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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4 views

HOW TO PLOT DAG (DIRECTED ACYCLIC GRAPH) in BNT toolbox for matlab.

I have used markov chain monte carlo (MCMC) in BNT toolbox for matlab, from which i have got one output "sampled_graphs " which is cell array. Now how to plot DAG (Directed acyclic graph ) from ...
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0answers
7 views

Equality of two conditional expectations

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

How do you calculate P(A/B), when event B occurred after event A?

There's really only one question I can't begin to handle when it comes to probability, literally. It's not the only type of question I struggle with, though it's the type of question where I can't ...
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0answers
9 views

Which properties are stable under convergence in distribution?

I'm currently looking at convergence in distribution (i.e. weak convergence) of random variables. A question is bothering me since quite a while and I hope I could express it properly: Given weak ...
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0answers
23 views

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 ...
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0answers
11 views

Two random vectors converge does this mean that the entries converge?

Suppose you are given the following two equalities $\mathbf{\delta }^{n}=\left( \begin{array}{ccccccc} \delta _{n,1} & \delta _{n,2} & \cdots & \delta _{n,n} & 1 & 1 & ...
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0answers
16 views

Multiplication rule and regular conditional probability

I've been studying the conditions of existence of the regular conditional probability and have a question about it. Let's $(\Omega, \mathcal{B}, P)$ be a product probability space, and let's say the ...
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0answers
18 views

$E(X_T; T < \infty) \leq E(X_0)$ with $T$ stopping time

I'm doing this exercise: $(X_n)$ is a non-negative supermartingale and $T$ a stopping time, then $$E(X_T; T < \infty) \leq E(X_0)$$ My attempt: $(X_n)$ is a negative supermartingale, and so ...
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4answers
39 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($X,Y$) = ...
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0answers
11 views

marginal likelihood survivor function [on hold]

Suppose ($x_1$,…,$x_m$) and ($y_1$,…,$y_n$) are drawn from survivor function $S(x;e^{θ/2}) $ and $S(y;e^{-θ/2}) $ respecitively , where $S$ is a defined above. (i) The form of the marginal ...
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0answers
14 views

How to compute the covariance matrix of a random variable uniformly distributed in an ellipsoid

Suppose that x is a random variable uniformly distributed in an ellipsoid \begin{equation} x^{T}Mx\leq\delta, \end{equation} where $x\in \mathbb{R}^{n}$. Clearly, the mean of $x$ is zero. The ...
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0answers
14 views

Showing that $\mathcal{A}$ being countable $\Rightarrow f(\mathcal{A})$ is countable - (Algebras/Sigma Algebras)

For the first question my idea was to show that $\sigma(f(\mathcal{A})) \subseteq \sigma(\mathcal{A})$ and $\sigma(\mathcal{A}) \subseteq \sigma(f(\mathcal{A}))$. As for the second question I am at ...
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0answers
7 views

Conditional expectation with disjoint $\sigma$-algebras

Let $(B^1,B^2)$ be independent Brownian motions with corresponding filtration $\mathcal{F}_t$. Let $\mathcal{F}^2_t$ be the filtration generated by $B^2$. How does one prove that for any $s<t$ and ...
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1answer
16 views

Find and sample minimum of two exponential distribtions

I have two (or more) independent exponential variables $ X_1 \sim \exp(\lambda_1) $ and $ X_2 \sim \exp(\lambda_2) $. I want to get both the value of $ \min(X_1, X_2) $ and $ \arg\min(X_1, X_2) $. Can ...
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1answer
36 views

Showing that if $A_{1},A_{2},…$ are all algebras then the union of all of them is an algebra [duplicate]

I am not sure how to show this. It seems obvious but maybe its not. The help would be greatly appreciated!
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1answer
13 views

Is this measure finite, $\sigma$-finite, or a probability measure?

I was a little unsure on this problem. I do have some ideas though. The way I thought of translation invariant is that you can take an interval and shift it, and in the process is will still be the ...
2
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1answer
46 views

Do not exist IID random variables $X, Y$ such that $X-Y \sim U[-1,1]$

This is an exercise from Williams, Proability with martingales. Prove that if $Z$ has the $U[-1,1]$ distribution, then $$\phi_Z(t) = \frac{\sin t}{t}$$ Then prove that do not exist IID random ...
3
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1answer
30 views

Playing the St. Petersburg Lottery until I lose everything

This question continues the following question: Calculating the probability of winning at least $128$ dollars in a lottery St. Petersburg Paradox Here is a lottery: A fair coin is flipped repeatedly ...
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0answers
34 views

Equivalence of sigma algebras on the set of probability measures.

