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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Let $f$ be a bounded uniformly continuous function in $R^1$. Then $X_n\to 0$ in pr. implies $E(f(X_n)) \to f(0)$.
The following is problem 4 from Section 4.2 of "A Course in Probability Theory" by Kai Lai Chung.
Let $f$ be a bounded uniformly continuous function in $R^1$. Then $X_n\to 0$ in pr. implies ...
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1answer
18 views
Conditional Expectation with independent sub-sigma fields
Let X and Y be bounded random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider two independent sub-$\sigma$ fields $\mathcal{G}$ and $\mathcal{H}$ of $\mathcal{F}$. We ...
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1answer
31 views
Rigorous proof that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$
I'm trying to prove rigorously that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$. Where $f$ is the pdf of the random variable $X$.
I can't find a proof on the wikipedia article, or if it's ...
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11 views
How to derive this new Poisson to characteristic function
i rewrite Poisson depending on p and q, but further derive with maple, takes a very long time to calculate, i am afraid that it can not be calculated, i guess the key is using substitution
what is ...
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2answers
18 views
A basic doubt on independence of events in probability
Let $X_1$ and $X_2$ be i.i.d random variable. Now, in a book I see the following steps to calculate $P(X_1 < X_2 < x)$
$P(X_1 < X_2 < x)$
= $P(X_1 < x, X_2 < x, X_1 < X_2)$
...
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2answers
27 views
A basic probability doubt on independence
Let $X_1$ and $X_2$ be two i.i.d continuous random variable. I need to find the probability that $P(X_1 < X_2)$. I know how to formally find the probability by integrating over appropriate region ...
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26 views
Please show me that with which formula, I can calculate pooled variance for unequal population variance?
When equal population variances, I can calculate pooled variance (as like part-b)
But when unequal population variances, how to calculate pooled variance ( as like part-e)(also I underlined it ...
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13 views
sampling schemes for binomial distribution
Two acceptance sampling schemes, A and B, are proposed for deciding whether or not to accept a large batch of items from a production process in which 5% of the items produced are defective. Scheme A: ...
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0answers
35 views
Random Walk probability
I currently have a probability class tutorial question that I have no idea where to begin. At first instinct, I thought it may have been a CTMC question or branching question, but now I have no idea, ...
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0answers
24 views
Convergence in distribution and convergence of expectation.
Let $\{X_n\}_{n\geq1}$ and $\{Y_n\}_{n\geq1}$ be two sequences of uniformly integrable iid random variables with distributions $F_n(x)$ and $G_n(x)$, respectively. If
$$|F_n(x)-G_n(x)|\leq ...
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1answer
33 views
A generalization of the conditional expectation to kernels
Let $\left(\Omega_1,\mathcal{A}_1, P\right)$ be a probability space, let $\left(\Omega_2,\mathcal{A}_2\right)$ be a measurable space and let ...
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24 views
Trying to understand formula for the Survival Function (survival analysis) [migrated]
I'm trying to learn the Cox Proportional Hazards Model on my own, and found this link that describes it in clear terms. But when I get to formula (5) (S(t) = exp(−H(t))) I can't figure out where ...
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1answer
37 views
Probability on product spaces
I am having some trouble, more of an argument with someone else, about a simple question regarding product spaces.
Let $X_1,X_2,\dots,X_n$ a set of independent and identically distributed random ...
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2answers
20 views
(simple) Expectation of random variable as a multipart function
Let the random variable $Y \in [0, \infty)$, a real number $\theta >0$, and the random variable $X$ such that $X = \theta - \min(\theta,Y)$, thus, $X \in [0, \theta]$. That is, $X = 0$ if $Y > ...
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0answers
34 views
Can it be confirmed in this state when state transition probability >= 25%
if have a 2 * 2 state transition matrix,
can it be said that if one of cell probability >= 25% then it is confirmed in this state
if it is < 25%, then it is not in this transition state
if this ...
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23 views
Probability Space and Conditions
maybe a bit of a basic question but I don't see how to formally solve it right now.
In order to analyze a stochastic process, I want to change my probability space to a more convenient one (we can ...
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1answer
37 views
Order of convergence of a sum
Let $(X_t)_{t\geq 0},\;X_0=0$, be a positive stochastic process such that
\begin{align*}
\mathbb{E}\left[\sum_{n=1}^{\infty}X_t^n\right]=\sum_{n=1}^{\infty}\mathbb{E}[X_t^n]<\infty.
\end{align*}
...
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1answer
39 views
Help me solve the invariant measure of $Q$
My $Q$ matrix is given by:
\begin{bmatrix}
-\lambda &0 &\lambda &0 &0 &... \\
\mu&-(\lambda+\mu) &0 &\lambda &0 &... \\
0&\mu ...
