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 ...

learn more… | top users | synonyms (1)

2
votes
0answers
26 views

Is a limit of a sequence of distribution functions is necessarily a distribution function

I got a question: "is a limit of a sequence of distribution functions is necessarily a distribution function". My answer is NO. I have a counterexample: $F_n(x)=0$ if $x<n$ and $1$ otherwise. It ...
3
votes
1answer
46 views

Limit of measures is a measure

I know the following theorem (see exercise 1.3.3 from Achim Klenke: »Probability Theory — A Comprehensive Course«): Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of finite measures on the ...
0
votes
0answers
32 views

What is the interval for an exponential random variable?

Suppose that I have generated a random number $x$ using an exponential distribution with rate parameter $\lambda$. How can I find an interval $[a,b]$ such that $x$ is in this interval with probability ...
1
vote
2answers
51 views
+50

Conditions for uniqueness of the median

A median of a random variable is defined as any $m \in \mathbb{R}$ such that $P(X \le m) \ge 1/2$ and $P(X \ge m) \ge 1/2$. Alternatively, in terms of the CDF $F$ of $X$ defined by $F(x) := P(X ...
0
votes
1answer
22 views

Conditional probability -conditioning on a random variable

Let $ (\Omega, \mathcal{F}, \mathbb{P}) $ be a probability space, $A \in \mathcal{F}$ and $X$ a random variable. What does it mean $$ \mathbb{P} (A | X) $$ when $X$ is not discrete? Thank you!
5
votes
1answer
28 views

Does this sequence converge almost surely or not?

I have a sequence of independent random variables $X_1, X_2,...$ such that $P(X_n = 1) = \frac{1}{n}$ and $P(X_n = 0) = 1 - \frac{1}{n}$. Using the second Borel-Cantelli Lemma, we have $\sum P(X_n ...
2
votes
1answer
19 views

Clarification - DeGroot Proof on Transitivity Property of Subjective Probability

In developing axiomatic foundation for subjective probability DeGroot (Optimal Statistical Decision, 2004, p71) gives two axioms/assumptions: SP1: For any two events A and B, exactly one of the ...
3
votes
3answers
71 views

Probability that a size $d$ sample will contain all $k$ colours present

I tried looking for this question, but couldn't find it exactly... apologies if this is a repeat! Imagine an urn with $m$ balls. Each ball has a different colour and there are $k$ colours (obviously, ...
2
votes
1answer
19 views

Expectation of maximum of Binomial RVs

Given an iid sample $X_{1}, \ldots, X_{n} \sim Bin(n, p)$ I'm trying to find $$E(X_{(n)})$$ that is the expectation of the sample maximum. Unfortunately I don't know where to start. It seems that ...
0
votes
2answers
27 views

Finding probabilities - Elementary probability.

Suppose $\mathbb P(A) = 0.3$, $\mathbb P(B) = 0.5$, and $\mathbb P(B |A) = 0.6$. a. Find $\mathbb P(A \text{ and } B)$. b. Find $\mathbb P(A \text{ or } B)$. c. Find $\mathbb P(A|B)$. ANSWER: a. ...
1
vote
1answer
38 views

Conditional independence of sigma-algebras

If ${\mathcal{H}_1}$ and ${\mathcal{H}_2}$ are conditionally independent given $\mathcal{G} \subseteq {\mathcal{H}_2}$, are they conditionally independent given $\mathcal{F}$ such that $\mathcal{G} ...
0
votes
1answer
28 views

What does (0+) mean?

I'm currently learning from a script (which is written in German and not publicly available, sorry) for introduction to stochastics, where the topic is the Laplace transformed function for random ...
0
votes
1answer
30 views

Product topology and uniform topology on C[0,T]

Is the product topology on $\mathbb{R}^{[0,T]}$ restricted to $C[0,T]$ (T finite) the same as the topology induced by the uniform norm on $C[0,T]$? I am curious because I saw a claim on wiki saying ...
1
vote
0answers
33 views

A maximal inequality on distance to median, so called Lévy's inequality?

Problem (Kai Lai Chung, A Course in Probability Theory, section 5.3, ex6) Suppose $X_1,\dotsc,X_n$ are i.i.d. random variables, and $S_j:=X_1+\dotsb+X_j,S_j^0:=S_j-m_0(S_j)$, where $m_0(S_j)$ is a ...
1
vote
1answer
9 views

Expectation of distance from centre of a circular scattering

A random point $(X,Y)$ has a normal distribution on a plane with circular scattering with $E[X]=E[Y]=0$ and var$[X]$=var$[Y]$=$\sigma^2$. The distance of the point $(X,Y)$ from the centre of ...
0
votes
0answers
27 views

Probability Of 2 consecutive dice throw for 4 players

Consider a fair dice. 4 players are throwing the dice one after another. If some one gets 6 he will get extra chance to throw it again. So he will throw it twice and then the next player will get the ...
0
votes
0answers
43 views

Chebyshev's Inequality and the length of a random vector

Suppose that we take $n$ iid random variables $X_1,\dotsc, X_n\sim \operatorname{Unif}[-1, 1] $ and define $Y_n=\lVert (X_1,\dotsc, X_n) ^\intercal \rVert $. By what means can we employ ...
3
votes
1answer
29 views

Natural Filtration and Sigma-Field Generated by path function

Suppose we have a continuous real-valued stochastic process $X=(X_t;t\geq 0)$ defined on a probability space $(\Omega,F,P)$. Usually one defined the filtration to be $F_t=\sigma(X_s;s\leq t)$. But on ...
1
vote
1answer
32 views

How do you represent a random choice of random variables mathematically? What is its mean, variance, etc.?

