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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Two definitions of recurrent and transient stochastic states

On a finite state space $\{1,\cdots,M \}$, I have seen two definitions of a transient/recurrent state for a Markov Process. Let $p_n(i,j) = \mathbb{P}(X_n=j | X_0=i)$. Then $\sum\limits_{n\ge 0} p_n(...
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1 answer
25 views

determine $f_Y$ given $f_X$, cdf then pdf or inverse function?

I have this problem to determine a density function $f_{Y}(y)$ of $Y$ given $X$ as a random variable with uniform distribution on $[−1, 1]$ and $Y = X^2$. What is the simplest way to approach this ...
0 votes
0 answers
17 views

Limit of Logarithm In Probability Space

Cannot solve $$S(f)=\lim_{\epsilon \to 0} \int f \log(f+\epsilon) d\mu$$ for $\mu$ probabilty measure and $\int f d \mu =1$ How can I apply Jensen's to show $$\int fg d\mu - S(f) \leq \log \bigg[\int ...
1 vote
1 answer
24 views

Sample a specific number of elements from a list with matching probabilities

Say I have a list of $n$ elements, $x=[x_1,\dots,x_n]$, and a list of $n$ probabilities $p=[p_i]$. I want to sample $c$ elements of the list without replacement, where $1<c<n$, and I want the ...
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0 votes
0 answers
7 views

Show that $c$-approximately strongly universal hash function implies $c$

Let $[m] = \{0,1,\dots,m-1\}$ and $1< m < |U|$. Define a $c$-approximately universal hash function $h: U \to [m]$ as a function where $$ \text{Pr}\left[h(x) = h(y)\right] \leq c/m $$ for some ...
1 vote
0 answers
24 views

Birthday paradox - variance, parallelisation, simple proofs?

Suppose we sample uniformly random elements from a set of cardinality $n$, and save them in a table. We continue doing this process (each sampling is one step) until we get a collision. What is the ...
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1 vote
0 answers
21 views

Inequality in i.i.d random variables

Let $X_1, X_2, \dots, X_n$ be i.i.d random variables. Suppose $(\mathbb{E}|X_i|^p)^{1/p}=M < \infty, 1 < p < 2$. Let $X_i=X_i\mathbf{1}_{|X_i| \leq c}, Y_i = X_i\mathbf{1}_{|X_i| > c}, \mu=...
2 votes
0 answers
11 views

Transformation of Variables as Marginalisation of Joint Distribution - Where am I going wrong?

I am trying to derive some equivalent of the transformation of variables formula. That is, given a random variable $Y=g(X)$, where $g$ is an invertible function, then the pdf of $Y$, $f_Y(y)$, is ...
0 votes
1 answer
16 views

Inequality between random variables and its implications

Let $0 \leq X_1, X_2 \leq 1$ be two random variables I have an expression that looks like this: $$\left| |t| - |X_1 -X_2|\right|$$ where $t$ is some constant. For any particular realisation of $X_1, ...
1 vote
0 answers
53 views

Toy example of superdeterminism using Rule 30

From what I understand of Bell's Theorem, it requires giving up local realism or embracing superdeterminism. I still haven't been able to understand why superdeterminism gets such a bad rap, so I've ...
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1 vote
2 answers
27 views

Conditional probability with statistics

$X_1, X_2,..., X_{16}$ are observations with normal distribution $N(\mu, \sigma^2)$. We have two statistics: $$ \overline{X}=\frac{1}{16}\sum_{i=1}^{16}X_i$$ $$ S^2=\frac{1}{15}\sum_{i=1}^{16}(X_i-\...
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0 answers
13 views

How to select next state in Q-learning based on Markov chain?

