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 [tag:probability-...

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1answer
47 views

Upper bounds on $E[{\rm var}^2(X|Y)]$

I am looking for an upper bound on the quantity \begin{align*} E[{ \rm var}^2(X|Y)] \end{align*} where ${\rm var}(X|Y)=E[(X-E[X|Y])^2|Y]$. Getting a lower bound is rather easy using Jensen's ...
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0answers
42 views

Define r.v. by partition of $\Omega$ and other integrable r.v., and show its integrable

Let $(\Omega, \mathcal F, P)$ be a probability space, and let $\{ A_i \}_{i\in I}$ be an at most countable partition of $\Omega$ (i.e. $I = \{1,\ldots, n\}$ or $I = \mathbb N$). Also let $X : \Omega \...
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2answers
46 views

Am I using these Probability Theory Terms Correctly?

Let $\Theta$ be a random variable with sample space $\{\theta_1, \ldots, \theta_n\}$. Questions: Is it terminologically correct to refer to $\Theta$ as a "parameter"? What about the members $\...
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2answers
47 views

Computing a conditional expectation for uniform RVs

Suppose $X_1, ..., X_n \sim U[0, 1]$ are iid uniform RVs. How would I go about computing $E[X_n | X_{(n)}]$ where $X_{(n)}$ is the nth order statistic, i.e. $\max\{X_1, ..., X_n\}$ ? I'm stuck ...
0
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1answer
64 views

Variance of an integral of Brownian Motion

Let $W(u)$ $(u \geq 0)$ be a Brownian motion on a probability space $(\Omega, \mathscr{F}, \mathbb{P})$. Let $I(T) = \int_0^T W(u) du$. One can use integration by parts to show that $I(T) = \int_{0}^...
0
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1answer
30 views

Expectation of normal and log normal distribution

Let $X \sim N(\mu_x, \sigma_x^2)$ and $Y\sim N(\mu_y, \sigma_y^2)$, with correlation $\rho$. How do I find $$E[Xe^Y]$$? I tried a bunch of things without result. I'm also interested in "general" ...
0
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1answer
38 views

an application of bounded convergence theorem

I have read the following statement: $P\{Y_{t}\neq Y_{t+u} \mid Y_0=i\}=\sum_{j\in E}P_t (i,j)[1-P_u(j,j)]$, where $E$ is a countable set. As $u\downarrow 0$ we have $1-P_u(j,j)\to 0$. This implies ...
0
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1answer
28 views

If $\lim_{n \rightarrow \infty} E(X_t| \mathcal{F}_{t-n}) = 0 $ then $E(X_t) = 0$?

Suppose I have a sequence of random variables $X_t$ adapted to a filtration $\mathcal{F}$ when is it true that if $\lim_{n \rightarrow \infty} E(X_t| \mathcal{F}_{t-n}) = 0 $ then $E(X_t) = 0$ ? ...
2
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1answer
194 views

Infinite Sample Space

I came across this line in a textbook I'm reading, When $\Omega$ is infinite, its power set is too large a collection for probabilities to be assigned reasonably to all its members. I'm not ...
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0answers
51 views

Conditional expectations related to Black-Scholes formula

While computing the price at time $t$ of a European call option with strike $K>0$ and maturity $T>0$ for $t$ in $[0,T]$, I encountered the following conditional expectation which I cannot ...
0
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2answers
122 views

Bayes' Theorem, Rigorously

For a given probability space $(\Omega, \mathcal{F}, P)$ Bayes' rule is given by $$ P(A|B) = \frac{P(B|A)P(A)}{P(B)} \quad \text{for all } A,B \in \mathcal{F}. $$ However, in many examples (in fact, ...
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2answers
52 views

Explaining the form of the Gaussian measure

The Gaussian density $\mu(dx)=e^{-x^2/2}\ dx$ is fundamental in probability theory. Does anyone have a (non-computational) heuristic why this function should be special? (By non-computational, I mean ...
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4answers
2k views

Facebook Question (Data Science)

