Tagged Questions

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 ...

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3
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1answer
33 views

How to calculate the characteristic function of compound Poisson random variable?

Let $\phi_X(t)$ be the characteristic function of $X$. Let $N$ be a Poisson random varivale with mean $1$ and $(X_i)_{\in\mathbb{N}}$ be i.i.d. copies of $X$. Then how to derive the charactersitics ...
0
votes
1answer
30 views

A basic question on weak convergence of measures

Why do we need separability of the space to talk about weak convergence of measures ?
-2
votes
1answer
57 views

In a game you receive three cards, ω, from a well-shuffled deck.. [closed]

In a game you receive three cards, ω, from a well-shuffled deck. You then receive \$30 per face card contained in the hand. That is if the hand contains 1 face card you get \$30, 2 you get \$80, 3 ...
1
vote
0answers
31 views

probability: continuous uniform distribution mean by symmetry

I am trying to show that the mean of the uniform continuous distribution is $(b+a)/2$ by symmetry. The direct method is fairly simple but, for some reason, I cant get this one. \begin{align} E[X] ...
1
vote
2answers
23 views

Conditional Coin Probability:Will The Decision Change

A decision making problem will be resolved by tossing $2n + 1$ coins. If Head comes in majority one option will be taken, for majority of tails it’ll be the other one. Initially all the coins were ...
2
votes
2answers
64 views

An example for conditional expectation

A factory has produced n robots, each of which is faulty with probability $\phi$. To each robot a test is applied which detects the faulty (if present) with probability $\delta$. Let X be the number ...
1
vote
0answers
59 views

Rotational invariance and distributions

Let $k\leqslant n$ denote two positive integers, $A$ an $n \times k$ matrix with $A'A = I_k$, and $X$ and $Y$ two independent random variables on $\mathbb R^n$, each rotationally invariant (that ...
4
votes
0answers
87 views

How to model this easy problem as sum of indicator random variables in order to apply Chernoff bound

Do you have an idea how I could model the following process somehow as a sum of independent indicator random variables? I have given a grid of size $n \times n$ for $n \rightarrow \infty$. Now I ...
2
votes
1answer
66 views

Martingale Transforms and quadratic variation

Let $M$ be a martingale with $\mathbb{E}M_{n}^2<\infty$ for all $n$. Let $C$ be a bounded predictable process and set $X=M\cdot C$. Show that $\mathbb{E}X_{n}^2<\infty$ for all $n$ and that ...
2
votes
1answer
71 views

What does “taking expectation w.r.t some random variable” mean in this probability calculation?

I am trying to calculate the following probability $$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$ where, $$A_i \sim \exp(\lambda), \quad S_i \sim ...
2
votes
1answer
37 views

Using Levy-Khintchine representation theorem to prove the following theorem

Let $(X_{n,i})_{1\le i\le n}$ be a triangular array of independent random variables, satisfying the uniform infinitesimality condition $$\lim_{n\rightarrow\infty}\max_{1\le i\le ...
0
votes
2answers
56 views

Simple formula on $X_n^{(k)} = \sum_{1 \le i_1 < … < i_k \le n} Y_{i_1} \cdot \dots \cdot Y_{i_k}$ (to show $X_n^{(k)}$ is martingale)

Let $$X_n^{(k)} = \sum_{1 \le i_1 < ... < i_k \le n} Y_{i_1} \cdot \dots \cdot Y_{i_k}$$ If I take $k=2$ and $S_n = Y_1 + \dots + Y_n$ I have of course: $$X_n^{(2)} = \frac{1}{2} (S_n^2 - ...
3
votes
0answers
43 views

prove the tightness of probability measures [closed]

Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of probability measures on $\mathbb{R}$, such that the Laplace transform $\phi_n(\lambda)=\int e^{-\lambda x}\mu_n(dx)$ converges pointwise to a limit ...
1
vote
1answer
53 views

Sum converging a.s.

Let $X_k$ be independent random variable s.t. $\sum_{k=1}^nX_k\rightarrow_{a.s.} X$. So, $$X=\sum_{k=1}^\infty (X_k^+ -X_k^-)$$ Is it true that $X=\sum_{k=1}^\infty (X_k^+)-\sum_{k=1}^\infty ...
0
votes
2answers
40 views

Pick two points on the perimeter of a circle of radius 1. Find the expected value of the length of the shortest arc.