I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and ...
2
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1answer
41 views

$\frac{S_n}{n} \to -1 \ \ a.e.$, exercise from probability book

I'm stuck with this exercise from Williams, probability with martingales. Let $X_1, X_2, \ldots $be independent random variables with $$P(X_n = n^2-1 )= \frac{1}{n^2}$$ $$P(X_n = -1 )= ...
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2answers
15 views

Value of lambda in poisson distribution

I am currently studying statistical estimators and I came across a question that asks to give an estimate of the parameter λ of a Poisson distribution (using the method of moments), given that the ...
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0answers
2 views

Haris recurrent markov chain

What is the definition of haris recurrent markov chain ?. And, when the $\Phi$ irreducibility implies haris recurrent ? Thanks a lot
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1answer
25 views

probability of X+Y which are two independent random variable & uniform distribution[0,1] [duplicate]

Two random variables X, Y are independent and both uniform-distributed in[0, 1]. How to calculate the probability density function Z=X+Y ? I tried below, $$f_X(x) = \begin{cases} \frac1{1-0} \\ ...
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1answer
46 views

Exercise from Williams book Probability with martingales

I'm doing this exercise from Williams book Probability with martingales Let $(X_n)$ be a sequence of IID random variables with $E(|X_n|) = \infty $ for all $n$. Then prove that $$1)\ \sum_n ...
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0answers
17 views

Does a state which is passed at least 3 times had to be passed 5 times in Markov chain

Prove of disprove: Let $\{X_n\}_n$ be homogenous Markov chain. if we start from state $i$, there is a positive probability that we pass at least 3 times at state $j$. Does it follows that exists ...
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1answer
26 views

Showing that a collection of intervals (see problem) generates the Borel sigma algebra on $(0,1]$

I would be very appreciative if someone could show me how to do this problem so that I can try to get a better understanding of what a Borel sigma algebra is. Examples are how I learn best so seeing ...
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1answer
31 views

Probability of two elements falling within a certain interval

a, b are selected at random and 0 <= a <= 3 and -2 <= b <=0. What is the probability the distance between a and b is greater than 3. I did this and said that $a$ must be in $(2, 3)$ ...
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1answer
19 views

Applying Ito's formula to a complicated expression

I am faced with some (predictable) process $(r_t)$ and let $0 \leq t \leq T$. I am baffled with the issue of applying Ito's formula to the process $$ \bigg\{ \int_{t}^{T} G(s-t, r_t) \,ds \bigg\}_{t ...
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1answer
21 views

How does a change of measure affect covariance?

Suppose I have the three random variables $X,Y,M$ where $E[M] = 1$ under the measure $P$. Now, suppose I define a new measure $\widetilde P$ so that $\widetilde E[X] = E[M X]$ and $\widetilde E[Y] = ...
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0answers
11 views

local martingales and dividend processes

Consider a $d+1$ asset, continuous-time model where asset $0$ is a riskless numeraire. Assume that the asset prices are modelled by a $(d+1)-$dimensional Ito process (B,S). Further, let $D$ be the ...
0
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1answer
19 views

How do I prove that a given probability distribution is Gaussian

I am trying to plot the distribution of a random variable $x$. I got this distribution by marginalising a wishart distribution. When I plot the distribution curve of $x$, it looks like bell shaped ...
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0answers
13 views

Find pdf for solution of Stochastic DE

I have some troubles learning with Stochastic DE. There is a problem. Find the probability density function f(x,t), of $X_t$ where {$X_t$} is a solution of SDE: $dX_t = mdt + \sigma dW_t, X_0 = 0$ I ...
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2answers
37 views

Calculation of a characteristic function

Suppose $X_1, X_2, \ldots X_n \ldots$ are independent random variables with $$P(X_n = 1) = \frac{1}{2}$$$$P(X_n = -1) = \frac{1}{2}$$ Then $$\sqrt{\frac{3}{n^3}}\sum_{k=1}^n kX_k$$ converges to ...
2
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0answers
18 views

Convergence of Uniformly Distributed Random Variables (n-dimensional)

Suppose that ${U_n} = ({U_{n1}},{U_{n2}},...,{U_{nn}})$ is uniformly distributed over the n-dimensional cube ${C_n}={[0,2]^n}$ for each $n=1,2,...$ That is, that the distribution of ${U_n}$ is ...
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0answers
6 views

