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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4
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2answers
8k views

Expected Value of a Hypergeometric Random Variable

How do you show, that the expected value of a hyper-geometric random variable X with parameters $r$,$w$, and $n$ (a box contains $r$ red balls, $w$ white balls and $n$ balls are drawn without ...
4
votes
1answer
689 views

Expectation of $TS_T$ where $T$ is the absorption time at $\{a,-a\}$ of a simple symmetric random walk $\{S_n\}$

I was trying to calculate the expectation of $T^2$ using some martingale and got that I needed the expectation of $TS_T$. Any idea?
3
votes
1answer
203 views

Is a probability of 0 or 1 given information up to time t unchanged by information thereafter?

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_n\}_{n \in \mathbb{N}}, \mathbb{P})$, let $A \in \mathscr{F}$. Suppose $$\exists t \in \mathbb{N} \ \text{s.t.} \ E[1_A | \...
3
votes
1answer
1k views

$K$ consecutive heads with a biased coin?

You toss a coin repeatedly and independently. The probability to get a head is $p$, tail is $1-p$. Let $A_k$ be the following event: $k$ or more consecutive heads occur amongst the tosses numbered $2^...
3
votes
2answers
175 views

Show that $\lim\limits_{n\rightarrow\infty} e^{-n}\sum\limits_{k=0}^n \frac{n^k}{k!}=\frac{1}{2}$

Show that $\displaystyle\lim_{n\rightarrow\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}=\frac{1}{2}$ using the fact that if $X_j$ are independent and identically distributed as Poisson(1), and $S_n=\sum\...
2
votes
2answers
560 views

The distribution of barycentric coordinates

Let $\mathcal{X} = \{x_1,\dots,x_n\} \subset \mathbb{R}^d$ and let $Z$ be a random variable uniformly distributed over convex hull of $\mathcal{X}$, denoted as $\text{conv}(\mathcal{X})$. Assuming ...
2
votes
1answer
82 views

Does $E[e^{it(aX + bY)}]=E[e^{itaX}]E[e^{itbY}]$ for every $a,b\in\mathbb{R}$ imply that $X$ and $Y$ are independent?

Let $X, Y$ be two random variables such that for every $\alpha, \beta \in \mathbb{R}$, $$E[e^{it(\alpha X + \beta Y)}]=E[e^{it\alpha X}]E[e^{it\beta Y}]$$ for all $t\in\mathbb{R}$. Does it follow ...
1
vote
1answer
86 views

Assume $A,B,C,D$ are pairwise independent events. Decide if $A\cap B$ and $B\cap D$ are independent

Assume A,B,C,D, are pairwise independent events. Decide if $(A \cap B)$ and $(B \cap D)$ are independent events? Then repeat this assuming the four events are mutually independent. Well, what I'm ...
1
vote
2answers
143 views

If $\{X_n\}$ converges in probability to $1$, where does $\{1/X_n\}$ converge to?

Without using the continuous mapping theorem, I want to show that, given $\{X_n\}$ is a sequence of random variables converging in probability to $1$, $\{1/X_n\}$ converges in probability to $1$. The ...
0
votes
1answer
42 views

Geometric random variables $X_1:G(p_1)$ $X_2:G(p_2)$ $X_3:G(p_3)$ are independent, prove the following :

$$P(X_1 < X_2 < X_3)= \frac{(1-p_1)(1-p_2)p_2p_3^2}{(1-p_2p_3)(1-p_1p_2p_3)}$$ To be frank I do not know where to start with this question, I would like an idea to get me going, or better yet ...
0
votes
2answers
283 views

Is a non-negative random variable with zero mean almost surely zero?

We have proven the following in class: If $X$ is a finite random variable with $X\geq 0$ then $$E(X)=0 \iff P(X=0)=1$$ (By finite I meant that the range has finitely many elements). Does it ...
9
votes
2answers
1k views

Completeness of a finite direct sum of closed subspaces of $L^2$

Let $X_1$ and $X_2$ be real-valued square-integrable random variables defined on a probability space $(\Omega, {\cal F},P)$. For $i=1,2$, set $$ A_i := \{g(X_i)\in L^2 \mid g \text{ is some Borel ...
8
votes
0answers
256 views

Uniqueness of the random variable from its distribution [closed]

Moderator's Note: This question has been put on hold due to the version over at MathOverflow having received better attention and produced an accepted answer. Interested readers are advised to visit ...
7
votes
1answer
883 views

Hölder Continuity of Fractional Brownian Motion

I would like to prove the following theorem: Let $H\in (0,1)$. The fractional Brownian motion $B_H$ admits a version whose sample paths are $a.s.$ Hölder continuous of order strict less than $H$. ...
6
votes
2answers
315 views

Proof of the infinitude of primes by probabilistic methods.