Pick two points uniformly randomly on the perimeter of a circle of radius 1. This divides the circle into two pieces. Find the expected value of the length of the shortest piece. I have no idea to ...
3
votes
1answer
39 views

Bounding $P(X \le \tau)$

I am trying to upper bounding $P(X \le \tau)$ where $X$ is non-negative r.v. and where $\tau \le 1$. I have become aware of the Reverse Markov inequality that says that, if $P(|X|\le a)=1$ then for ...
1
vote
0answers
35 views

Moments of $|ax-by|$

Suppose that $X$ and $Y$ are independentr.v. uniform on $[0,1]$. What is the $E[|aX-bY|^p]$ for some constants $a,b,p>0$? What I did. \begin{align*} E[|aX-bY|^p]=\int_0^1\int_0^1|ax-by|^p dx ...
2
votes
3answers
33 views

Simple question about random walk with stopping time

I was reading a book and stuck with one line as follows: $$\sum_{m=1}^\infty E[X_m] P(T \ge m) = EX_1 ET$$ where $\{X_m\}$ is i.i.d. with $EX_m < \infty$ and $T$ be discrete stopping time ...
1
vote
2answers
71 views

Soft Question: Are sigma fields, fields?

I'm sorry if this is a foolish question but: Is a $\sigma$-field (of sets) a field (in the sense of algebra) if we only consider finite intersections and finite unions?
5
votes
1answer
76 views

SLLN with dependent Bernoullis implies convergence of sum with conditional means?

Suppose I have $n$ bernoulli (values zero or one), possibly dependent and nonidentically distributed, random variables (like the generalized binomial model), where a law of large numbers holds. Let ...
1
vote
0answers
32 views

Does the system of equations always have a nontrivial solution?

$f:[0,1]^2\to R_+$ is a continuous conditional density function. For $g,h\in C$ on $\{(x,y)\in [0,1]^2|x\geq y\}$, the system of equations is given by$$ \frac{\partial g}{\partial x}\leq ...
0
votes
1answer
27 views

Large deviation in relation with Wishard matrix

I try to prove the following fact. Let $A$ be an $m\times n$ matrix with iid standard normal random variable. Then $B:=A^{t}A$ is a $Wishart$ matrix with $m$ degrees of freedom and covariane ...
-1
votes
0answers
18 views

Estimation in Multiplicative and additive Noise problem

I have unsolved problems below for random process homework. Consider the problem of estimating a random vector $\underline{S}(u)$ from an observation $\underline{X}(u)$, where ...
3
votes
0answers
19 views

“Uniform” Convergence in Distribution (bounded Lipschitz metric)

I have been thinking about the following problem. Let me know if the notation below makes sense. Let $\mathcal{P}$ denote the set of Borel probability measures on a metric space $(\mathbb{R}^{k}, ...
2
votes
0answers
125 views

Derivative of conditional expectation

Let $\left( {{X_t}:t \in \left[ 0 \right.\left. {, + \infty } \right\rangle } \right)$ be a continuous time Markov chain on a probability space $\left( {\Omega ,\mathcal{F},\mathbb{P}} \right)$ with a ...
-3
votes
0answers
9 views

Cost based Risk calculation to obtain the correct decision making path

Here is the example scenario which I want to apply to real situation I have a damaged house, Attempt to repair the House cost $555 Chance of success 25% (if repair successful no additional cost ...
0
votes
0answers
3 views

Expected number of distinct nodes visited in a directed bipartite graph

Let $G = (V,E)$ be a directed bipartite graph with $V = \{I \cup O\}$ where $\left\vert{I}\right\vert = n$ and $\left\vert{O}\right\vert = m$. All the edges start from a vertex in $I$ and end on a ...
0
votes
1answer
36 views

How to derive this pdf?

I understand how to find the pdf for the sum of $N$ exponentially distributed random variables, but how do I find the pdf when $N$ is also an independent random variable. Here is the problem: Let ...
0
votes
1answer
51 views

Probability that some that m points are probable given the probability of subsets.

Working in a problem in analysis I came across this combinatorial/discrete probability question. I would appreciate if someone knows how to approach this problem or knows if this problem is related ...
0
votes
1answer
19 views

Probability Density Function of Scaled Gamma Random Variable

Assume we have a Gamma Random Variable $X$ with the following pdf $$ \frac{m^mx^{m-1}}{\Gamma(m)}\text{exp}(-mx)$$ If I am asked to find the distribution of the following $$Y= aX$$ where a is ...
1
vote
1answer
36 views

“With high probability” statement from CLT

Suppose $X_1,X_2,..,X_n$ i.i.d. with mean $\mu$ and variance $\sigma^2$, so that \begin{equation} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\longrightarrow ...
0
votes
1answer
22 views

Calculation of PDF of derived multivariate random variables?

Let we have $X$, $N$ dimensional vector of independent random variables. If we multiply this vector by some matrix $V$ with size $r\times N$, with property $V*V'=I$, where $I$ is identity matrix, and ...
0
votes
0answers
17 views

Is there a unique tilted measure with specified marginals?

Suppose $\mathcal{A},\mathcal{B}$ are finite sets and $\mu_{A,B}(a,b)$ is a probability measure on the product set $\mathcal{A}\times \mathcal{B}$ so that $\mu_{A,B}(a,b)>0$ for each $a\in ...
1
vote
0answers
63 views

Are there probability density functions with the following properties?