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33 views
On using fourier transforms to solve the root of a convolution
In continuation of Lower bounds of laplace transform of characteristic functions.
My question is:
Can anyone point out where i'm going wrong in the derivation below.
It's been a while ...
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0answers
24 views
Is there a conditional version of the asymptotic equipartition property?
Let $X_i$ be independent random variables with $\operatorname{Pr}(X_i = x) = p_x$, and let $F_n$ be the empirical frequency distribution of $X_1, \ldots, X_n$: that is, $(F_n)_x$
For any frequency ...
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0answers
22 views
Proof that the radius is a sufficient statistic for a circle
How can I prove that the radius of a circle is the sufficient statistic for the probability of choosing random points in the area of the cricle?
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1answer
25 views
Using the Chebyshev Inequality
This is the Q:
A 20 fair coins tosses, (f means the "H" of the coin).
I have to block the probability that I will get n/2+n/100 "H"-s by Chebyshev Inequality. [n=20 in this case...], so:
n/2+n/100 = ...
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1answer
51 views
Stochastic process, Gaussian, with zero mean is a Wiener process
Let $(\Omega, \mathcal F , \mathbb P)$ be a probability space and let $\mathcal F = \{\mathcal F_t\}_{t\ge} $ a filtration. Let $W=\{W_t;t ≥ 0\}$ be a stochastic process adapted to $\mathcal F$. ...
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23 views
Continuity of a certain matrix-like function
Let $X$ be a finite set and let $M$ be a space of all probability measures on $X$. Let $f:X\to\Bbb R^{m\times m}$ be a random matrix and consider a function $c:M\to\Bbb R$ defined as
$$
c(\mu) := ...
2
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1answer
25 views
Ratio of PDF to complementary CDF
Let $f(x)$ be a probability density function, and $F(x)$ be the cumulative distribution function of $f(x)$.
$$F(x) = \int_{-\infty}^{x}f(u)du$$
Then intuitively, what does the following ratio ...
2
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1answer
43 views
Find the limiting distribution
Find the limiting distribution for $n\rightarrow \infty \text{ of} \prod\limits^n_{i=1}X_i$. Given is that $f(x)=\frac{1}{2x\sqrt{2\pi}}e^{-\frac{1}{8}(\ln x-\theta)^2}, x\geq 0$.
2
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2answers
57 views
Proving that $P\left(\bigcup_{j=1}^n E_j^{(n)}\right)\sim\sum_{j=1}^n P(E_j^{(n)})$ for independent events $E_j^{(n)}$
For arbitrary events $\{E_j, 1\le j\le n\}$, we have
$$P\left(\bigcup_{j=1}^n E_j\right)\ge\sum_{j=1}^n P(E_j) - \sum_{1\le j < k \le n} P(E_jE_k)$$
If $\forall n: \{E_j^{(n)}, 1\le ...
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1answer
31 views
Does a product measure on a product space constructed from two sub-fields of the same space determine a measure on the underlying space?
Let $\mathcal{A}_1,\mathcal{A}_2$ be $\sigma$-algebras on $\Omega$. Let $P$ be a probability on $\mathcal{A}_1$ and let $Q$ be a Markov kernel from $\mathcal{A}_1$ to $\mathcal{A}_2$. Set $K:=P\otimes ...
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1answer
53 views
$E(|X+Y|^p)\ge E(|X|^p)$
If $X$ and $Y$ are independent, $E(|X|^p)<\infty$ for some $p\ge 1$, and $E(Y) = 0$, then $E(|X+Y|^p)\ge E(|X|^p)$.
Is it maybe true that for each fixed $x$ that the following inequality is true?
...
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0answers
14 views
Levy process absolute moment
For a Levy process $(X_t)_{t\geq 0}$, we have $\mathbb{E}[X_t]=t\mathbb{E}[X_t^1]$ and $\text{Var}(X_t)=t\text{Var}(X_t^1)$. Does the same hold for the first absolute moment, i.e. does ...
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2answers
47 views
Probability Joint PDF
Every night Joe goes to the casino and takes with him an amount of money in dollars, X, that is distributed according to the pdf:
f(x) = Ax^2 for 0 < x < 10
where A is a constant that you need ...
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0answers
26 views
Calculating the probabilities of different lengths of repetitions of numbers of length 6
This question is similar to the question I asked here: Calculating the probabilities of different lengths of repetitions of numbers of length 4
except now I'm having problem with numbers of length 6.
...
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0answers
54 views
A basic doubt on CDF and pdf
Let $X$ be a random variable and $A$ be a Borel subset of $R$.
Then $$P(X \in A) = \int _{A}f_X(x) dx$$
where $f_X$ is the probability density function of the random variable. Now I have not ...
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2answers
38 views
If $E(|X+Y|^p)<\infty$, then $E(|X^p|)<\infty$ and $E(|Y^p|)<\infty$.
