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 ...

learn more… | top users | synonyms (1)

1
vote
0answers
19 views

Probability of occurrence of two events simultaneously

I have a question with probability of occurrence of two events simultaneously. I have a probability histogram for some events occurring individually. Is it possible to predict the simultaneous ...
0
votes
2answers
15 views

Calculating Probabilities for a cumulative distribution function within a given inequality

Given that K = 1/36, I require some help understanding (b) • Pr(1/2 ≤ X ≤ 1) Is re-written as such: Pr(X ≤ 1) - Pr(X < 1/2) I do not understand why! Is it because Pr(X ≤ 1) is solved as ...
1
vote
1answer
12 views

Calculating Probabilities using a cumulative distribution function

For (b) Pr(X greater than or equal to 2) = ? The textbook says as such but I am confused: Pr(X greater than or equal to 2) = 1 - pr(X less than 2) I do not understand why they re-write the ...
-1
votes
0answers
14 views

Determining the Cramer-Rao lower bound [on hold]

Let $X = (X_1,\dots,X_n)$ be a vector of iid variables from the smooth density $f(x,\theta_0), \theta_0 \in \Theta \subset \mathbb{R}$. Let $L(\theta)$ be the likelihood and $I(\theta)$ the ...
-1
votes
0answers
9 views

Product of Gaussian random variable with hermition of another independent gaussian random variable. [on hold]

If X∼ CN(0,1) and Y ∼ CN(0,1). X and Y are vectors independent of one another. How to find the E[(X†)Y]. What will be the probability density funtion of Z, If Z = (X†).Y ?
0
votes
0answers
33 views

Values of $\mu$ for which $S_n=e^{\sum_{i=1}^n X_i}$, is a martingale ($X_i ~ \mathcal{N}(\mu,1)$)

Let $(X_n)_{n\geq 1}$ a sequence of $\mathcal{N}(\mu,1)$ $\mathcal{F}$-adapted and $S_n=e^{\sum_{i=1}^n X_i}$. I have to write the conditions for which S_n is a martingale, then I have to show that ...
1
vote
1answer
22 views

Poisson process has independent and stationary increments

Being $N_t$ a Poisson process, defined as $$N_t:=\sum_{n\geq 1} n \mathbb{1}_{[T_n,T_{n+1}[}(t)$$ where $T_n$ are sums of independent exponential random variables, how can I prove it has stationary ...
2
votes
1answer
16 views

Why $E[X|\mathcal{G}]=X$ if $X$ is $\mathcal{G}$-measurable?

If $X$ is a $\mathcal{G}$-measurable random variable, why $E[X|\mathcal{G}] = X$? I know the intuition (basicly we're conditioning on the same informations on which $X$ is defined, $\sigma(X)$, we ...
0
votes
2answers
23 views

I'm not sure if I'm supposed to use a Poisson distribution or Conditional Probability (or both) to answer this question

I have a question that I'm trying to solve. I have the answer but I don't know how they arrived at the answer so I can't compare my work and see where I went wrong. The number of injury claims per ...
1
vote
2answers
25 views

Riemann Integral in Probability With Martingales by Williams

"Note on the Riemann integral" in chapter 5 of Probability with Martingales by Williams reads: If, for example, f is a non-negative Riemann integrable function on $([0,1],\mathcal{B}[0,1],Leb)$ ...
0
votes
0answers
22 views

Pi-System Independence in Williams

In David Williams' Probability with Martingales, $\exists$ this exercise. Given a probability space. Let $\mathfrak{I}_1, \mathfrak{I}_2, \mathfrak{I}_3$ be $\pi$-systems on $\Omega$ such that for k ...
0
votes
2answers
10 views

Expected Profit for Binomial Variable

Part (a) I am familiar with: (a) P(batch is rejected) = P(X greater than or equal to 3) and n = 15 and p(defective) = 0.1 This gives me the correct answer of 0.1841 I am stuck at part 2! I have ...
0
votes
0answers
18 views

Will the branching process go extinct with probability 1?