I'm looking if there is proof of the infinitude of prime numbers using probabilistic method. I am motivated by the answer of my question here. The answer is based on a relationship between ...
6
votes
1answer
2k views

Understanding the relationship of the $L^1$ norm to the total variation distance of probability measures, and the variance bound on it

I am trying to find a bound for variance of an arbitrary distribution $f_Y$ given a bound of a Kullback-Leiber divergence from a zero-mean Gaussian to $f_Y$, as I've explained in this related question....
5
votes
1answer
160 views

Uniform Integrbility

I need to show that if $|f_n|\leq g$, and $g$ is integrable, then $\{f_n\}$ is uniformly integrable, i.e., $\underset{a\rightarrow\infty}\lim\sup_n\int_{[|f_n|\geq a]} |f_n| d\mu=0$. Here is how I ...
5
votes
2answers
535 views

Prove that the maximum of $n$ independent standard normal random variables, is asymptotically equivalent to $\sqrt(2\log n)$ almost surely.

Lets $(X_n)_{n\in\mathbb{N}}$ be an iid sequence of standard normal random variables. Define $$M_n=\max_{1\leq i\leq n} X_i.$$ Prove that $$\lim_{n\rightarrow\infty} \frac{M_n}{\sqrt{2\log n}}=1\quad\...
5
votes
1answer
528 views

What is the significance of $\sigma$-fields in probability theory?

I'm an undergraduate level maths student and I've just done a number of exercises on elementary set theory and sigma fields. We've just started the courses and are discussing probability events as ...
4
votes
2answers
2k views

Explanation of Lyapunov condition of CLT

I found the Lyapunov condition for applying the central limit theorem, which is useful in settings where one has to deal with non-identically distributed random variables: Lyapunov CLT. Let $s_n^2 = \...
4
votes
2answers
117 views

Conditions under which the Limit for “Measure $\to 0$” is $0$

Let $\mu$ be a probability measure on $X$, so that $\int_X \mu(dx) = 1$. Say under which conditions on the function $f: X \rightarrow \mathbb{R}_{> 0} \ $ (that is measurable and integrable) we ...
4
votes
2answers
502 views

Let $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$. Find $EZ$.

Let $X,Y$ independent random variables with $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$. I already showed that $F_Z$ of $Z$ suffices $F_Z(z)=F(z)^2$. Now I need to find $EZ$. Should I start like ...
4
votes
0answers
564 views

Is a predictable process adapted?

Let us consider a measurable space $(\Omega, \mathcal{F})$, with a filtration $(\mathcal{F}_t)_t$ of sub $\sigma$-algebras of $\mathcal{F}$. The predictable $\sigma$-algebra $\mathcal{P}$ is the $\...
4
votes
1answer
245 views

X,Y,Z are mutually independent random variables. Is X and Y+Z independent?

X,Y,Z are mutually independent random variables. Is X and Y+Z independent? Please, give me a hint how to prove it?
4
votes
0answers
118 views

Why not defining a measure as a function on functions?

A measure $\mu$ is a function to $\left[0,\infty\right]$ on the sets belonging to a $\sigma$-algebra. Then for integrable functions $f$ the integral $\int fd\mu$ comes in, having nice properties ...
3
votes
1answer
186 views

Functions and convergence in law

Let $X$ be a random variable taking values in some metric space $M$. Let $\{\phi_n\}$ be a sequence of measurable functions from $M$ to another metric space $\tilde M$. Suppose that $\phi_n(X)$ ...
3
votes
2answers
212 views

Do moments define distributions?

Do moments define distributions? Suppose I have two random variables $X$ and $Y$. If I know $E[X^k] = E[Y^k]$ for every $k \in \mathbb N$, can I say that $X$ and $Y$ have the same distribution?
3
votes
1answer
144 views

Pairwise independence of Random variables does not imply indendence

Show by a counterexample that for a family $(X_i)_{i\in I}$ of random variables the independence of all pairs $(X_i,X_j)$ with $i,j\in I, i\neq j$ does not imply the independence of the family (It is ...
3
votes
1answer
199 views

Why is the following example of a Markov process not strong Markov

$X(t) := 0 \;\; (t \leq \tau),\;\; t - \tau\;\;(t \geq \tau)$ with $\tau$ exponentially distributed. Then X has the Markov property but not Strong Markov Property. But why ???? Can someone kindly ...
3
votes
2answers
3k views

Transform uniform distribution to normal distribution using Lindeberg–Lévy CLT

Currently i am developing a game which is based on many computations of random values and therefore i have implemented many algorithms like the Mersenne-Twister etc. Unfortunately, all generators ...
3
votes
2answers
353 views

Limit value of a product martingale

This question came from a problem i was solving for self-study. I'll state the problem first: Let $Y_n \sim \mathcal N(0,\sigma^2)$ be independent normally distributed variables, $X_n = Y_1+\cdots+...
3
votes
1answer
146 views

A question about a stochastic process being Markov

Let $(X_{s},\mathcal{F}_{s})$ be a stochastic process adapted to a given filtration. I was told that, in order to prove that $X$ is Markov, it suffice to prove that for any nonnegative, Borel-...
3
votes
2answers
948 views