I am trying to find the next state in order to update the $Q$-table based on Markov chain as shown in the attached image. For example, suppose my current state ($s_t$) is 2 then what would by my next ...
0 votes
2 answers
32 views

Probability the minimum is three when three dice are rolled

I want the probability that the minimum is three when three dice are rolled. First, we have $6 \cdot 6 \cdot 6$ configurations. Then, if the minimum is three we have to select $3,4,5$ and $6$ on the ...
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0 votes
0 answers
21 views

Expected number of elements greater than randomly chosen element

Let $S = \{x_{1},x_{2},...,x_{n}\} $ be a multiset of real numbers. Let $X$ be a number sampled uniformly from $S$ and let $Y$ represent the number of elements in $S$ that are greater than or equal to ...
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-2 votes
0 answers
16 views

$(X_{n})$ random walk, and $M_{n}=\text{max}\{X_{0}, ..., X_{n}\}$. Is it that $\mathbb{P}(M_{n}\geq k,X_{n}\leq k-j)=\mathbb{P}(X_{n}\geq k+j)$?

$(X_{n})$ is the simple random walk on the real line starting from $0$, and $M_{n}=\text{max}\{X_{0}, ..., X_{n}\}$. Is it that $\mathbb{P}(M_{n}\geq k,X_{n}\leq k-j)=\mathbb{P}(X_{n}\geq k+j)$?
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0 answers
9 views

An upper bound of the probability involving Bernoulli on the power (generalized)

I asked about a similar problem in this post yesterday. sudeep5221's answer is very neat. However, I'm working on a generalized version which seems cannot be solved using similar factorization ...
0 votes
0 answers
13 views

The relationship between Spearman coefficient and Pearson Coefficient

The Spearman coefficient is defined as following:$r_s = 1- \frac{6\Sigma d^2}{n(n^2 -1)}=1$ and the Person Coefficient is given by $r_p=\frac{\sum ^n _{i=1}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum ^...
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0 votes
2 answers
27 views

Using characteristic functions, find the pdf of $Z = X + Y$

A R.V. $X$ has a pdf of the form $f_{X}(x) = e^{-x} u(x)$ and an independent R.V. $Y$ has a pdf of $f_{Y}(y) = 3e^{-3Y} u(y)$ using characteristic functions, find the pdf of $Z = X + Y$. This is ...
0 votes
1 answer
27 views

How to write proof using the symetrical structure of the equation?

I encountered a problem recently, Let $f(x_1,x_2,x_3)=e^{-(x_1+x_2+x_3)},0<x_1,x_2,x_3<\infty$ be the joint pdf of random variables $X_1,X_2,X_3$. Find $P(x_1<x_2<x_3)$. I want to state ...
0 votes
1 answer
20 views

Multiple steps of branching probabilities

There are 4 urns. urn A has 2 black balls and 6 white balls urn B has 4 black balls and 4 white balls urn C has 6 black balls and 2 white balls urn D has 8 black balls You choose an urn at random with ...
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1 vote
2 answers
20 views

Joint distribution of $\max(X, Z)$ and $\max(Y, Z)$ where $X$, $Y$, $Z$ are independent exponential variables with mean $1$

Assume there are three independent exponential random variables $X$, $Y$, and $Z$ with mean $1$. I am trying to look at the joint distribution of $[U = \max(X, Z), V = \max(Y, Z)]$. I want to find $F_{...
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2 votes
0 answers
40 views

Covariance of $X$ and $Y$ is 0

Let $X$ be a random variable. Is it possible to construct a random variable $Y=g(X)$, with $g$ strictly monotone, such that $\text{cov}(X,Y)=0$?
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4 votes
1 answer
18 views

Convergence of moments

Suppose $\left(X_{n},n\in \mathbb {N}\right)$ is a sequence of random variables taking values in $[0,1]$. Suppose that for every $k\in \mathbb {N}$, $$ \lim_{n\to \infty}\mathbb {E}\left(X^{k}_{n}\...
0 votes
0 answers
20 views

How to select next state using Markov chain?

the Markov chain in my case is as shown in the image attached My query is I am not getting what will be my next state as per the given Markov chain if my current state is 2. Any help in this regard ...
0 votes
0 answers
14 views

Symmetric Random Walk of dimension $d \geq 3$ returns infinitely many times with probability 0

