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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All of Statistics example 2.11

Two people take turns trying to sink a basketball into a net. Person 1 succeeds with probability 1/3 while person 2 succeeds with probability 1/4. What is the probability that person 1 succeeds before ...
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Does the probability of a coin toss always stay 1/2 no matter the previous tosses

Assumption 1: The coin is unbiased Assumption 2: The tosses are independent Suppose i toss a coin 3 times and each time it came up heads then what is the probability that the next toss would be heads. ...
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Understanding the Probability Outcomes

A eight sided dice is rolled thrice and the first roll was an odd number. If the selected roll in random is odd what is the probability? Total number of possible outcomes when a die is rolled thrice = ...
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Given $Y_i $ with pdf $f(y) = my^{m-1}/n^m, 0<y\leq n, 1\leq i \leq k$, prove that $-k\ln (\frac{Y_{(k)}}{n})$ is Gamma$(1,1/m)$

Given $Y_i $ with pdf $f(y) = my^{m-1}/n^m, m>0, n>0, 0<y\leq n, 1\leq i \leq k$, prove that $-k\ln (\frac{Y_{(k)}}{n})$ is Gamma$(1,1/m)$. I tried finding the CDF of $f$. I used that to find ...
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Chernoff bound for binary cross entropy loss

Consider the binary cross entropy loss of an estimator of the posterior $\eta: \mathcal{X} \rightarrow [0, 1]$: $$\mathcal{L}_n(\eta) = \frac{1}{n} \sum_{i=1}^n Y_i\log(\eta(X_i)) + (1-Y_i)\log(1-\eta(...
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3answers
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Combinatorics approach for player $A$ and $B$ rolling a 20-sided die

Problem: Consider players $A$ and $B$ rolling a 20-sided die. Player $B$ is allowed to re-roll. Assume that player $B$ will re-roll a single time if his first dice roll is $\leq 10$, and will not re ...
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1answer
33 views

How to calculate the expected value of where the first dice roll is greater than the second dice roll?

Consider an $N$-sided die. I roll it twice. I want to find the expectation of the first dice roll given that it is greater than the second dice roll. So if we let $X$ and $Y$ denote the random ...
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1answer
35 views

If $\lim_{\alpha \to \infty}\alpha P[X > \alpha] = 0$ then $E[X] < \infty$?

Let $X$ be a positive random variable. Suppose that $\lim_{\alpha \to \infty}\alpha P[X > \alpha] = 0$ Does this implies that $X$ has finite expectation? that is $E[X] < \infty $ I know that if $...
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Need help with determining exact formula [closed]

We form a random sequence of length n, by selecting, with replacement from a set of 49 numbered balls (each numbered from 1-49). Each ball has an equal probability of being selected. We repeat this ...
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Finding MGF of Geometric Random Variable -1

Suppose $Y$ has a Geometric distribution with success probability $p$ and consider $W = Y - 1$. Find the MGF of $W$.
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Using coin toss to decide winner among 3 children

I have 1 chocolate and 3 children. I want to decide which child gets the chocolate using a toss of a fair coin. I also want each child to be equally likely to get the chocolate. How do I do this?
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Probability bounds using CLT [closed]

Let $X\sim Bin(100,0.5)$. Use CLT to bound the probability that $X\ge 60.$ You may use the following fact about $Z\sim N(0,1)$: $[|Z|\le1]\approx0.680$ $[|Z|\le2]\approx0.950$ $[|Z|\le3]\approx0.997$ ...
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24 views

Given a joint Laplace-Stieltjes transform, is it possible to find a joint PDF?

If we have a joint Laplace-Stieltjes transform (or MGF or characteristic function) of some independent nonnegative continuous random variables $X = (X_1, X_2, ..., X_n)$ defined on $[0,\infty)$, the ...
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1answer
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Show that $S \leq \sup _{x \geq(1+\varepsilon) M_{n}} x^{-1 / r} \cdot x^{1 / r-1} \sum_{k=1}^{n} E|X_{k}|^{r} I(|X_{k}|>x^{1 / r}) $

Let $1<r<2$ and let $\left\{X_{n}, n \geq 1\right\}$ be a sequence of pairwise independent random tariables with $E X_{n}=0$ and $E\left|X_{n}\right|^{r}<\infty$ for all $n \geq 1 .$ Set $M_{...
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Question regarding stochastic independence

Let $X,Y,Z$ be random variables and pairwise independent, i.e. $X$ independent from $Y$, $Y$ indepedent from $Z$, and $X$ independent from $Z$. I am interested in a rigorous argument, why $ (X,Y) $ is ...
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26 views

Random variable conditioned on event

What does it mean to condition a random variable on an event Z? More specifically take two random variables $(X,Y)$ taking values in $(U,V)$ and let $Z \subset U \times W$. In Villani "Optimal ...
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Combinations in Two sets of the same amount of numbers [closed]

So if you have two sets of, let’s say 3 numbers 1-2-3, one set is red and one blue, what you would do is multiply 3 by 3 to get the amount of combinations. My question is do you count red1 then blue1 ...
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Getting samples or subsets from a distribution

I probably missed some basic basic math class, but I'm struggling a bit with finding a solution for this marble problem. Suppose you want to divide $Y$ marbles into $Z$ categories. Every category has ...
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How can I prove $\left|\frac{e^{it_p x_j}-1}{t_p}\right| \leq 2|x|$?