I am trying check whether the branching process goes extinct with probability one. Single Type Branching Process with Pk = (1/2n)(n/k), for k = 0,.....,n with n > 2. Assuming, i can be able to ...
0
votes
2answers
28 views

The relation of $P(X=x+1)$ and $P(X=x)$ in binomial distribution

If I substitute the values to the binomial probability theory, it appears as such $${n \choose x+1} p^{x+1} (1-p)^{n-x-1}$$ I don't know how to move on... What am I doing wrong, or are you ...
0
votes
1answer
35 views

How to apply the strong Markov property in this case?

I'm trying to understand the following proof: Theorem: Let $(X_n)$ be an irreducible $(\alpha, \mathbf p)$-Markov chain with a finite state space $S$. Then $(X_n)$ is positive recurrent. ...
0
votes
0answers
19 views

Let $X_t$ be a Brownian motion find $X_2>2, x_1>x_2,$ and $x_t<4$ for all $2\leq t\leq 3 $ [on hold]

Let $X_t$ be a Brownian motion find $X_2>2, x_1>x_2,$ and $x_t<4$ for all $2\leq t\leq 3 $ Can you help me with tips and bibliography... I don't understand very good the topic, and I can't ...
0
votes
1answer
26 views

Transformation of random variable

I want to prove the following: $$\text{Let F be a distribution function of any random variable $\\$ and G(x) the quantile function (or inverse) of } \frac 1 {1-F(x)}$$ $$\text{Then, for a standard ...
0
votes
1answer
20 views

What are possible theories/models to describe social influence based on frequency?

I am working on social influence based on frequency, meaning if a person repeatedly appears at a location many times then he/she is more influential at that location. In other words, he/she can ...
0
votes
0answers
13 views

How to get closed form solutions to stopped martingale problems?

Way back when, I took a course in stochastic processes in college. I remember being frustrated by the plethora of abstract proofs without much in the way of how to use them to get actual results. It ...
2
votes
2answers
34 views

How to maximize pay with repeated toss of coin

repeated toss a coin and you can stop anytime and payoff is just #times you got head divided by total number of throws, how do you maximize your pay. Does anyone have a clever strategy for this? ...
0
votes
3answers
59 views

Transition Matrix of M/M/1 Queue

We know that for an M/M/1 queue the state space is $S=\{0,1,2,... \}$. Further the probability to go from state $i$ to $i+1$ is $\lambda$ for all $i$ in $S$. Moreover, to go from $i$ to $i-1$ is the ...
1
vote
1answer
52 views

Confusion with real numbers and random variables; Integration and Independence in Williams

In David Williams' Probability with Martingales, $\exists$ this exercise. Let $X_n$ be iid RVs with the same continuous dist function. Let $E_1 = \Omega$ and for $n \geq 2, E_n = (X_n > X_m ...
2
votes
1answer
62 views

“Fair” game in Williams

In David Williams' Probability with Martingales, $\exists$ this exercise. What's fair about a fair game? Let $X_i$, i = 1, 2, ... be indp RVs s.t. $X_i = i^2 - 1$ with prob $1/i^2$ and $-1$ with ...
0
votes
1answer
23 views

Probability of highest common factor in Williams

In David Williams' Probability with Martingales, $\exists$ this exercise. Let s > 1 and let $\zeta(s) = \sum_{n=1}^{\infty} {n^{-s}}$. Let X and Y be independent $\mathbb{N}$-valued random variables ...
2
votes
2answers
36 views

Let $X$ be the value of the first die and $Y$ the sum of the two dice. Find $E(X / Y)$ [on hold]

Consider an experiment of rolling two dice. Let $X$ be the value of the first die and $Y$ the sum of the two dice. Find $E(X / Y)$, ie, obtain the value of $E(x/y) (y)$ for all $y$ Good evening, I ...
0
votes
0answers
36 views

