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Questions tagged [probability]

For basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using [tag:measure theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

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8 views

Random Walk $\mathbb P(T_0>n $ and $S_n=a) = \mathbb P(T_a=n) =\frac{a}{n} \mathbb P(S_n=a)$

Consider the random Walk $S_n$ on $\mathbb Z$ starting in $x=0$. Let $a\in \mathbb Z$. Define $T_a(\omega)=\min\{n\in \mathbb N : S_n(\omega)=a\}$. Show for $a> 0$ $\mathbb P(T_0>n $ and $S_n=...
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11 views

How to calculate discrete probability distribution of number of particles in a sample?

I have a statistical problem related to taking samples of liquid with microparticles in it. I have a solution of let’s say 1000 microparticles in 1 mL of water (large amount of this solution). The ...
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1answer
8 views

Is $P_{s,t}f(X_s)$ the same than $P_{s,t}(X_s,\mathbb R )$?

In the Book of Yor-Revus (continuous martingales and Brownian motion), they define $$P_{s,t}(X_s,A):=\mathbb P(X_t\in A\mid \sigma (X_u : u\leq s)).$$ And after they say that one can prove with usual ...
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1answer
41 views

Norm of vector with entries sent to zero

I have a vector $x $ sampled uniformly at random from the sphere $\mathbb{S}^n$, with $n+1$ being an even number. Let's now assume I have a vector $y$ with the same dimension of $x$ whose first $\frac{...
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0answers
32 views

Finding infinite sum

I am really stuck as to how I find this infinite sum: $$\sum_{n=o}^\infty1-p(1-q)^{n-1}$$ The restrictions on p and q is that they both must be less than 1 but greater than 0, as this is the ...
2
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1answer
26 views

At least two girls sit together probability

Three girls $G_1,G_2,G_3$ and three boys $B_1,B_2,B_3$ are made to sit in a row randomly. The probability that at least two girls are together is .... My try: Probability of no girl together: _B_B_B_ ...
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2answers
30 views

Convergence of Student's t-distribution to a standard normal

I was looking at this question where it is shown that a Student's t-distribution converges to a standard normal distribution as the degrees of freedom tend to infinity.We start with the Student's t-...
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0answers
9 views

Deriving mean of posterior when the variance-covariance matrix is known

Let's consider the bayesian estimation of multivariate normal distribution when the variance-covariance matrix is known. Let $y_n$ be $$y_n \sim N(\mu, \Sigma), n=1,...,N ~~~~~~ \mu \sim N(\mu_0 , \...
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2answers
21 views

Find the probability that no two among $A$, $B$, and $C$ are together when $12$ people are arranged in a circle

There are $12$ people including A,B and C. They are arranged in a circle. Find the probability that no two among A, B and C are together. I have solved problem where cases involving two person ...
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1answer
14 views

Probability of card distribution of lowest four cards in a suit.

Inspired by a game of solo Spades I recently lost. In a four player game where all 52 cards are distributed evenly. If the lowest card I have in a suit is the 5, what is the probability the 2, 3, and ...
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1answer
22 views

proportion of the voters/ Central limit theorem

I want to compute the proportion of the voters p. Therefore I consider random variables $X_k$ for $k=1,...,n$: $$ X_k=\left\{\begin{array}{ll} 1, party \ is \ elected: "p" \\ 0, party \ ...
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1answer
29 views

Duration to guess password

I have been super stuck on this problem for a while and thought I turn to some expert help. My problem question: A password has length $8$ with a mix of $1$ uppercase letter (from $A$..$Z$), $5$ ...
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0answers
33 views

Proof by induction for a pdf

I am trying to solve this question, but I do not understand how to do induction on min(K,l). Can somebody explain this to me? Thank you!
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2answers
25 views

probability distribution proof $P(a < X \leq b) = F(b) - F(a)$

Let F be the distribution function of the probability $\mathbb{P}$ on $\mathbb{R}$ (induced by some random variable $X$). Prove: $\mathbb{P}((a,b]) =\mathbb{P}(a < X \leq b) = F(b) - F(a)$ This is ...
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0answers
19 views

Solution Verification. How to find $P(X + Y \le tZ)$?

