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 ...

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Exercise about martingale convergence

Let $p \in [0, 1]$, consider a stochastic process $(X_n)_{n\in\mathbb{N}_0}$ with $X_0 = x_0 \in [0, 1]$ and the following dynamics: For $n\in \Bbb{N}_0$, conditional on $X_0, X_1, \ldots, X_n$, we ...
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1answer
36 views

The Pólya urn model describes a martingale

Suppose an urn contains one blue and one red ball and that we perform the following random experiment: In each round $n\in\mathbb{N}$ we randomly draw a ball If the drawn ball is blue, we replace it ...
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1answer
46 views

Are $X$ and $X+Y$ independent, if $X$ and $Y$ are independent? [closed]

As asked in the title? Does the independence of two random variables $X$ and $Y$ imply the independence of $X$ and $X+Y$? If so, what's the easiest way to prove that?
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1answer
34 views

Bias of $\sigma^2$ estimator

I need to find the bias of $\frac{\sum(x_{i}-\bar{x})^2}{n+1}$ for $\sigma^2$. To do so, one must take its expectation but add and minus $\mu$ from the summation part so we can bring $\sigma^2$ into ...
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2answers
39 views

How to compute the following covariance?

$\alpha(t),\beta(t)$ are two stochastic process. How to prove the following equation: ...
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1answer
46 views

Coin weighting problem

There are n coins, among which there may or may not be one counterfeit coin. If there is a counterfeit coin, it may be either heavier or lighter than the other coins. The coins are to be weighed by a ...
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0answers
22 views

calculating expected gains.

A game costs \$100 to play. Toss a coin repeatedly, and win \$1 if you get heads for the first time, \$2 if you get heads both of the first two times, \$4 all of the first three times, \$8, and so ...
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0answers
32 views

Convergence theorem for uniformly integrable martingales

This is a theorem in my textbook: Why "for all $n\in\mathbb{N}$" and not "for all $n\in\mathbb{N}_0$"? What's wrong with setting $n=0$, e.g. $$ X_0 =\mathbf{E}[X_\infty| \mathcal{F}_0] \; ?$$
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16 views

Do you think this transition Matrix is correct?

Here is the situation we are trying to model: given a car that has 3 states, labeled 1, 2 and 3. state 1: is when the vehicle is in good operating condition. state 2: repairs may be required to ...
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0answers
18 views

Find Likelihood function of $\theta$ for $f(y|\theta) = \frac{1}{2\theta + 1}$

So, I know the likelihood function is the product of the density from $1$ to $n$. So then it would be $\displaystyle \left(\frac{1}{2\theta + 1}\right)^n$. I just want to see if I am doing that ...
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1answer
46 views

Coin Flipping Game - Expected Number of Tails

Can someone help me with this problem? In this game, let $S_{t}$ denote your earnings at time $t$. Your initial earnings is one dollar ($S_{0} = 1$). For each subsequent time, $t = 1, 2, ..$, flip a ...
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0answers
29 views

Riesz decomposition of a nonnegative supermartingale

$X=(X_n,\mathcal{F_n})$ is a nonnegative supermartingale, and moreover $EX_n\to 0$, i.e., it is a potential. If $X_n=M_n-A_n$ is the Doob decomposition, then $$EX_n=EM_n-EA_n=EX_0-EA_n,$$ so by the ...
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1answer
8 views

Independence of subalgebra in the limit

I am solving problem 9.4.3. from Theory of Probability and Random Processes (Koralov, Sinai). The problem says the following: Let $\xi_1,\xi_2,\dots$ be a sequence of r.v. on a probability space ...
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2answers
31 views

Pointwise convergence and L1 convergence in bounded mass case

I have a question regarding convergence modes and their relationships, my problem is actually an application to probability, How to prove that : ${f_n}$ and $g$ are probability density functions such ...
2
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1answer
45 views

If $N$ is an integral random variable, can $S_N$ be expressed in terms of the $S_n$ and $P(N=n)$?

Suppose $X_1,X_2,...$ is an infinite sequence of i.i.d random variables and let $N$ be a positive integer valued r.v independent from $\{ X_i \} $. Let $S_n = \sum_{i=1}^n X_i $ and $S_N = X_1 + ... + ...
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1answer
28 views

New characteristic functions from old

I am doing an exercise which says: If $f$ is a characteristic function, then show that $$ F(t):= \int_0^{\infty} f(ut)e^{-u}du $$ is again a characteristic function. Is this answer correct? Let ...
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1answer
27 views

Toss an unbalanced coin twice; Y is the number of heads. Depending on value of y actually oberved what is the MLE of p?

The original question says, "It is known the probability $p$ of tossing heads on an unbalanced coin is either $1/4$ or $3/4$. The coin is tossed twice and $Y$ is the number of heads. For each value of ...
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0answers
27 views

Quadratic Variation of Increasing Process?

