence of certain integrals under weak convergence

As I work on certain problems that are relevant to me, I come across questions that make me realize how much just knowing the basic apparatus of results is not enough.. To simplify as much as ...
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81 views

finding n in binomial distribution

A player tosses a coin $2n$ times. The probability of head is $0.48$. The player wins if he gets head more than $n$ times. But he can choose the number $n$ before the game. What $n$ should he choose ...
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40 views

Function of probability measures representation

Let $\mu$ be a in $\mathbb M(\mathbb R)$, the space of probability measure on $\mathbb R$. Let $F$ be in $C_b (\mathbb M(\mathbb R), \mathbb R)$, the space of continuous bounded function on $\mathbb ...
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1answer
25 views

Utility Theory: Risk Averse, which should I choose?

Question: If I am slightly risk averse which do I prefer. (Give a mathematical justification for your conclusion): [0.5, \$450; 0.5, \$400] or [0.1, \$4375; 0.9, \$0] Okay so I get that the risk ...
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0answers
22 views

Find the density for the random variable $Y=X_1+X_2$

Problem: Let $X_1, X_2$ be independent variables with uniform density on (-1,1). Find the density for the random variable $Y=X_1+X_2$. My attempt using convolution: $y_1=x_1+x_2, y_2=x_2 \Rightarrow ...
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0answers
22 views

Integrating a function of measures

I've been reading John Baez's series of posts on Information Geometry. I'm currently on part 6... Midway through the post he discusses Radon-Nikodym derivatives: The formula for information gain ...
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1answer
28 views

Necessary and sufficient conditions for convergence almost surely and in probability

I have difficulties in solving following problem in Rick Durrett's "Probability Theory and Examples" This is the problem 2.3.15 in the 4th edition and problem 1.6.15 in the 3rd edition: Let $Y_1$, ...
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17 views

Can a randomized rule induce a random measure on the action space?

$D = \{d_i: X\to Y, i=1,\dots,n\}$ is a finite set of mappings from $X$ to $Y$, $(\Omega, \mathcal F, P)$ is a probability space, and $\delta: \Omega \to D$ is a measurable mapping. Can $\delta$ ...
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1answer
16 views

Domain in marginal density functions

I am so confused about the domain for marginal density of this problem... Here is the joint density function : Here is the solution: I perfectly understand how we get those two marginal ...
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0answers
32 views

Proving an identity involving expectation

Let $S_t$ be a stochastic process satisfying $S_t = S_0 \exp \{ (r- \frac{ \sigma^2}{2})t + \sigma W_t \}$, where $S_0 >0$ and $W_t$ denotes a Brownian motion. Also, let $Z$ be a $N(0,1)$ random ...