Prove that almost surely a symmetric random walk defined in $Z^d$, $d \geq 3$, will return finitely many times (including the case that the random walk does not return). It is well-known that a 3-...
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0 votes
1 answer
29 views

notaion of joint distribution of mixed random variables

I have some random variables $\mathbf{X}_1,...,\mathbf{X}_N$, where $N$ is a discrete random variable and $\mathbf{X}_i$ are i.i.d coutinuous random variable. What notation should I use for the joint ...
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0 votes
1 answer
46 views

Limit of log in positive measure space

Let $\mu$ be a probability measure, $f$ is non-negative integrable function such that $\int_X f d \mu = 1$. Prove that the limit exists for: $S(f) := \lim_{\varepsilon \downarrow 0} \int_X f \log(f+\...
0 votes
0 answers
27 views

For $V = \min(X, Y)$, calculating the CDF, is it $\mathbb{P}(V \leq k)$ or $\mathbb{P}(V \geq k)$

I have this variable $V = \min(X, Y)$, where $X$ and $Y$ are random variables geometrically distributed and independent. I want to compute the CDF. My attempt: Let $k \in V(\Omega) = \mathbf{N}^*$. We ...
1 vote
0 answers
60 views

Getting the wrong answer on what seems like the most basic probability test

Source (Appendix 2). In 5 interactions between individuals i and j, i wins 4. $n_{ij} = 5$ (number of interactions), $s_{ij} = 4$ (number of successes), and $P_{ij} = 4/5$ (proportion of successes). $...
1 vote
1 answer
40 views

Understanding a proof in Probabilities with Martingales A1.5.

I have trouble understanding a detail in the proof of the following theorem: In particular, the proof (which I adjoin at the end) establishes the identities \begin{equation} \begin{split} \lambda(L^c\...
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0 votes
2 answers
27 views

Express probability of an event in terms of some variable

Three people $A,B,C$ go to a shop and buy either pen or notebook. Three choices are mutually independent and each person buys pen with probability $p$. Let $X$ be the event that at least two of them ...
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0 votes
1 answer
17 views

How does one create a discrete function where the probability of an outcome continuously changes with respect to time?

For example, a fair coin would have a sample space = {H, T}. A particular probability function could take on different probability values (in the interval $[0,1]$) relative to time, such function ...
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0 votes
1 answer
48 views

Upper Bound on Expectation of a Random Variable

Let $X_1, X_2, \dots, X_n$ be i.i.d random variables. Suppose $(\mathbb{E}|X_i|^p)^{1/p}=M < \infty, 1 < p < 2$. Let $X_i=X_i\mathbf{1}_{|X_i| \leq c}, Y_i = X_i\mathbf{1}_{|X_i| > c}, \mu=...
0 votes
1 answer
29 views

Can we find $C>1$ so that $ P(|X|\le \frac{\epsilon}{C})\ge 1-\delta $?

Fix $\epsilon>0$. Consider two $n-$dimensional random vectors $u$ and $v$ uniformly distributed on the sphere. Define $X_n :=u\cdot v$. Note that as $n\to \infty$, $\sqrt{n}X_n \to N(0,1)$ by Why ...
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4 votes
2 answers
73 views

Optimal stategy for 2-players d6 game?

I came across the following problem when preparing for an interview: Two players each roll a d6, and are not able to see each other's rolls. The player with the higher value wins 1\$ (no win in case ...
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0 votes
2 answers
44 views

Coin toss where $Pr(H)=1/3$

Question: A coin is tossed where the probability it lands on heads is $1/3$. What is the expected number of heads before tails? My answer: number of heads before tails = $\frac{1}{3}^1+\frac{1}{3}^2+\...
1 vote
0 answers
35 views

Probability of partially sortedness of a perfectly shuffled array of unique numbers

Sorry that I don't know any better way to express this question. Assuming that we have a perfectly shuffled array of unique numbers, we know that the array is in fact consisted of ascending or ...
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0 votes
0 answers
19 views

Series vs parallel system reliability

I'm working on system reliability with dependent components and I'm not sure if my formula is mathematically correct. We consider that a series system as the system reliability given by $$Rs =Pr(Y<...
0 votes
0 answers
24 views

Probability of A, given P(A∩B), P(A'∩B), P(A∩B') and P(A'∩B')

I have been given: P(A ∩ B) P(A' ∩ B) P(A ∩ B') P(A' ∩ B'). I have to derive the formula for P(A) from the given probabilities, which are independent. I assumed: P(A ∩ B) = a P(A ∩ B') = b P(A' ∩ B) = ...
1 vote
0 answers
28 views

Can $P(X \leq M)$ be expressed in terms of $P(X \geq M)$?