Background: I am trying to understand why, if $X$ is a random variable with $\mathbb{E}\big[|X|\big]< +\infty$ and $\mu=\mathbb{P}^{X}$, then $$\frac{\partial}{\partial x_j}\overline{\mu}(u)=i\int ...
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please help me. I have probability generating function as follows, how can get the mixture distribution of $X$ from it?

$$ G_x (s)=\frac{s \cdot (1-q(1-α+α \cdot s))}{(1-q \cdot s)(1-α+α \cdot s)} $$ Or, in another way, how can I say that a random variable follows a certain distribution with a certain probability and ...
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Conditional variance of the sum and difference of two independent variables?

Suppose that $X$ and $Y$ are independent with mean 0, variance 1. Suppose $ A = X+Y \\ B = X-Y $ I read that $Cov(A, B)=Var(X)-Var(Y) = 0$. Can I say anything about the terms in the conditional ...
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2answers
37 views

Probability of getting a specific sequence of length $4$ in $10$ coin tosses

This is more of a thinking question maybe, hope that's ok. Suppose I toss a coin $10$ times. What is the probability that within these 10 tosses I get the sequence THHT. My attempt: If I have 10 coin ...
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1answer
29 views

Two players rolling a $20$-sided die; player B can re-roll; how to decide when to reroll

This is somewhat related to my earlier question What is the probability that player A rolls a larger number if player B is allowed to re-roll (20-sided die)? and somewhat related to 30 sided die and ...
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1answer
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How do you write the law of total expectation for a conditional expectation?

I would like to compute $E[N|D]$, expectation of some random variable $N$ given that event $D$ occurs. In the outcome space, there are a set of disjoint events $B_1, B_2, B_3$. How can I apply the law ...
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1answer
38 views

Understanding the mean function for time series analysis

I am studying time series and came across the Mean function which the textbook defines as: $$ \mu_{xt}=E(X_t)=\int_{-\infty}^{\infty}xf_t(x)dx $$ I don't understand what this function does. I looked ...
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1answer
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Probability of catching covid - statistical model

I have been thinking about this approximate model of what is the probability of you catching covid based on the assumption how many people you were in contact with that would transmit it to you, if ...
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2answers
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What is the probability that player A rolls a larger number if player B is allowed to re-roll (20-sided die)?

The problem statement is: 2 players roll a 20-sided die. What is the probability that player A rolls a larger number if player B is allowed to re-roll a single time? The question is a bit ambiguous, ...
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3answers
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Given joint pdf, how can I find $\operatorname{Var}(X)$ without using the marginal distribution of $X$?

$(X,Y)$ has joint pdf $\frac{1}{y}$ for $0<x<y<1$. I would usually use the marginal pdf to get the expected value. But the question doesn't let me use the marginal distribution of $X$. I ...
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Two players until one player wins three games in a row. Each player will win with probability $\frac{1}2$. How many games will they play?

QUESTION: Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with ...
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Ball Selection without replacement

A box contains $2$ white balls, $3$ red balls and $4$ black balls. Now I have to select $2$ balls one white and one red from the box without replacement. I'm confused between whether the probability ...
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How can I model a difference in probabilities?

I am trying to optimize the strategy to choose while taking a penalty kick. I have identified two strategies- keeper independent and keeper dependent and I am trying to find the best one using a ...
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2answers
60 views

Calculating expected value of $X$ with the density function $f(x)=16xe^{-4x}$

Suppose, $X$ be a random variable with probability density funciton, $$ f(x) = \begin{cases} 16xe^{-4x}, & x \geq 0; \\ 0, & \text{otherwise} \end{cases} $$ (source) I tried to find the ...
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1answer
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Convergence of Uniform random variables

Let $U$ be a Uniform[0,1] random variable. If we consider $[nU]$ where [] is the greatest integer function, then we know that $[nU]$ is a discrete uniform $\{1,2,...,n-1\}$ random variable. We know a ...
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1answer
44 views

Bayes' theorem and law of total probability with CDFs

Suppose $X$ has Gamma(2, λ) distribution, and the conditional distribution of $Y$ given $X = x$ is uniform on $(0, x).$ Find the joint density function of $X$ and $Y,$ the marginal density function of ...
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Gamma Random Variable Confusion

I know that the gamma random variable can be thought of as an extension to the exponential random variable. The exponential random variable measures the time it takes for one event to appear, while ...
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Box containing 4 black and 6 white marbles.