Mean and variance of a stochastic process

Let \begin{equation} \begin{array}{l} y_1(t)=e^{-\kappa_1 t}y_1(0)+\displaystyle\int_0^t\kappa_1 e^{\kappa_1(s-t)}\theta_1ds +\sigma_1\displaystyle\int_0^te^{\kappa_1(s-t)}\sqrt{y_1(s)}dZ_1(s),\\ ...
-2
votes
0answers
11 views

probability distributions [on hold]

![ Question 1. 1. Using the probability distribution table, what is the value of P(X = 2 or X = 0)? X 0 1 2 3 4 5 P 0.3 0.05 0.1 0.15 0.15 0.25 P(X = 2 or X = 0) = _____ (Points : 1) ...
0
votes
2answers
70 views

If bridges between islands collapse independently with probability $p$, what is the probability that islands remain connected?

This is a follow-up to Probability Question: Bridge problem. There are $n$ islands in the ocean. Each island is linked by a single bridge to each other island. The probability of each bridge ...
1
vote
1answer
43 views

Probability Question: Bridge problem

There are $n$ islands in the ocean. Each island is linked by a single bridge between each and every unique pair of islands to ensure no island is isolated from the others. The probability of each ...
1
vote
1answer
25 views

How to determine long-run probability using conditional probability?

How to determine long-run probability on a calculator and manually? For example: Ben plays a tennis match every day. If he wins on one particular day, the probability that he wins the next day is ...
1
vote
0answers
30 views

Independence Exercise in Rosenthal

In Rosenthal's, "A First Look At Rigorous Probability Theory", $\exists$ this exercise: Exercise 3.6.19. Let $A_1,\ A_2,\ldots$ be independent events. Let $Y$ be a  random variable which is ...
1
vote
1answer
33 views

Proving that $ \frac{1}{n}\int_{-\infty}^{\sqrt{n}w}k(v/\sqrt{n})\phi(v)dv$ is $O(n^{-1})$

Suppose that $h:\mathbb{R}\to\mathbb{R}$ is infinitely differentiable. Define \begin{equation} k(w)=\left\{ \begin{array}{ll} \frac{d}{dw}\left(\frac{h(w)-h(0)}{w}\right)&w\neq 0,\\ ...
2
votes
1answer
28 views

Independence of Events in Rosenthal

$\exists$ this exercise in Rosenthal's A First Look at Rigorous Probability Theory: For letters d and e, how do you show that the ff events are independent? My attempt: It suffices to show that ...
1
vote
1answer
24 views

Question on a variation of Borel Cantelli Lemma

In this question, what is the purpose of the summation? If the limit of the sequence is zero, the corresponding series is convergent. Does the desired conclusion not then follow from BC1?
0
votes
1answer
45 views

Using Borel-Cantelli Lemma

Let $X_1, X_2,\ldots$ be iid Geometric(p) where $p \in (0,1)$. Thus if $q=1-p$, then $P(X_n > k) = q^k$ for $k\geq 0$. Prove that for any fixed $\epsilon \in (0,1)$, $P(X_n > ...
0
votes
0answers
23 views

Entropy derivation from Multiplicity

Multiplicity(W)= N!/(n1!*n2!....ni!) Entropy = 1/N * ln W = 1/N*ln N! - 1/N*sigma_for_all_i(ln ni!) As N->infinity, By Stirlings approximation ...
-1
votes
0answers
14 views

Percolation - measure - Cylinders [on hold]

$\Omega = \{0,1\}^{\mathbb{Z}^{2}}$ is the set of open and closed $\mathbb{Z}^{2}$ $\Omega_{\wedge} = \{0,1\}^{\wedge}, \forall \wedge \subset \mathbb{Z^{2}}$ finite. $C:=$Cylinders (That is, local ...
1
vote
1answer
19 views