In seismology (note that I am not an expert in this field), the occurrence of a magnitude M earthquake or greater can be modeled using the Gutenberg-Richter law: $N_M = 10^{a-bM}$ Where: $M$ = ...
0 votes
0 answers
30 views

Can prove $ P\left(\frac{X_1^2+X_2^2+\dots+X_k^2}{X_1^2}>\frac{1}{\epsilon^2}\right)>1- \delta $?

Given i.i.d. Gaussian random variables $X_1,\dots, X_n$ with $N(0, 1/n)$. Fix $\epsilon\in (0,1)$, can prove that for any $\delta>0$ there exists $1\le k=k(\epsilon, \delta)\le n$ so that $$ P\...
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2 votes
1 answer
31 views

Show that $\sqrt{n}(\overline{X}_n^2 - \mu^2) \to 2\mu|\sigma Z$ in distribution

(Full disclosure--this is a homework problem.) Let $(X_n)_{n=1}^{\infty}$ be a sequence of i.i.d. random variables with mean $\mu$ and variance $\sigma^2 < \infty$. I want to show that if $\mu \neq ...
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-1 votes
0 answers
58 views

What is $P(X \leq M)$ given $P(X \geq M)$?

In seismology (note that I am not an expert in this field), the occurrence of a magnitude M earthquake or greater can be modeled using the Gutenberg-Richter law: $N_M = 10^{a-bM}$ Where: $M$ = ...
0 votes
1 answer
27 views

Computing expected values with regard to beta distributions

Can someone help me compute the following expected value? Considering $q_1(\theta_1) = Beta(\alpha_1,\beta_1)$,$q_2(\theta_2) = Beta(\alpha_2,\beta_2)$ and $q_3(\theta_3) = Beta(\alpha_3,\beta_3)$. ...
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0 votes
0 answers
9 views

Definition of neighbouring dataset (Differential Privacy), dependent on scale of dataset?

I am having trouble understanding how the # of features (i.e. columns) and the scale of those features in a dataset doesn't impact the definition of 'neighbouring dataset' in differential privacy (DP)....
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3 votes
3 answers
81 views

Game "Guess who" with two cards

I was playing "Guess who" with my sons and a friend, and she told me that she used to play with two cards to get a "not so simple game". The questions have to be formulated in this ...
0 votes
0 answers
20 views

An upper bound of the probability involving Bernoulli on the power

I would like to find an upper bound of $\text{Pr}\{a^{u_1+u_2+u_3+u_4}+a^{l_1+l_2+l_3+l_4}\leq a^{u_1+u_2+l_1+l_2}+a^{u_3+u_4+l_3+l_4}\}$, where $a>1$ and $l_1,\cdots,l_4$ follows i.i.d Bernoulli$(...
0 votes
0 answers
22 views

Maximizing payout given two piles, followup

Here's a followup to my question here: Strategy for game with two piles, maximize the payout Say I'm playing a game with two piles, both of which are initially empty. On each turn I can perform one of ...
10 votes
2 answers
349 views

About problem A4 2022 of Putnam

I'm not passing the William Lowell Putnam competition (I live in France) but I'm still fascinated by some of the problems. This year the A4 problem challenged me and I wanna know your thoughts about ...
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1 vote
2 answers
60 views

Convolution, sum of two independent random variables $X+Y$

Let $$ f_{(X,Y)}(x,y) = 2x $$ for $x \in (0,1), y \in (0,1)$. I need to compute density of $X+Y$. So, I know that $X \perp Y$, because \begin{align} f_X(x) &= 2x, \ \ x\in(0,1)\\ f_Y(x) &= 1,...

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