I am looking through my notes and I came across the following statement: Let $X_s$ be a positive local martingale and let $M_t = max_{0 \le s \le t} X_s$. Then since $M_t$ is an increasing process, ...
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0answers
15 views

Cumulative beta density calculation. [duplicate]

The beta distribution of y, w.r.t $\alpha,\beta, min - a \text{ and } max - c $ is. $$f(y; \alpha, \beta, a, c) = \frac{ (y-a)^{\alpha-1} (c-y)^{\beta-1} }{(c-a)^{\alpha+\beta-1}B(\alpha, \beta)}$$ ...
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29 views

every pdf can be regarded as a marginal distribution of a joint pdf

suppose we have functions $g\ge0,h\ge0$ that $\int g \, dx=1 , \int h \, dy = 1$. it means $g$ is pdf for random variable $X$ and $h$ is pdf for $Y$. now how we can prove that there is function ...
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0answers
28 views

Marginal Distribution of the sum of Bernoulli rv

consider the conditional on probabilities $p_1, \ldots, p_n$, with independent Bernoulli random variables $Y_1, ..., Y_n$ given that $P(Y_i = 1\mid p_1, \ldots, p_n) = p_i, \ P(Y_i = 0\mid p_1, ...
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17 views

How can we have $T_n \xrightarrow{\mathbb P_\vartheta} \vartheta$ if $T_n$ are defined on different spaces?

Here is how I understand the standard parametric model in statistical inference: We have a r.v. $X:\Omega \to \Psi$ which has some known to us distribution yet the exact parameter is unknown to us. ...
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1answer
38 views

When does the variance of a consistent estimator go to zero?

I came across the following statement (marked as true) in multiple-choice section of an old exam: The variance of a consistent estimator goes to zero with the growing sample size. As far as I ...
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18 views

If 12 identical balls are to be placed in 3 identical boxes, then the prob that one of boxes contains exactly 3 balls is? [duplicate]

If 12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is? Please help. I can't understand this problem. I am bad at ...
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22 views

Hammersley–Chapman–Robbins bound for Rice distribution

I am trying to evaluate the Hammersley–Chapman–Robbins bound for the variance of an unbiased estimate $\hat{\alpha}$ of $\alpha$ (for a given $\sigma$) for the Rice distribution: $$p(x|\alpha,\sigma) ...
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1answer
74 views

Wiener process - proof of independent increments

I have defined the Wiener process to be a stochastic process $X_t$ with values in $\mathbb{R}$ such that $X_0=0$, the paths $t \mapsto X_t$ are continuous, and for any times $0<t_1<\dots<t_n$ ...
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1answer
165 views

Understanding probability

I'm stariting to study probability and some really interesting questions starting to bother me. Let's consider the unit circle $C$ and $D$ - the circle with radius $\frac{1}{2}$. I know that the ...
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1answer
57 views

Series of independent random variables are independent again

Let $\{X^i_j : i=1,..n, j\in\mathbb{N}\}$ be an independent set of random variables on a probability space $(\Omega, A, \mathbb{P})$, $$X^k_l: \Omega \to \mathbb{R^+} := \{x \in \mathbb{R} : x \ge ...
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1answer
69 views

Profile likelihood: Box-Cox transformation

I'm trying to prove a result that shows that the maximum likelihood estimator reduces the number of parameters in a Box-Cox model. In essence, we're trying to prove that $\bar{z}$ is the nuisance ...
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1answer
51 views

Ito's Integral's definition: Importance of isometry

I'm reading Oksendal's Stochastic Differential Equations (5th edition). He defines the Ito integral of $f$ as the limit $$\lim_{n\to\infty} \int^T_S \phi_n(t,\omega) dB_t(\omega)$$ Where $\{\phi_n\}$ ...
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25 views

Probability generation function example

let all the $X_i$ (for $1 \leq i \leq n$) be i.i.d. non-negative int-valued rand. variables. let each $X_i$ with probability generating function G(s), determine the generation function of the sum of ...
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13 views

Big O p Question about Eigenvalue of Random Matrix

Suppose $S_1, S_2, \dots$ are a sequence of random symmetric matrices in $\mathbb{R}^{d\times d}$. Suppose we know that $|\lambda_\max(S_n)| = O_p(b_n)$ and also that $|\lambda_\min(S_n)| = O_p(b_n)$ ...
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57 views

Identical objects into identical boxes

$12$ identical balls are to be put into $3$ identical boxes.Find the probability that one of the boxes contain exactly $3$ balls. The question would have been easy had both balls and boxes were ...
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1answer
23 views

Would the joint distribution of Normal Random Variable and the distribution of a X bar from the same sample be bivariate Normal?