Centered independent increments process is a martingale

Let $(X_n)$ be an centered integrable process with independent increments (which as far as I understand means that $(X_{n+1}-X_n)_{n\in \mathbb N}$ is independent). While showing that $(X_n)$ is a ...
-1
votes
2answers
36 views

Random Variable Problem with unrestricted Parameters Worded Problem

I have no idea how to go about solving (a) -> (c) For (a) Is $k=0.2$, because $\frac{k}{1-0.8}=1$ Hence, $P(Z=z) = 0.2(0.8)^x$ But How do we determine the mean or variance with unrestricted z ...
1
vote
1answer
23 views

if $X,Y$ i.i.d $\mathcal{N}(0,1)$, then $X+Y$ is independent of $X-Y$

I found on another thread* that if $(X+Y)$ is independent $(X-Y)$, and if $X,Y$ are i.i.d., then $X,Y$ are $\mathcal{N}(0,1)$ distributed. Is also the opposite true? Being $X,Y$ i.i.d ...
0
votes
1answer
16 views

Expectation of Random Variable - Probability Worded Problem

The part I am confused with is (c) I found part (a) which is: p(0) = 7/24, p(1) = 21/24, p(2) = 7/40 and p(3) = 1/120 How do we find the values for a and b, for part (c) ?
0
votes
1answer
29 views

Expanding the expected value

How to expand: $E(Y+1)^2$ my working out: $E(Y^2)+E(1^2) = E(Y^2)+1$ (I'm not sure why this is though..) Can someone link to or list the rules for expanding the expected value ......
0
votes
2answers
39 views

Evaluating the integral to find the expected value of the exponential random variable

I want to find the expected value of the exponential random variable. I have $E(X)=\int_{0}^{\infty }xae^{-ax}dx$. I use Integration By Parts (IBP). Let $v=-e^{-ax}\Rightarrow dv=ae^{-ax}dx$, and ...
1
vote
2answers
19 views

Finding values of a constant in a probability distribution

A probability distribution for the random variable $X$ is defined by: $$\mathbb{P}[X=x] = K\cdot(0.9)^x,\quad x = 0,1,2,\ldots$$ It is asked to find $\mathbb{P}[X\geq 2]$. When there is a domain for ...
-5
votes
0answers
23 views

Adding liquid to container that is never emptied: how much of the liquid is old?

Suppose I had a 10-liter container that is full of liquid. Every day I pour out half the liquid and fill it back up. At the end of n days, what is the probability that the container contains at least ...
0
votes
2answers
14 views

coin toss game with know intitial resources on each side

This question relates to the end game in poker when there only two players left. Suppose player A has 5 times as many chips as player B. If player A bets the max amount on each hand, the max being ...
1
vote
1answer
30 views

Do we need $\tau \leq \nu$ to show $E(X_\tau)=E(X_\nu)$?

My lecture notes claim that if $(X_n)$ is a martingale and $\tau$ is a stopping time bounded by $N$ then $$E(X_\tau)=E(X_{\tau \wedge N})=E(X_{\tau \wedge 0})=E(X_0)$$ and then remarks that if $\tau$ ...
1
vote
1answer
35 views

Expectation of uniform distribution with unknown parameter, given maximal (minimal) observation.

Let $x_i \text{ be} ~ i.i.d. ~ \sim Uni[0,\theta]$ $(\theta \text{ unknown})$. Denote $M_n = \max x_i$. So, through circumferential means, I can show that $E(x_1|M_n) = \frac{n+1}{2n} M_n$. The ...
0
votes
0answers
19 views

Ross probability models questions [closed]

I am studying for a course and have no professors to talk to live, so I hope some members here can be kind enough to help me. Rather than writing everything out, and splitting it up into different ...
3
votes
2answers
80 views

How to give rigorous proofs of these two limit statements?

Let $X$ be a random variable with cumulative distribution function $F(x)$. Then how to rigorously prove the following two limit statements? $\lim_{x \to - \infty} F(x) = 0$. $\lim_{x \to + \infty} ...