Four marbles are drawn from the box at random without replacement. Calculate the probability that at least two white marbles are drawn given that at least one of each colour is drawn. I have ...
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1answer
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If $2^d\,\mathbb{P}(\text{Bin}(n,\frac{1}{2})<k)<1$, there is a binary linear code of dimension $d$, length $n$ and minimum distance at least $k$.

Please give some comments and hints for the problem. A binary linear code of dimension $d$ and length $n$ is a code over the alphabet $A=\mathbb{F}_2^d$ defined through a matrix $M \in \mathbb{F}_2^{...
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1answer
38 views

What does area under t distribution give?

My question is really simple but i have not been able to find any satisfactory answer anywhere, hence asking here. What does area under t distribution mean? Example : for normal distribution, P(z< ...
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Justification for the calculation of expectation on the first indicator random variable of a series of mutually dependent experiments.

I am not asking for a proof of linearity of expectation, which is available in multiple posts. Instead I would like to develop a more robust understanding as to why the calculation of the expectation ...
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1answer
43 views

Can I always write $Y=E[Y|X]+\varepsilon$ where $E[\varepsilon|X]=0$?

Suppose that we have two random variables $Y$ and $X$. Is it true that we can always write $$ Y=E[Y|X]+\varepsilon $$ where $\varepsilon$ is some random variable such that $E[\varepsilon|X]=0$? I ...
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1answer
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Let $X \sim P_1$, $Y \sim P_2$, can we find $f$ such that $f(X,Y) \sim P_1 P_2$?

Let $X \sim P_1$, $Y \sim P_2$ be two random variables with respective probability laws $P_1,P_2$. We define $P(A)=P_1(A)P_2(A)$ for all $A$'s in the sigma algebra. $P$ is a probability law. Can we ...
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Finding probability of Sample standard deviation given population is normally distributed

For a random sample of size n, if the values are taken from the N(a, b ) population, what is the probability that S (where S^2 is sample variance) will exceed a particular value? I had 2 approaches in ...
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2answers
34 views

How is a Gaussian random process different from a Gaussian random variable?

I am from a physics background with lousy mathematical training. I am trying to delve into the subject of stochastic processes as applied to physics. I have a pretty naive question. A random variable $...
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Calculating the population size given a probability

I'm not too sure how to even start this question. Could someone explain how to solve this question :) Question: The minimum number of times that a fair coin can be tossed so that the probability of ...
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2answers
41 views

Finding Expected Value of Conditional Poisson Distribution

Consider a random variable X ~ Poisson (1). Namely, $P(x=k) = \frac{e^{-1}}{k!}$ , k=0,1,2,... I'm trying to solve for $\mathbb{E}\{X|X\geq 1\}$. My approach: Given that $X\geq1$ then we know that at ...
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1answer
34 views

Application of law of total probability

Consider the discrete random variables $X,Z,W$ with supports $\mathcal{X}, \mathcal{Z}, \mathcal{W}$ respectively. Let $\mathcal{W}\equiv \{w_1,w_2,w_3\}$. Take any $x\in \mathcal{X}, z\in \mathcal{Z}$...
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1answer
61 views

Markov chain magic trick (Kruskal Count)

I am trying to understand why this works. We have a Markov chain here with 10 states which are the numbers 1,..,10. There is a none zero probability to pass between any two states, so we can show that ...
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2answers
29 views

Variance of sum of independent random variables - case of undefined density

When the density $f_{X, Y}$ is not defined for independent random variables $X, Y$, is it possible to say anything beyond \begin{align} Var(X + Y) &= Var(X) + Var(Y) + 2 \int_{\Omega} (X - E(X))(Y-...
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28 views

Does a probability distribution exist given a finite number of moments?

Given a finite set of moments $(m_n)$ for $n\in V \subset \mathbb Z^+$, is there a way to determine that there exists a probability distribution $d\mu(x)$ such that $m_n = \int x^n d\mu(x)$ for $n\in ...
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1answer
44 views

Proving the variance of the distribution of $m$-fold products of elements of a generating set is asymptotic to $c^m$ without advanced tools

Let $G$ be abelian with $n$ elements and let $G' = \{g_1 = e, \dots, g_k\} \subsetneq G$ be a (not necessarily minimal) set of generators. An element $g \in G$ is obtained by independently, uniformly ...

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