I know this question is somewhat redundant... but here goes: My text asserts that the joint distribution of $$X_1=N(\theta, 1)\text{ and } \bar X = N(\theta, \frac 1n)$$ is Bivariate normal with ...
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1answer
20 views

Equivalent events in proof of Central Limit Theorem for Sample Median

Let $X_1,X_2,\ldots,X_n$ be i.i.d. random variables with cdf $F$ on $\mathbb{R}$ and $M_n$ be the sample median. The proof starts off with defining for any $a\in\mathbb{R}$, $S_n = \#\text{ of ...
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19 views

DTMC: repairing the machine

A machine works for $Y_0$ time then fails and takes $X_1$ time to repair. Then again works for $Y_1$ time and then fails and takes $X_2$ time to repair and so on. All the $X_n$'s and $Y_n$'s ...
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23 views

Law of Large Numbers - IID Assumption

My first question here! I was doing some probability review and was just wondering why exactly we need the IID assumption for the law of large numbers to work? Intuitively it makes sense of course, ...
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1answer
19 views

If $x$ is a $\chi^2_{N-n}$ RV. what is $x/N$ as N goes to infinity

We know that if we have $N-n$ gaussian iid RVs $\{e_i\}$ with mean $0$ and variance $1$, the RV $x = \sum e_i^2$ is $\chi^2$ distributed with $N-n$ degrees of freedom. We have $N$ larger than $n$. I ...
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0answers
47 views

Image of probability measure

Let $(\Omega,\Sigma,P)$ be a probability space. What is known about $P(\Sigma)$, the set of probabilities of events in $\Sigma$ by $P$? Clearly, $P(\Sigma)$ contains $0$ and $1$ since ...
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17 views

pairwise independent

I came across this example: Suppose Sam has to revise for an exam. He loves the sunshine so much that he will revise only if its a rainy day, which will happen with probability 0.3, and evan then ...
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23 views

Conditional Borel-Cantelli lemma

Let $A_1, A_2, \ldots$ be events with $A_n\in\mathcal{F}_n$. Show that $$\biggl\{\sum_{n=1}^\infty \mathbf{P}[A_n|\mathcal{F}_{n-1}]=\infty\biggr\} = \limsup_{n\rightarrow\infty} A_n \text{ a. ...
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1answer
34 views

Normal Distribution Probability with known mean and variance

I believe I am quite close to solving this, but I would just like to double check some of these answers. Two species have different size toes. Lengths of toes of species X is normal distributed with ...
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1answer
29 views

Poisson and Cumulative Distribution Double Check

this is simply me double checking my answer. Let Y be number of fish caught on a trip with Poisson distribution and $\lambda=4$. What is prob of catching 3 or fewer trout on trip? I said this was ...
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22 views

Martingale with bounded increments converges or diverges to $\pm \infty$ [duplicate]

Let $(M_n)$ be a martingale with $|M_n - M_{n-1}| \leq c$ for some fixed $c < \infty$. Check that the two disjoint events $$C:=\{M_n \text{ converges to a finite limit}\}, \; F:=\{\limsup M_n ...
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1answer
43 views

Negative Binomial

Let $X$ be a negative binomial with parameters $r$ and $p$. So $X$ is the number of trials $k$ till the $r^{th}$ success. My first question is determine which values of $k$ the ratio ...
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1answer
21 views

Conditional Probability with Ordered Stats

I am very close to solving this, but this last part is killing me on how to solve it. I dont really know where to begin. Scores run from 0 to 5 with density $f(x)=c(x^2-6x+10)$. An intermediate score ...
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30 views

Proof of the optional sampling theorem

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}$ be a Filtration on $(\Omega,\mathcal{A})$ $X=(X_n)_{n\in\mathbb{N}_0}$ be a nonnegative $\mathbb{F}$-supermartingale ...
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1answer
31 views

Two questions about Almost sure convergence and Uniform integrability

Let $X_n$ and $Y_n$ be two sequences of random variables such that $X_n\stackrel{n}{\rightarrow}C$ almost sure and $Y_n\stackrel{n}{\rightarrow}C$ almost sure, $C\in \mathbb{R}$. Suppose that $X_n$ ...
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1answer
46 views

Why is the first equality of this proof valid?

Proposition: Given a sequence $(Y_n)_{n \in \mathbb{N}}$ of real-valued random variables on $(\Omega, \mathcal{F}.\mathbb{P})$, if $Y_n \to Y $ in probability and $\sup_{n \in \mathbb{N}} ...
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26 views

where can I probability proofs for discrete/continuous rv

What book can I find probability proofs like linearity of expectation i.e. proof E(X+Y) = E(x) + E(Y) even though X and Y might NOT be independent ? Please note X and Y can be discrete/continuous RV