Out of curiosity, here's a question from Glassdoor (Facebook Data Science Interview) You're about to get on a plane to Seattle. You want to know if you should bring an umbrella. You call 3 ...
1
vote
1answer
62 views

Probability inequality problem about discrete random variable

Here is the problem. Let X be a discrete random with $\ E(X) = 0$ and $\ \text{Var}(X) = \sigma^2 < \infty $. Show that $$ P(X \geq a) \leq \frac{s^2}{s^2 + a^2} $$ for all $\ a > ...
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0answers
38 views

covariance of a function of Wiener processes

Consider two independent Wiener processes, $W_1$ and $W_2$. The covariance of certain functions of Wiener processes is simple, for example $$\text{cov}\Big(W_1(r)-\int_0^1W_1(r)dr,W_1(s)-\int_0^1W_1(r)...
0
votes
1answer
54 views

Expected number of dice throws to fill out a table.

Say I have a table of numbers ${1,2,3,4,5,6}$. Every time I throw a fair die, if the position in the table is unchecked, it becomes checked and if it is already checked it becomes unchecked. For how ...
0
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0answers
23 views

Expected value of signals

I am trying to learn DPS. A couple of explanations are based on statistics. I would like to understand what is and how coherence works, but I am stuck on its definition. I found the following ...
0
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2answers
73 views

Random equation-does it make sense?

What is the probability that the equation $$x^2+2bx+c=0$$ has real roots? Answer is exactly $1$. (or $100$%) For example: if $b=1$ and $c=2$ roots are complex. Does it make sense? If $P(A)=0$, then ...
1
vote
1answer
53 views

Relation between sum of indicators and average of probability for independent sequence

Let $A_1, A_2, \ldots$ be independent events, set $N_n := \sum_{i=1}^n I_{A_i}$ and $\overline p_n := n^{-1} \sum_{i=1}^n P(A_i)$, then $$ P\left( \lim_n \left( n^{-1} N_n - \overline p_n \right) = ...
1
vote
1answer
33 views

Expected time to fill a table [duplicate]

Say I have a table of numbers 1-6. I throw a 6 sided die a number of times. Each time I get a number I have not already had, I mark it in the table. What is the expected number of times to throw the ...
2
votes
0answers
40 views

Describing convergence with probability $1$ in “finite” terms, proof correct

I tried to solve the following exercise: Show that $Z_n \to Z$ with probability $1$ if and only if for every $\varepsilon$ there exists some $n$ such that $P(|Z_k - Z| < \varepsilon, n \le k \...
2
votes
2answers
41 views

Var$(X) = \mathbb{E}((X - \mathbb{E}(X))^2) = \mathbb{E}(X^2) - \mathbb{E}(X)^2$

I have a question about something my teacher told us: let $\mathbb{E}$(X) donate the expected value of a certain random variable $X$. Then Var$(X) = \mathbb{E}[(X - \mathbb{E}(X))^2] = \mathbb{E}(X^...
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votes
4answers
58 views

Roll a die until, for the first time, the same side shows up two times in a row. Let $X$ be the number of rolls needed. compute $\mathbb{E}(X)$.

I'm having trouble with solving this problem: Roll a die until, for the first time, the same side shows up two times in a row. Let $X$ be the number of rolls needed. compute $\mathbb{E}(X)$. I know ...
0
votes
2answers
51 views

Take $k$ shoes ($k \leqslant n$) from a wardrobe. What is the expected value of the number of pairs ($X$) you take?

I'm having trouble with this question: Let there be $2n$ shoes ($n$ pairs) in a wardrobe, arbitrary ordened. Take $k$ shoes ($k \leqslant n$) from that wardrobe. What is the expected value of the ...
0
votes
1answer
83 views

The quadratic variation of $B \cdot B$, where $B$ is a Brownian motion

Let $B$ be a standard, one-dimensional Brownian motion. Can I show that $[B \cdot B] = B^2 \cdot [B]$, using the "fundamental identity of stochastic integration", namely that $[H \cdot X, Y] = H \cdot ...
1
vote
1answer
44 views

Determining the population size of a branching process

Suppose that I have the following branching process. Each parent can have up to two children, the number of which is determined by two independent fair coin flips. I know that this branching process ...
0
votes
1answer
69 views

In throwing 6n dice, what is the probability of getting each face n times? Use Stirling's formula to estimate this probability for a large n.