If $X$ and $Y$ are independent and for some $p>0$: $E(|X+Y|^p)<\infty$, then $E(|X^p|)<\infty$ and $E(|Y^p|)<\infty$.
How can I go from $E(|X+Y|^p)<\infty$ using independence to ...
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0answers
34 views
B.F.'s generated by disjoint subfamilies are independent
Problem 5 of Section 3.3 from "A Course in Probability Theory" by Kai Lai Chung
If $\{X_\alpha\}$ is a family of independent r.v.'s, then the B.F.'s generated by disjoint subfamilies are independent.
...
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1answer
12 views
Analytic tools in the theory of Galton-Watson processes
The questions basically aims at discussing the relative power of using probability generating functions, moment generating functions and characteristic functions as an example for ...
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1answer
32 views
Calculating the probabilities of different lengths of repetitions of numbers of length 4
I'm trying to calculate the probabilities of different lengths of repetitions of X length number however I know I'm doing it incorrectly since when I add all the probabilities together they don't ...
0
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0answers
39 views
If the fields $F_\alpha^0$ are independent, then so are the B.F.'s $F_\alpha$.
Problem 4 of Section 3.3 from "A Course in Probability Theory" by Kai Lai Chung
Fields or B.F.'s $F_\alpha(\subset F)$ of any family are said to be independent iff any collection of events, one from ...
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1answer
41 views
On unions of independent events
If the events $\{ E_{\alpha}, \alpha\in A\}$ are independent, then so are the events $\{F_\alpha,\alpha\in A\}$, where each $F_\alpha$ may be $E_\alpha$ or $E_\alpha^c$; also if $\{A_\beta, \beta\in B ...
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37 views
Orthogonalization of a set of random vectors
Suppose $w_1$ and $w_2$ are zero-mean jointly Gaussian random vectors. Further suppose that they have a covariance matrix given by
$$
\mathbf{cov}\begin{bmatrix}w_1 \\ w_2\end{bmatrix}
= ...
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1answer
19 views
Expectation of function of stochast
I've got a general question regarding a certain sticking point I often encounter. When tackling questions where for example an UMVUE (uniformly minimum-variance unbiased estimator) has to found I get ...
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1answer
40 views
Martingale and indicator
Exercise comes from "1000 exercices in probability" (12.9.6). Let $X_1, X_2, \dots$ be independent random variables with
$X_n=\begin{cases}
1, & \text{with probability} & (2n)^{-1}, \\
0, ...
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60 views
How does this violate probability theory?
Given: $X = Y^2 + Z^2$ (hence $E[X] = E[Y^2] + E[Z^2]$)
$p(X = 1) = .52$, $p(X = 4) = .24$, $p(X = 16) = .24$
$p(Y = -1) = .5$, $p(Y = 3) = .5$
Question: Despite not being handed any information ...
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2answers
30 views
Uniform distribution on the n-sphere.
I have the next RV:
$$\underline{W}=\frac{\underline{X}}{\frac{||\underline{X}||}{\sqrt{n}}}$$
where $$X_i \tilde \ N(0,1)$$
It's a random vector, and I want to show that it has a uniform ...
1
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1answer
18 views
Joint distribution of multiple binomial distributions
In the picture below, how do they arrive at the joint density function? I understand how Binomial distributions work, but have never seen the joint distribution of them.
The original file can be ...
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1answer
32 views
Fisher information of a Binomial distribution
The Fisher information is defined as $\mathbb{E}\Bigg( \frac{d \log f(p,x)}{dp} \Bigg)^2$, where $f(p,x)={{n}\choose{x}} p^x (1-p)^{n-x}$ for a Binomial distribution. The derivative of the ...
1
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2answers
68 views
Law of large numbers?
Given random variables $Z_1,Z_2,Z_3,\ldots$, which are uniformly distributed for $[8,10]$:
If $X_k =\min\{Z_1, Z_2,Z_3,\ldots,Z_k\}$, prove convergence in probability and find the constant. ...
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1answer
36 views
Continuous Non negative martingale converging to 0
Is there any (non trivial) continuous non negative martingale which converges to 0?
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1answer
31 views
Approximation of a random variable by a sequence of simple random variables
It said in a probability book that any non-negative random variable $X$ can be approximated by a sequence of simple random variables (finite range) $X_1,X_2,\dots,X_n$ such that ...
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2answers
29 views
Generalization of Doob Dynkin for Stochastic processes
Let $\{X_t\}_{t\geq 0}$ be continuous time stochastic process and $\{\mathcal{F}_t^X\}_{t \geq 0}$ be the filtration generated by it. If the process $Y$ is $\{\mathcal{F}_t^X\}_{t \geq 0}$ adapted, is ...
