Questions tagged [probability]

For basic questions about probability and the questions associated with the calculation of probability, expected value, variance, standard deviation, or similar statistical quantities. 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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1answer
38 views

What is the probability to randomly choose a specific area in a square?

We have a square on $(0,0),(0,1),(1,0),(1,1)$. That's the $\Omega$. and we randomly choose a point, $M(x,y)$, in it. What is the probability that $M(x,y) \in D=\{(x,y)\in \Omega \quad :min(x,y)<\...
6
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1answer
67 views

Easy question on probability

I know this is a trivial question but I want to make sure I'm not missing anything: We have a biased 6-sided die, which brings any of the 6 numbers with equal probability in the first roll, but in the ...
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0answers
11 views

Conditionality, joint probability.

We have: P[Xt+h = j, T0 > t | X0 = i], where, Xt+h = state of process at t+h, T0 is waiting time in state i, X0 = I is the current state. They then say that this is equal to: P[Xt+h = j, T0 > t |...
2
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1answer
54 views

At what kind of rate does $e^{\log(x)^2}$ increase?

If you look at the LogNormal distributions PDF, take $\mu=0$ and $\sigma=1$ and get rid of constant multipliers, you're left with: $$\frac{1}{x e^{\log(x)^2}} \tag{1}$$ We can also see from the ...
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1answer
59 views

What is the probability of at least one pair of people who share a birthday and whose mothers share a birthday?

Problem $71$ of Chapter 4 from Introduction to Probability by J. Blitzstein and J. Hwang. In a group of $90$ kids, what is the approximate probability of there being at least one pair of kids born on ...
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1answer
35 views

Derive pdf of Z=XY where X and Y are two independent uniform r.v. on [0,1]

First, let pdf X and Y be, $f(x)$ and $g(x)$, respectively. I start by considering the a similar problem - derive the pdf of Z=X+Y from the cdf. Consider the cdf of X+Y and condition on the value of Y:...
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0answers
20 views

$l_\infty$ error bounds for linear regression

Consider the fixed design multidimensional linear regression: $Y = X \beta^* + \epsilon$, where $\beta^* \in \mathbb {R}^d$, $X \in \mathbb {R}^{n\times d}$, and $\epsilon \sim \mathcal{N}(0, \sigma I)...
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0answers
33 views

Is it possible to have $\limsup_{n\to \infty}A_n = \emptyset$? [closed]

Given a sequence of events $(A_1, A_2, A_3,\ldots)$, is it possible that: $$\limsup_{n\to \infty} A_n = \{𝜔∈Ω:∀𝑛∈N,∃𝑘≥𝑛 | 𝑠.𝑡:𝜔 ∈𝐴_k\} = \emptyset$$ If it's possible kindly show me an example.
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71 views

The Plausibility of Cavemen "Inevitably Discovering" Mathematical Principles

Recently, I had the following idea about the inevitableness of a prehistoric caveman "discovering" fundamental principles of mathematics based on the laws of physics and nature of the ...
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1answer
28 views

Prove that $P(X<a)=\lim_{x_{n}\to a}F(x_n)=F(a^{-})$

Prove that $P(X<a)=\lim_{x_{n}\to a^-}F(x_n)=F(a^{-})$ So I want to use this property if $A_1\subset A_2\subset...$ and $\cup_{k=1}^\infty A_k=A$ then $\lim_{k\to\infty}P(A_k)\to P(A)$ So I have $...
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1answer
16 views

Finding a probability given a box of 50 fuses where 10 are bad

Below is a problem I did. I believe I did it right. However, I would like somebody to check me. I am concerned about the possibility of a computational error. Problem: A box contains $40$ good and $10$...
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43 views

Central limit theorem : manufacture of chocolates

In the manufacture of chocolates of a specific variety with a nominal weight of 20.4 grams, there may be fluctuations in the actual weight of a praline. We describe the weight of a praline of this ...
4
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1answer
107 views

Markov chain returning to its starting position

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $(X_n)_{n \in \mathbb{N_0}}$ be an $I$-valued (homogeneous) Markov chain, where $I \subset \mathbb{R}$ is countable. For $i \in I$...
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1answer
43 views

Confidence interval for $X \sim U(0,\frac{1}{\theta})$

If $X \sim U(0,\frac{1}{\theta})$ and $Y= \theta X_{(n)}$ where $X_{(n)}=\max \{X_1, X_2, ..., X_n \}$. $F_{Y}(y)=y^n \quad \ f_{Y}(y)=n z^{n-1}$ And then $P(a<\theta X_{(n)}<b)=1 - \alpha \...
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1answer
49 views

Central limit theorem : game of chance

We consider the following game of chance with independent rounds: In each round we can win ten euros with probability $0.1$, we can lose one euro with probability $0.7$ and two euros with probability $...
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0answers
51 views

Intuition behind conditioning to events with probability zero

What’s the intuition behind why the conditional expectation w.r.t. a $\sigma$-algebra allows us to condition to events with zero probability? For example, let’s say we have two continuous random ...
2
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0answers
56 views

Confidence interval for the variance of exponential distribution?