This question is taken from Probability Theory: A Concise Course by Y. A. Rozanov. My attempt at a solution is as a following: I think of $6n$ dice rolls as $n$ groups of 6 rolls. The probability ...
3
votes
1answer
214 views

Uniform distribution on a sphere

Consider the unit ball $S_n$ (centered at the origin) in $\mathbb{R}^n$ for $n \ge 1$ and a stochastic process $(X_t)_{t\ge 0}$ taking values in $\mathbb{R}^n$. Let $T = \inf\{t > 0 \colon X_t \...
0
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0answers
91 views

Impossible Events, Probability Zero Events, Change of Sample Space, Invariant, Canonical Sample Space?

I am reading this post about probability theory and its foundations by T. Tao, and also this and this post, and they say in essence that the underlying sample space is not that much important. Often ...
0
votes
1answer
51 views

If $P(a_n\leq X_n\leq b_n)\to1$ and $a_n\to0,b_n\to0$ then $X_n\to\delta_0$ in distribution

If $P(a_n\leq X_n\leq b_n)\to1$ and $a_n\to0,b_n\to0$ then prove that $X_n\to\delta_0$ in distribution. Here $\delta_0$ is the degenerate random variable putting all its mass at the point $0$. I do ...
0
votes
3answers
265 views

Cumulative distribution function of Cauchy distribution

Let X be a Cauchy distribution with X~Cauchy(1) (so a=1). Prove that Y=1/X has the same cumulative distrubtion as X. Now I've tried taking F_X(x) for a=1 combined with the identity arctan(x)+...
1
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2answers
268 views

time it takes to service a car with exponential random variable with rate 1

Need help with this question here. Ill post exactly what it says then show my ideas so far. "The time it takes to service a car is an exponential random variable with rate 1. (a) If A brings his car ...
6
votes
1answer
263 views

Berry-Esseen bound for binomial distribution

From the Berry-Essen theorem I can deduce $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right| \le \frac{C(p^2+q^2)}{\sqrt{npq}}$$ with $C \le 0.4748$. My ...
0
votes
0answers
25 views

The length of the set which will be covered by Brownian motion in a time $t$

I have the following question in mind which I wanted to answer: what is the measure of the set which will be covered by a standard Brownian motion $B(t)$ in a time $t$? Call this random variable $M(t)$...
0
votes
1answer
49 views

Law of large numbers for nonnegative random variables [closed]

I'm struggling with specific variation of Strong Law of Large Numbers. Suppose $X_1,X_2,\ldots$ are independent, identically distributed, nonnegative random variables and $\mathbb{E} X_1 = \infty $. ...
1
vote
1answer
76 views

Show that $(W_t)_{t \geq 0}$ and $(W_t^2-t)_{t \geq 0}$ are not uniformly integrable

I'm considering the following martingale $M_t:=W_t^2-t,\ t\geq 1$, where the $W_t$ is a Brownian motion. I want to prove that this martingale and the Brownian motion are not uniformly integrable. I ...
3
votes
1answer
180 views

Why is the supremum a random variable in the Glivenko–Cantelli theorem

According to wikipedia: Assume that $X_1,X_2,\dots$ are independent and identically-distributed random variables in $\mathbb{R}$ with common cumulative distribution function $F(x)$. The empirical ...
0
votes
1answer
67 views

mutual information adds along path

Is it true that $I(X;Y)+I(Y;Z)=I(X;Z)$ for $X \to Y \to Z$? $I(X;Z) = H(X)+H(Z)-H(X,Z)$ and $I(X;Y)+I(Y;Z) = H(X)+H(Z)-H(Z|Y)-H(X|Y)$ Hence, we would require $-H(X,Z)=-H(Z|Y)-H(X|Y)$ -- is it true? ...
2
votes
3answers
61 views