Let $X, Y, Z$ ~ $U(0, 1)$. For a fixed parameter $t$ find $P(X + Y \le tZ)$. I believe the problem requires handling different possible values of $t$. To begin with, I will assume $t$ is positive, ...
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0answers
21 views

Marginal density function comes out negative

I am having a problem understanding marginal functions in a class problem. The exercise says: Given the bidimensional density function $f(x,y)=6(3x-y)$ when $0<x<y\leq 1$ Find the probability ...
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0answers
27 views

Bernoulli's trial

In Bernoulli's trial with success probability $p$. Mean is (expected value) $p$. Expected value of first success should be $>1$. How can have success in less than $1$ trial which is $p$
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1answer
22 views

show normality with non-linear transformation [duplicate]

This is one of the problem in the Allan Gut's Second Course for Probability. Let $X_1$, $X_2$ be independent standard gaussian e.g. N(0,1). Let $Y_1 = \frac{X_1^2 - X_2^2}{\sqrt{X_1^2 + X_2^2}}$, $...
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3answers
23 views

Given the bivariate standard normal density, how to show that $X$ and $Y$ are standard normal densities? [on hold]

Given the standard bivariate normal density with correlation coefficient $\rho$ for $X$ and $Y$: $$f_{X,Y}(x,y)=\frac{1}{2\pi\sqrt{1-\rho^2}}e^{-(x^2-2\rho xy+y^2)/2(1-\rho^2)}$$ Is there a way to ...
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1answer
21 views

Prove $2Cov(X_1,X_2)=E[(X-\bar{X_1})(X_2-\bar{X_2})]$

I'm trying to prove Hoffding's identity and I have trouble proving $2Cov(X_1,X_2)=E[(X-\bar{X_1})(X_2-\bar{X_2})]$. Where $\bar{X_1}$ and $\bar{X_2}$ are the independent copy of $X_1$ and $X_2$. Note $...
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1answer
26 views

Finding the Probability of A

In a sample space, events $A$ and $B$ are such that $P(A) = P(B)$ and $P(A\cap B) = P(A'\cap B') $. What is $P(A)$? I came across this question. They pose it like there is only one correct answer is ...
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1answer
22 views

What is the probability that John and Michael sit next to each other at a round table? that they do not sit next to each other? [on hold]

John and Michael go to a cafe with $3$ friends. They get a round table with $5$ slots. a) What are the probability that John and Michael are sitting next to each other? b) Probability of NOT sitting ...
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1answer
22 views

Use of conditional expectation in problem

A cricketer bats until he is out. For any given shot he takes, he either plays an attacking shot with probability $\frac{2}{3}$ or a defensive one with probability $\frac{1}{3}$. The type of each shot ...
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0answers
30 views

Probability and statistics: [on hold]

A uk distributor is to import 1000 mobile phones which will be sold with an insurance package for £400 each. They will be sent out to a number of shops in batches of 10. The distributor noted that, in ...
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0answers
19 views

Kolmogorov Complexity of a distribution

I read a paper that use a Kolmogorov complexity for a distribution, i.e., $K(p)$, where $p$ is a density function. However, I did not find the definition of it in this paper. Could anyone provides me ...
2
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1answer
42 views

Validity of Proof of Wald's identity

$\newcommand{\E}{\mathbb{E}}$ Theorem (Wald's identity): Suppose $\{X_i\}_{n \in \mathbb{N}}$ is a sequence of i.i.d random variables with $\E X_1 < \infty$. Let $\tau$ be a stopping time with ...
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0answers
39 views

Genius prediction [on hold]

It's a long story, but trust me, it's worth it. Matt is a detective and there's news of 3 murders about to happen in the upcoming week. Matt's usual success rate is 60%, i.e. He'll be successful in a ...
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0answers
18 views

Probability of choosing certain numbers [on hold]

What is the probability having the numbers 1,2,3,....,80 and choosing 20 of them to have exactly 10 odds and 10 even numbers.
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1answer
11 views

Convergence to infinity of distinct partial sums

I am currently working what I thought would be a simple exercise, but am actually having trouble getting any foothold. For $n\in\mathbb{N}$ let $\{X_{n,k}\mid 1\leq k\leq n\}$ be i.i.d. random ...
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1answer
29 views

Confused about the expression relating the CDF and expected value of a random variable

Let X be a random variable that takes on nonnegative values and has distribution function $F(x)$. $E(X) = \int_{0}^{\infty}1-F(x)dx$ The proof my book gives is: This statement was included at the ...
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0answers
14 views

How do you calculate error bars on elo rating?

In season 14 of the Top Chess Engine Championship, Stockfish defeated Leela 50.5-49.5 in a 100-game match (the closest possible winning margin). Using an elo calculator such as this one, we can see ...
4
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1answer
31 views

Series Converging Almost Surely But Diverging in Mean

I am looking for an example of independent, non-negative random variables $X_1, X_2, \dots$ such that $$ \sum_{n=1}^{\infty} X_n \, \lt \, \infty $$ almost surely but $$ \sum_{n=1}^{\infty} \...
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3answers
35 views

In determining probability using 2 dice rolls why are permutations (x,x) not counted twice?

So I've been working in probability regarding dice rolls. I came across this problem: If you roll 2 dice, what is the probability the first die is a 6 given that you rolled an 8? This is clearly a ...
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0answers
24 views

Regression on trivariate data with one coefficient 0

Suppose {$(x_i,y_i,z_i):i=1,2,...,n$} is a set of trivariate observations on three variables:$X,Y,Z$, where $z_i=0$ for $i=1,2,...,n-1$ and $z_n=1$.Suppose the least squares linear regression equation ...
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0answers
36 views

Gaussian process is a Brownian motion.