I know that if $X \sim \operatorname{Exp}(\theta)$, and $Y=\theta X \implies Y \sim \operatorname{Exp}(1)$. I want a confidence interval for $E(X)= \frac{1}{\theta}$ and $\operatorname{Var}(X)= \frac{...
2
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0answers
36 views

Master Equation for Population Growth

The growth of a population $Q$ is described by the following evolution equations: $$ \boxed{ \begin{array}{cccc} (\textrm{i}) \hspace{0,3cm} \textrm{Q} \xrightarrow[]{\sigma} \textrm{Q} +\textrm{...
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1answer
26 views

loop bus waiting time with randomly stopping

A bus runs on a loop. It takes to the bus 10min to complete the loop. You go to the bus station at random times. Along the way the bus driver randomly stops for 10 min with a $p$ chance (only one time ...
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1answer
57 views

How to prove these two properties?

For two independent r.v. $X\sim N(\mu_1, 1)$ and $Y\sim N(\mu_2, 1)$. Show that $$ P(\min\{X^2, Y^2\}>\chi_1^2(\alpha))\le \alpha $$ given that $\mu_1\mu_2=0$ where $\chi_1^2(\alpha)$ is the $1-\...
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1answer
33 views

Finding type II error for testing mean in normal distribution

Let's consider variable $X \sim N(\mu, 4)$. I want to check hypothesis $H_0:\mu=-1$ versus $H_1: \mu = 1$. Critical region is given as $R = \{\overline{X}_n > c\}$ I want to find such $c$ that this ...
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0answers
27 views

Probability of a sum of random variables exceeding a threshold given that it exceeds a smaller threshold [closed]

given random variables $X_1...X_n$ and constants $0\leq\delta,\rho\leq1$ such that for any $t$-tuple $y=\{X_{i_1}...X_{i_t}\}$ we have that $P[A_y]=P [X_{i_1}=...=X_{i_t}=1]\leq \rho^t$ for $t=\frac{\...
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0answers
19 views

Posterior density calculation

Consider the normal linear regression model $y = x \beta + \epsilon $ where $\epsilon$ is normally distributed as $N(\epsilon|0,\sigma^2 I_n)$ and $\beta$ with a prior distribution $N(\beta|0,A)$ and ...
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0answers
24 views

Confusion about empirical distributions notation that uses a dirac delta function in paper

I've been reading this deep learning paper https://arxiv.org/pdf/2110.12567.pdf in section 2.3.1 the authors define $pQ$ and $pK$ as two empirical distributions as: $pQ = \sum_{i=1}^{w}\frac{1}{w}\...
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2answers
51 views

Name of the formula used to get the moments of a random vector

I'm lloking for the name of the following formula used to get the moments of a random vector ($j \in \lbrace 1,...,n \rbrace$; $ k_1,...,k_j \in \lbrace 1,...,n \rbrace $; $r_1,...r_j \in \mathbb{N}$)....
1
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1answer
43 views

Probability that in an endless series of independent Bernoulli trials, the pattern of $101$ will appear infinite times?

What is the probability that in an endless series of independent Bernoulli trials, with probability $p$ for getting $1$ and probability $q=1-p$ for getting $0$, the pattern of $101$ will appear ...
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26 views
+50

Bivariate negative binomial distribution for 2d count data

Bivariate negative binomial distribution The probability mass function (PMF) of a bivariate negative binomial distribution ($\mathsf{BNBin}$) is given by [1]: $$P(X=x, Y=y) = \frac{(a + x + y - 1)!}{(...
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0answers
34 views

Expected value and correlation

I have the following table, where the first row are the intervals of $y$ variable and the first column are the intervals of the $x$ variable. After getting 4000 samples, I placed their count into the ...
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0answers
50 views

Are these notations about convergence in distribution standard?

I wanted to ask if the following notations make sense and/or are used. Convergence in distribution of a sequence $X_n$ of real random variables to the random variable $X$ is often indicated like this: ...
0
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1answer
19 views

Transforming multivariate normal distribution

Let $X$ be multivariate normal distribution with $ \begin{bmatrix} \mu_1 \\ \mu_1 \end{bmatrix} $and covariance matrix given as$ \begin{bmatrix} \Sigma \;\Sigma \\ \Sigma \;\lambda \Sigma \end{...
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0answers
26 views

$n$ couples sitting around a table. Let $X$ be the number of couples that sits next to each other.

Alright, so we have $n$ couples sitting around a table, randomly. Hence, $|\Omega|=(2n-1)!$ Let $X$= number of couples that sits next to each other. I was asked to find $\sigma^2(X)$ and it took me ...
0
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1answer
23 views

Calculating covariance and correlation

Suppose we have a bird that lays two eggs. Probability for one egg to hatch is $0.8$ so the probability for it not to hatch is $0.2$. If the egg hatches, there's a $0.5$ probability for it to be a ...
1
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1answer
64 views

Direct way to check if a random variable is $W_T$ measurable

Inspired by this: Calculate stochastic integral $\int_0^T s^2 W_s dW_s$ , I was asking myself this question: Given a stochastic integral: $I=\int_0^T f(W_s) ds$ is there a direct way to check if it is ...
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1answer
25 views

Conditional Probability with possible unnecessary informatio?