Suppose that I have six random variables $X_1, X_2,\ldots,X_6$ (say, e.g., six coins with different biases). We should be able to get a new random variable $Y$ by rolling a die to get a number $n\in ...
-1
votes
1answer
14 views

Confusion about calculating probability of at least one event occurring

The probability that Tom will win the Booker prize is 0.5, and the probability that John will win the Booker prize is 0.4. There is only one Booker prize to win. What is the probability that at least ...
2
votes
2answers
54 views

Estimating $\mathbb P\{\max_{1\le j\le n}\lvert S_j\rvert\le t\}$, so called Charles Stein's theorem?

Problem (Kai Lai Chung, A Course in Probability Theory, section 5.5, Ex6) Suppose $\{X_n\}_{n>0}$ is a sequence of i.i.d. random variables. $S_n:=X_1+\dotsb+X_n$. For each $t>0$, define ...
1
vote
1answer
23 views

Normal approximation of binomial distribution

Problem: On average, every 50th shell has a pearl. What is the minimum amount of shells you have to open to get at least one pearl with probability greater or equal to 0.95. Calculate using the ...
0
votes
0answers
14 views

Partial derivative of a random vector

If $x$ indicates a $1\times n$ random vector of any distribution, then is the partial derivative of $x$ w.r.t $x$ equal to the derivative of the individual elements in the matrix, or are they just the ...
3
votes
1answer
20 views

Martingales and different definitions

Are there any differences between the following definitions of Martingales and if so what are they? Let $(X_{i})_{i=1}^{n}$ and $(Z_{i})_{i=1}^{n}$ be sequences of random variables then $(X_{i})$ ...
1
vote
1answer
79 views

Find the characteristic function of combination of random variables

I have the same problem as here: Find characteristic function of random variable. Could you explain last equality? Can I get it without using law of total expectation? Update I have some idea, $E $$ ...
0
votes
0answers
23 views

Cramer-Rao-Bound for squared parameter

I have a random sample $X_{i}, \ldots, X_{n} \sim \mathcal{N}(\theta,1)$. I would like to find the Cramer-Rao lower bound for the variance of unbiased estimators of $\theta^{2}$. Here's what I have: ...
2
votes
0answers
36 views

Expected magnitude of a vector of $n$ i.i.d. random variables as $n\to\infty$

Suppose that $X_i$ are i.i.d. real valued random variables with probability distribution $f(x)$ for $i=1,2,3,\ldots$. Let $Y_n=\left(\sum_{i=1}^nX_i^2\right)^{1/2}$. Assuming that ...
2
votes
1answer
67 views

Showing that lim sup of sum of iid binary variables $X_i$ with $P[X_i = 1] = P[X_i = -1] = 1/2$ is a.s. infinite

Let $(X_i)_{i\in\mathbb{N}}$ be an i.i.d. sequence of binary random variables with $$P[X_i = 1]=P[X_i = -1] = \frac{1}{2}$$ and let $$S_n = \sum_{i=1}^{n} X_i.$$ I'd like to show that $$P[\lim ...
0
votes
1answer
18 views

Malliavin Derivative

Motivation : We know that, if the randomness in the system is due to Brownian Motion then any contingent claim with mean zero can be written as Ito integral. (Of course, we need to have boundedness ...
2
votes
0answers
46 views

Rigorously, what is the goal of (machine/statistical) Learning and why is that the goal?

After some time doing machine learning and statistical learning theory, I decided to return to my foundations and make sure that the goal of what I am doing makes sense. First let me define $I(f)$ as ...
-1
votes
1answer
34 views

Is square of Wiener process an orthogonal process?