Product of independent random variables

The following is a classic example that pairwise independent does not necessarily imply mutually independent: Let $X_1$ and $X_2$ be independent r.v.'s with distributions $$P(X_i=1)=P(X_i=-1)=\...
2
votes
1answer
138 views

Limit of a stochastic integral

Let $W_t$ be a one-dimensional Brownian motion and I would like to prove $$\lim_{\beta\rightarrow+\infty}\sup_{0\leq t\leq T}\left|e^{-\beta t} \int_0^te^{\beta s}\mathrm dW_s\right|=0$$ This is an ...
2
votes
5answers
4k views

Some case when the central limit theorem fails

If I understand correctly, for various versions of the central limit theorems (CLT), when applying to a sequence of random variables, each random variable is required to have finite mean and finite ...
2
votes
1answer
2k views

Prove that if $X$ and $Y$ are Normal and independent random variables, $X+Y$ and $X-Y$ are independent

If $X \sim \mathrm{Normal}(\mu,\sigma^2)$ and $Y \sim \mathrm{Normal}(\mu,\sigma^2)$ are independent random variables, how do I prove that $X+Y$ and $X-Y$ are also independent? What happens with the ...
2
votes
2answers
5k views

What is linearity of Expectations?

In reading about the average case analysis of randomized quick sort I came across linearity of expectations of indicator random variable I know indicator random variable and expectation. What does ...
2
votes
0answers
2k views

Expected value of max/min of random variables

I am trying to solve the following problem. Let there be $n$ urns and let us have $k$ balls. Assume we put every ball into one of the urns with uniform probability. Denote by $X_i$ the random ...
2
votes
1answer
188 views

Martingale formulation of Bellman's Optimality Principle

Related question: Deducing an optimal gambling strategy (using martingales). What I tried: For no 2, if $\ln Z_n - n \alpha$ is a supermartingale, then for $m < n$, $$E[\ln Z_n - n \alpha | \...
2
votes
1answer
147 views

Computing the expectation of conditional variance in 2 ways

Same as here. Let $X$ be a square integrable random variable on $(\Omega,\mathcal{F},P)$. Let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$. Define the conditional variance of $X$ given $\...
2
votes
3answers
439 views

How to express $E[\max(x,y)]$ as an integral?

In Hull (2008, p. 307), the following equation is found (Eq. 13A.2): $$E[\max(V-K,0)]=\int_{K}^{\infty} (V-K)g(V)\:dV$$ Where $g(V)$ is the PDF of $V$ and $V,K>0$. I'd like to extrapolate from ...
1
vote
2answers
1k views

Sum of two independent random variables converges in distribution [closed]

Show that if $X_n$ and $Y_n$ are independent random variables for $1 \le n \le \infty$, $X_n \Rightarrow X_{\infty}$, and $Y_n \Rightarrow Y_{\infty}$, then $X_n + Y_n \Rightarrow X_{\infty} + Y_{\...
1
vote
1answer
85 views

$E|X-m|$ is minimised at $m$=median

For a continuous random variable $X$, I want to show that $E|X-m|$ is minimum implies $m$ is the median of the distribution. Assume that the distribution function is $F$ and the density function is $f$...
1
vote
1answer
82 views

Markov property question

In every book I can find, the Markov property for ito diffusions, $E[f(X_{t+h})\mid F_s] = E^{X_t}f(X_h)$ is stated for $\textbf{bounded}$ Borel functions. However, I have the following statement ...
1
vote
1answer
132 views

Asymmetric Random Walk / Prove that $E[T:= \inf\{n: X_n = b\}] < \infty$

Given random variables $Y_1, Y_2, \ldots \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \...
1
vote
0answers
47 views

Why do people all the time exploiting almost sure properties of a stochastic process as if they were sure properties?

All the time, I see people working with a given Brownian motion $(B_t)_{t\ge 0}$ on a fixed probability space $(\Omega,\mathcal A,\operatorname P)$ and suddenly exploiting its almost sure properties ...
1
vote
1answer
182 views

Stopping time question $\sigma$

If $S$ and $T$ are stopping time, $S \vee T$ is $\max ({S,T})$, $F_S$ and $F_T$ are stopped sigma algebra, show that $F_{S \vee T} = \sigma(F_S,F_T)$. My thinking : I should take a set $A$ in $F_{S \...
1
vote
1answer
146 views

What is the joint distribution of two random variables?

Today I was thinking about this and I have the feeling I am missing something obvious, but I can't seem to solve it. Suppose we have a continuos random variable $X$ with density $f_X(x)$. Let $Y = g(...
1
vote
1answer
525 views

convergence in probability induced by a metric

Let $M$ be the set of all random variables from a fixed probability space to $\mathbb R$ with its borel sets. Let's define a metric on $M$ by $d(X,Y)=E(\frac{|X-Y|}{1+|X-Y|})$ I want to prove that $...
1
vote
1answer
121 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 ...