Conditional Probability in Multivariate Normal

Given a tri-variate Normal, the conditional probability of an element given others truncated information is Now if I know that the mean vector u is (-0.91,-1.31,-1.39) and R is ...
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0answers
24 views

Probability density function definition

The definition above is given in my lecture notes. However there is no further reference/explanation given for what $o(h)$ represents. Can anyone explain this in this case?
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1answer
20 views

Relation between convergence in distribution and in probability

Does convergence in distribution imply convergence in probability ? I suppose no, but I need a counterexample. Does anyone know any counterexamples ?
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0answers
18 views

Exercise on Markov chains

I'm preparing my Probability exam and I'm having trouble with exercise 2 here. The question is to consider the random walk on $E$ with transition matrix $p$ and find the communication classes (or ...
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0answers
16 views

Does martingale model work for betting football matches?

Imagine I have 1 million USD and will be betting 1.000 USD on the win of FC Barcelona each time they play a match in La Liga (Spanish Tier 1 football league). If FC Barcelona loses or ties their last ...
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1answer
22 views

Probability of drawing balls from an urn with variable composition

A coin is tossed $k$ times, with probability $p$ of heads. In an urn, as many white balls are introduced as the amount of heads obtained, and as many black balls are introduced as the amount of tails ...
2
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2answers
62 views

Is it true that E [ X | E [ X | Y] ] = Ex [ X | Y] ? Does this law have a name?

Let $X$ and $Y$ be two random variables (say real numbers, or vectors in some vector space). It seems to me that the following is true: E [ X | E [ X | Y ] ] = E [ X | Y] Note that E [ X | Y ] is a ...
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0answers
19 views

Proving criterion for a transient state in Markov Chain

Let $\{X_n\}_n$ be a homogenous Markov chain. Prove that if exist a connected subset of states (means set of states which exist positive probability to move between them), $S$ which is not closed, ...
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3answers
85 views

If $G(x)=P[X\geq x]$ then $X\geq c$ is equivalent to $G(X)\leq G(c)$ $P$-almost surely

Suppose $[\Omega,\mathcal{F},P]$ denotes a probability triplet and $X:\Omega\to\mathbb{R}$ is a real-valued random variable. Define $$ G(x)=P[X\geq x]. $$ Claim: for any constant $c$, the event ...
2
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0answers
30 views

$\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated iff there is a random variable $X$ such that $\mathcal{G} = \sigma(X)$.

Where can I find a reference to the proof of the fact that a $\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated if and only if there is a random variable $X$ such that ...
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0answers
18 views

Clarification on product space, conditional probability measures

Suppose we have two random variables $X_i$ on the probability space $(\Omega_i, \sigma(\Omega_i),P_i)\; i = 1,2$. Now, $P_i$ is just a technical tool and we consider directly the distribution of $X_i$ ...
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1answer
16 views

Showing that a sequence converges, in distribution, to a normal r.v.

Let $X_1, X_2, \dots$ be independent, with $$P(X_n=1) = P(X_n = -1) = \frac{1}{2} \left( 1 - \frac{1}{n^2} \right),$$ and $$P(X_n = n) = P(X_n = -n) = \frac{1}{2n^2}.$$ Let $S_n = X_1 + \cdots + ...
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0answers
17 views

Suppose that there are two cells in a parallel system. In order for the system to work, at least one of the two parallel subsystems must work.

Consider a particular lifetime value $t_0$, and suppose we want to determine the probability that the system lifetime exceeds $t_0$. Let $A_i$ denote the event that the lifetime of cell $i$ exceeds. ...
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0answers
23 views

Finding a random variable.

Let $X,Y$ be two non negative random variables such that density of $Y$ is the same as the survival function of $X$. Is there any way we can find $Y$? Thank you for your time and help.
3
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1answer
43 views

$X = E(Y | \sigma(X)) $ and $Y = E(X | \sigma(Y))$

Suppose $X, Y$ are random variables in $L^2$ such that $$X = E(Y | \sigma(X)) $$ $$Y = E(X | \sigma(Y))$$ Then I want to show that $X=Y$ almost everywhere. What I've done: By conditional Jensen ...
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1answer
26 views

Probability Theory: Expectation Problem [on hold]

Need help. (1) Can we mathematically proof that the difference between $\mathbb E[\min(X,Y)]$ and $\min(\mathbb E[X],\mathbb E[Y])$ is minimal / negligible? (2) Can we say that $\min(\mathbb E[X], ...