Computing $p(d|e_1,e_2)$ from $p(d|e_1)$ and $p(d|e_2)$

I know the probability $p(d|e_1)$ and $p(d|e_2)$, how to compute the $p(d|e_1, e_2)$ if $e_1$ and $e_2$ are independent? What if $e_1$ and $e_2$ is dependent, how to compute?
0
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0answers
36 views

Is it true that $\mathbf{P} (X \leqslant t) =\mathbf{P} (F (X) \leqslant F (t))$ for continuous $X$

For continuous $X$ with distribution $F$, is it true that $\mathbf{P} (X \leqslant t) =\mathbf{P} (F (X) \leqslant F (t))$? Also is continuity required? I've attempted a proof: Since $\mathbf{P} (F (...
-1
votes
1answer
78 views

Constructing a new Markov chain from another Markov chain

I have a very simple problem, but it seems I have difficulty to prove it rigorously. Suppose random variables $X, Y$ and $Z$ form the following Markov chain: $X\leftrightarrow Y\leftrightarrow Z$. My ...
3
votes
3answers
69 views

A proof that $EX_n\to EX$ for uniformly integrable $\{X_n\}$ with $X_n\to X$ a.s.

I'm having some trouble following someone's proof of the following result: Assume that $\{X_n\}$ are uniformly integrable and that $X_n\to X$ a.s.; then $EX_n\to EX$. First, the author shows that $...
0
votes
1answer
36 views

A convergent sequence of normal random variables

Say $\{X_n\}$ is a sequence of normal random variables with means $0$ and variances $\sigma_n^2$. Also suppose that $X_n\to X$ (everywhere) and $\sigma_n^2\to \sigma^2.$ Then, using characteristic ...
3
votes
0answers
62 views

Charaterize the $\mathcal{F}_\tau$ a sigma algebra for the stopping time $\tau$

Consider a stochastic process $X: [0, \infty) \times \Omega \to \mathbb{R}^d$ We define $\mathcal{F}_t = \sigma(X(s), 0 \leq s \leq t)$ The sigma algebra generated by the sets $\{\omega: X(s,\omega) \...
0
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0answers
50 views

Why would one not want their probability space's sigma algebra to be the power set?

Let $(\Omega, \mathcal{F}, P)$ be a probability space. It seems like we would want to be able to measure the probability of each $\omega \in \Omega$, for which we must require that each $\{\omega\} \...
0
votes
1answer
26 views

From pairwise P(A > B), to P(A > all distributions in set)

$\{D_0,…,D_n\}$ is a finite collection of independent (but not necessarily identically distributed) random variables. Define $f(x,y)=P(D_x≥D_y)$ and $g(x)=P(∀y:D_x≥D_y)$. Does $f$ determine $g$, and ...
0
votes
1answer
66 views

How many loops? Expected value

I have a problem with this exercise. I completely do not know hot to tackle it. Please help. A bin contains $N$ strings. You randomly choose two loose ends and tie them up. You continue until there ...
2
votes
1answer
55 views

Show that $|X_n|\leq Y$ a.s. implies $\sup_n |X_n|\leq Y$ a.s.

Given a sequence $(X_n)_{n\geq 1}$, show that $|X_n|\leq Y$ a.s. implies $\sup_n |X_n|\leq Y$ a.s. Here is my attempt: $|X_n|\leq Y$ a.s. means that $P(|X_n|>Y)=0$, $\forall n\geq 1$ $P(\sup_n |...
1
vote
2answers
75 views

Why second moment about mean is better at describing spread than the first one?

Dispersion is usually used as a measure of inaccuracy of a measurement. It's defined as second moment about mean. Why not define dispersion as cubic root of third moment about mean or as first moment ...
1
vote
1answer
41 views

One Martingale problem

In the setting of Kolmogorov's maximal inequality, I need to prove the following $$P(\max_{1\leq m \leq n}|S_m| \leq x) \leq \frac{(x+K)^2}{var(S_n)}$$ Hint: Use the fact that $S_n^2 -s_n^2$ is a ...