Given two distinct and continuous probability density functions on real numbers, $f_0$ and $f_1$ consider the following set of density functions: ...
0
votes
1answer
22 views

Double-formula for the expectation

Let $(\Omega,\mathcal{F},P)$ a probability space and $X$ a continuous random variable with density $f(x)$. In probability theory, how to prove that \begin{gather*} \int_{\Omega} X(\omega)dP(\omega) = ...
1
vote
1answer
19 views

Independence between random variables and set

Let $(\Omega,\mathcal{F},\mathbb P)$ be a probability space, and $X$ is a random variable in this space. For a set $A\in \mathcal F$, can we conclude that "$X$ is independent with $A$" ...
1
vote
1answer
18 views

Why are these variables not conditionally dependent given 'active triplets' and the 'explaining away' effect?

I'm following the Udacity Intro to AI course. This quiz gives the following Bayes network and asks whether different variables are conditionally independent or not. (The explanation of the nodes, ...
2
votes
1answer
45 views

Prove that $S_n^2-s_n^2$ is martingale

Let $(X_i)$ be iid such that $EX_i = 0$ and $\operatorname{Var}X_i = \sigma_i^2$. Let $s_n^2 = \sum_{i=1}^n \sigma_i^2$ and $S_n = \sum_{i=1}^n X_i$. Prove that $S_n^2 - s_n^2 $ is martingale. My ...
0
votes
0answers
21 views

Gamma distribution

Suppose X is Gamma ~ (2,.5) and Y is Gamma ~ (2,.5) suppose X and Y are independent. Let Y =(1/8)(X-Y) what is the distribution of Y? I know X+Y~(2+2,.5) what if I apply the same logic I get (0,.5), ...
0
votes
2answers
21 views

mutually exclusive event vs independent event

Can you illustrate with examples, what is "mutual exclusive event" and what is "independent event". Without math equations, please elaborate it.. Thanks in advance
0
votes
1answer
25 views

The independence between stochastic integral and sigma-algebra

Let $(\Omega, \mathcal{F}, \mathbb{P} )$ be the probability space, and {$W_t,0\leq t\leq T$} is a Brownian motion and $\mathcal{F_t}^W$ is the canonical filtration. For the $f(t)\in L^2([0, T])$(a ...
8
votes
3answers
142 views

Conditional expectation equals random variable almost sure

Let $X$ be in $\mathfrak{L}^1(\Omega,\mathfrak{F},P)$ and $\mathfrak{G}\subset \mathfrak{F}$. Prove that if $X$ and $E(X|\mathfrak{G})$ have same distribution, then they are equal almost surely. I ...
0
votes
2answers
18 views

Moment generating function of bounded variables

According to the answer of this question a moment generating function exists if the random variable $X$ is bounded. The proof is not quite obvious to me. More formally, let $(\Omega,\mathcal{A},\mu)$ ...
1
vote
0answers
29 views

Convergence equivalent random sequences

Suppose we have a sequence of independent random vars $X_n$ and consider a sequence of truncated random variables $Y_n=X_n1_{X_n\le n}$ s.t. $E[Y_n]=0$. We know that $X_n$'s and $Y_n$'s are ...
1
vote
1answer
16 views

Let Z ∼ N(0,1). Find the probability density function of |Z|

So I can easily find the pdf for the standard normal(though its too complicated to write out here), but how would the absolute value sign change the outcome?
1
vote
1answer
18 views

$f_X(x) = e^{-x} x \geq 0,$ 0 otherwise. Calculate $P(|X-m| \leq k \sigma)$ and compare to Chebyshev's bound.

So first, $$E(X) = \int_0^\infty xe^{-x} = 1 = m.$$ $$Var(X) = E(X^2) - 1^2 = -1 + \int_0^\infty x^2 e^{-x} = 1.$$ So does $\sigma = \sqrt{(Var(X))} = 1$ or do I need to calculate something else? ...
0
votes
2answers
31 views

Suppose X ∼ Exp(λ) and Y = ln(X). Find the probability density function of Y .

I'm to a point on this problem where I don't know what to do from here, so any direction would be appreciated. P(Y ≤ y) P(lnX ≤ y) P(X ≤ e^y) I feel like I need to use the X ∼ Exp(λ) now, but I'm ...
2
votes
2answers
25 views

What is the expected value of the payment if T has exponential distribution with mean 5?

The initial value of an appliance is $700, and its future value is given by: \begin{align}v(t)=100(2^{3-t}-1),&&0\leq t\leq 3.\end{align} If the appliance fails in the first 3 years, the ...
1
vote
0answers
14 views

How to calculate expectation of Cantor distribution without using p = 0.5

The question goes like this: given F(x) is the distribution function of Cantor distribution, how to calculate $E(x) = \int xdF(x)$ without using the argument of the following: X is characterized by ...
1
vote
1answer
27 views

Why does convergence in law not worry about points of discontinuity?

Wikipedia states that the definition of convergence in law only requires that the cumulative distribution functions of the sequence of variables converges pointwise to the CDF of the limit variable at ...