Let's suppose event $A$ has $5\%$ chances of happening. The chances that event $A$ is of type $B$ is $70\%$. Event $A^c$ has $5\%$ chances of being of type $B$. Given that event is of type $B$, what ...
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0answers
34 views

Bernoulli probability problem [closed]

Toss a fair coin n times in a row, how calcute probability of heads being at least k consecutive times.
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0answers
53 views

Luck metric proposal: is it useful?

Let S be set of events {X1,X2,..}. Let Good be a set of events {G1,G2,...}. Let Bad be a set of events such that for every event in Good, the complement of that event is in Bad. The set S is ...
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0answers
23 views

How to adjust a probability over time while gaining more information? (bayesian statistics)

I'm struggling with a probability problem at work, which I need to understand more in order to devise an algorithm. It seems like a problem typically suited for a bayesian approach, but after hours on ...
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0answers
36 views

Is one SAT guessing strategy better than another?

The context is this paragraph from my SAT & ACT Prep book on page 11. "There is one thing to keep in mind: Pick one letter for the SAT or a two-letter combo for the ACT and stick to it ...
0
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1answer
28 views

$\sigma$-additivity and continuous random variables

Let $X(\omega)$ be a continuous random variable defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$. How does the measurability condition of this random variable (i.e. $\forall r\in\...
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1answer
47 views

Probability that a determinant is equal to zero about 2*2 or 3*3 Integer matrix.

In fact, it is an expansion of this problem. But I restricted the elements of the matrix to be integers only. Obviously this probability is related to the range of random $\text{item}\in[0,n]$. For ...
0
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0answers
40 views

Joint Probability Distribution, Find $c$

I am having a little bit of trouble with this practice question and I would like some insight of how to proceed. If $X$ and $Y$ are discrete random variables with joint probability: \begin{equation*} ...
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0answers
40 views

Expected value of $E[X_1X_2]$ of a stochastic process

Say I have a transition matrix $P$ of a stochastic process and an initial distribution vector $\nu$. Is the expected value $E[X_1X_2]$? $$E[X_1X_2] = \sum x_1.x_2P(X_1=x_1 \cap X_2=x_2).$$
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0answers
26 views

Properties of Expectations. [closed]

If X1, X2 are the results of rolling a fair, six-sided die twice. How do I show that E[min(X1, X2)] = 1/2 E[X1 + X2 − |X1 − X2|]. Also, how do I compute the value of this expectation E?
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0answers
36 views

Is it possible to calculate the odds for a one more goal to be scored in the first half from the odds of the teams winning the first half?

Let's say that at 35 minutes of the first half in one Bookmaker we find the following odds for the first half result: ...
1
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0answers
39 views

Approach of solving a probability question involving independent events not matching that given in my book.

Question: A and B are two independent events. The probability that both A and B occur is $\frac{1}{6}$ and the probability that at least one of them occurs is $\frac{2}{3}$​. The probability of the ...
2
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0answers
27 views

Higher derivatives of the log-partition function

I need higher derivatives of the log-partition function $Z(z)=\log \sum_i \exp(z_i)$, has anyone derived the formula? Looking at concrete values of derivatives up to order 8, evaluated at $z=(1,1,1)$ ...
5
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1answer
40 views

$|W_{t_n}| \to \infty$ a.s. as $n \to \infty$ for Brownian $(W_t)_{t \in \mathbb{R}_+}$ with $\sum_{n=1}^\infty t_n^{-0.5} < \infty$?

Let $(W_t)_{t \in \mathbb{R}_+}$ be a Brownian motion. Let $(t_n)_{n \in \mathbb{N}} \subset \mathbb{R}_+$ be a sequence of time points, such that $\sum_{n=1}^\infty t_n^{-0.5} < \infty$ (so, for ...
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0answers
33 views

Chances of a "Straight" in a 13 Card Hand [closed]

If dealt $13$ a card hand from a standard $52$ card deck; what is the probability that the $13$ card hand contains a $5$ card straight.
2
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1answer
47 views

Counterexample to Jensen's Inequality when the convex function admits values in the extended real set

Let $(X,A,\mu)$ be a set, a $\sigma$-algebra and a measure. Suppose that $\mu(X) = 1$. Let $u : X \rightarrow {\mathbb{R}}$ and $f : \mathbb{R} \rightarrow \mathbb{R} \, \cup \, \{+\infty \} $ be an ...
1
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0answers
32 views

simple walk on $\mathbb{Z}$

Consider the simple random walk $(X_n)_{n \in \mathbb{N}}$ starting from $X_0 = 0$. Consider $\varepsilon>0$, show that, for all $\delta>0$, $$ \lim _{n \rightarrow \infty} \mathbb{P}\left(\frac{...