I'm trying to prove: Let $t_1 < t_2 \leq t_3 < t_4$ and $(X)_t$ is the square of Wiener process. Then $E(X_{t_2} - X_{t_1})(X_{t_4}- X_{t_3}) \neq 0.$ Progress Maybe the fact $E(X_{t_2} - ...
0
votes
0answers
17 views

Sampling from general multivariate Gaussians

Suppose you have access to a sampler such as randn in Matlab/Octave that returns samples from a simple one-dimensional Gaussian distribution (a normal distribution) ...
2
votes
0answers
33 views

Questions regarding Martingales

I'm trying to learn about Martingales with specific focus on combinatorial problems. However i'm far from an expert in algebra and am having some trouble understanding the basic idea. I will write the ...
3
votes
1answer
31 views

Existence of a probability space

Let us assume that we are given a family of Markov chains $(X^\alpha_t)_{t\geq0}$ in continuous time. Kolmogorov's result ensures that for each $\alpha \in I$ there exists a probability space ...
1
vote
1answer
21 views

Convergence in distribution with exponential limit distribution

Let $X_1,X_2, \ldots$ be independent, identically distributed, positive random variables with probability density function $f$, which is continuous in $(0, \infty)$ and $\lambda :=\lim_{x \searrow 0} ...
1
vote
1answer
35 views

An inequality involving $\mathbb{E}[|X|],\,\mathbb{E}[X^2],\,\mathbb{E}[X^4]$

My question is an exercise which appears in a book on probability theory. Thank for helping. Let $X$ be a random variable with $\mathbb{E}(X^2)=1$, $0<\mathbb{E}(X^4)<+\infty$. Prove that: ...
1
vote
1answer
30 views

Elementary question on probability

Villages A,B,C, and D are connected by overhead telephone lines joining AB, AC, BC, BD, and CD. As a result of severe gales, there is a probability p(the same for each link), that any particular link ...
1
vote
2answers
34 views

Law of large numbers for Brownian Motion (Direct proof using L2-convergence)

In “Brownian Motion” by Schilling and Partzsc, they give a HINT to prove the Law of Large Numbers for Brownian Motion (not in their solutions, fyi) by (1) Noting that ...
1
vote
2answers
27 views

Value of constant k which makes the function $f(x)=\frac{k|x|}{(1+|x|)^4}$ a p.d.f.

Let $f(x)=\dfrac{k|x|}{(1+|x|)^4}$, $-\infty<x<\infty$. Then, what is the value for which f(x) is a probability density function ? f(x) will be a p.d.f. if $\displaystyle ...
0
votes
1answer
49 views

$\limsup$ and $\liminf $ of $\sum_{k=1}^n \frac{X_k}{\sigma \sqrt{n}} $

Suppose $(\Omega, \mathfrak{F}, p)$ is a probability space; $X_n$ are i.i.d. random variables defined on $\Omega$, with $E(X_i)=0$ and $Var(X_i)= \sigma$ for all $i$. Then $$ \limsup_n \sum_{k=1}^n ...
-2
votes
0answers
23 views

Is expectation also a distribution?

I'm confused about Expectation in probability theory. Is it also considered a probability law? if so, why?
4
votes
3answers
205 views

what is difference between probability and probability space?

I am a beginner in probability and started reading the relative material. I encountered the exercise question "find the probability space for tossing a fair coin till the first head is observed". so, ...
1
vote
1answer
51 views

A question on a certain transformation of a normal random variable

I have to solve the following exercise: Suppose $X\sim N(\mu,1)$ and consider $Y=\dfrac{1-\Phi(X)}{\phi(X)}$, where $\phi,\Phi$ are the pdf and cdf of the standard normal distribution with ...
0
votes
0answers
22 views

I have questionin from the stochastic differential equation merton model

in the following stochastic differential equation merton model we have $$\frac{ds}{s}=(\alpha-\lambda k)dt+\sigma dW+dq$$ where $\alpha$ is the instantaneous expected return on the stock; ...
0
votes
2answers
45 views

How to add two random variables?

Given that $$\begin{array}{cccc} \text{X} & -1 & 1 & 3 \\ \text{p} & 0.2 & ? & 0.3 \\ \end{array}$$ and $$\begin{array}{cccc} \text{Y} & 1 & 2 & 3 \\ \text{p} ...
1
vote
1answer
28 views

Variance of independent variables

Let $X_k$ and $Y_k$ be two stochastic variables whose joint distribution is the regular normal distribution on $(\mathbb{R}_2,\mathbb{B}_2)$ with mean 0 and variance matrix $\begin{align*} ...
3
votes
0answers
40 views

Estimating the support of a probability density function

The inverse moment problem deals with the reconstruction of a probability density function (PDF) of a random variable (RV) by means of its statistical moments. In the special case of the Hausdorff ...
1
vote
1answer
27 views

Fatou, Dominated Convergence, etc. for nets (in relation to stochastic processes)

In textbooks on Stochastic Processes, they always seem to assume that Fatou and DCT etc. can be applied to continuous-time stochastic processes $(X_{t})_{t\in\mathbb{R}_{+}}$. But in every book on ...
0
votes
1answer
19 views

Find the probability for … [duplicate]

Suppose we uniformly and randomly select permutations from the 20! Permutations of 1, 2, 3,..., 20. What is the probability that 2 appears at an earlier position than any other even number in ...
1
vote
0answers
48 views

Relation between a.s. and L_{2} convergence

I'm working through a proof, where I need to establish that $X_{t}\overset{a.s.}{\longrightarrow}0$. All I know is that $\left|X_{t}\right|\leq\left|Y_{n}\right|+\left|Z_{n}\right|$ for $t\in[n,n+1)$, ...