I am referencing from the book Probability from Dava Khoshnevisan. Thats my definiton of a brownian motion: 1) $W(0) =0$ and for all $t > 0$ is W(t) normal distributed with mean 0 and variance t. ...
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1answer
30 views

Bayesian network problem: third day rainy, given first day is

I've tried searching for this problem online but could not find a solution, hopefully you can help me. I have three random variables [r1,r2,r3], these three variables shows the probabilities of it ...
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1answer
20 views

Expectation of product of variables problem

A problem is asking to work out the total value of insurance claims made in a 400-day period. I am given that the expected value of any given claim is $£1000$. I am also given that the amount of ...
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2answers
19 views

Using the generating functions calculate the constant $C$.

Let $P (X=k, Y=j) = C \frac{ \binom{k}{j}}{2^{j+k}}$ , $k \in \mathbb{N}_{0}, 0\leq j \leq k$ be distribution of random vector $(X,Y)$. Using the generating functions calculate the constant $C$. I ...
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1answer
32 views

How $\mathbb{P} (A | B \cup B^c)$ is $\mathbb{P} (A | B) \cdot P(B) + \mathbb{P} (A | B^c) \cdot \mathbb{P} (B^c)$?

P(A) = $P(A|\Omega)$         = $ P(A|B \cup B^c)$ But how to reach P(A | B) * P(B) + P(A | B$^c$) * P(B$^c$) from P(A | B U B$^c$) ?
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1answer
17 views

Distribution of $Y = b \bar{X}$ doesn't depend on $b$ (Gamma distribution)

Assume a random sample $X_1,X_2, \dots , X_n$ following the Gamma distribution with pdf $$ f(x) = \frac{b^a}{\Gamma (a)} x^{a-1} e^{-bx}, \, x>0 $$ ($a$ known, $b$ unknown). The distribution of $...
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2answers
27 views

Inverse of sum of exponential random variables (mean and variance)

Assume $X_1,X_2, \dots , X_n$ following the exponential distribution with mean $\theta > 0 $ and the statistic: $$ T = \sum\limits_{i = 1}^n X_i $$ I know that the sum of exponential random ...
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1answer
10 views

A aperiodic state of a Markov chain has $N\geq 1$ such that $\forall n\geq N:p_{i,i}(n)>0$

The question I get asked is the following, I'm completely stuck on the problem: Let $i$ be an aperiodic state of a Markov Chain. Show that there exists $N\geq 1$ such that $p_{i,i}(n)>0$ for all ...
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1answer
18 views

How is P(N|$H_3$) derived?

An exercise with a solution attached below. I do not understand how is, in ii), $P(N\mid H_3$) derived. Could somebody please explain me?
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1answer
20 views

Expected number of selected subset for a weighted random sampling

We have a set of $\sum n_k$ items. The set includes $n_i$ items of $w_i$. And we will randomly choose $m$ items in the set according to the weights. Is there a formula for the expectation of number of ...
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1answer
38 views

Question about how to obtain the value of f X|Y(x|y)

In the following example question (from Bertsekas, edition 1), i have one question: Why the value of fY|X(y|x) is 1/2? Is it because Y is Y|X is either 0 or 1/6 (50% probability), or because some ...
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0answers
39 views

When do we have $E(p(Y|X)) = p(Y|E(X))$?

Let $p(Y)$ refer to the pdf of $Y$. I know that $\mathbb{E}_X(p(Y \mid X)) = p(Y)$. However, I am wondering if more can be said about the parameter of $p(Y)$ for parametric distributions. Are there ...
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2answers
434 views

Is the chance of winning rock paper scissors $1/2$ or $1/3$? [on hold]

Is it $1/2$ or $1/3$. How do people think it is $1/2$? And what is the true answer, $1/2$ or $1/3$?
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1answer
35 views

$\sum_{y=-\infty}^\infty E[X|Y=y]Pr[Y=y] $ algebraic manipulation

I am trying to understand the proof of $E[E[X|Y]]=E[X]$ and there is one part that I am not getting. $$\sum_{y=-\infty}^\infty E[X|Y=y]Pr[Y=y] $$ I know that the left side by definition equals to ...
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0answers
30 views

generalizing values for n [on hold]

I am trying to generalize for n on a bunch of related probabilities: n=2 P=1 n=3 P=1 n=4 P=5/6 n=5 P=7/10 n=6 P=9/15 n=7 P=11/21 n=8 P=13/28 ... I'm not quite getting how to describe P in terms of n....
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0answers
16 views

Proving independence between Beta estimated and Delta in OLS

I know that in ordinary least squares $b$(beta estimated) and $\delta^2$(variance estimated) are independent, but how do I prove that?