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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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how Abraham De Moivre proofed 2pqnb(n,p,np)

$E(|x-np|)= ∑|i-np|b(n,p,np)=2pqnb(n,p,np)$,why?
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About Weak union of conditional independence

Problem is: Let $(X,Y,W,Z)$ be disjoint sets of random variables each with finite space. Prove that : If: $X$ is independent of $(Y,Z)$ given $W$ then we have : $X$ is independent of $Y$ given $Z,W$...
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25 views

What is the probability that the second player will win in this random game?

I'm working through Stanford's CS109 Problem Set 1 on Probability for Computer Scientists, and on #15 there's an extra credit question I can't explain: Consider a game that uses a generator ...
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2answers
18 views

Put trivariate PDF in terms of bivariate PDFs

Let there be the random variables $X$, $Y$, and $Z$. Let all the bivariate PDFs $f_{X, Y}$, $f_{X, Z}$, and $f_{Y, Z}$ be known. Can we write the unknown trivariate PDF $f_{X, Y, Z}$ in terms of the ...
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1answer
22 views

Inconsistent answers from inferring probability of success from probability of failure

Alright so I was working on my previous post and stumbled into a problem. Say the $P(A$) failing is $0.02$, which translates to $2\%$ failure rate. Say the P(B) failing is 0.003, which translates to $...
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1answer
29 views

Why is $Y_n$ not a bernoulli process?

for $n \in \mathbb{N}$, let $X_n$ be a Bernoulli process with parameter $p = \frac12$, let $N = \min \{n \geq 2: X_1 \neq X_n \}$ for $n \in \mathbb{N}$, let $Y_n = X_{N +n -2}$. in a question it ...
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1answer
21 views

Probability of one alternatively numbered die rolling higher than another differently numbered die?

Given two alternatively numbered dice: Die A: $2\ 2\ 4\ 4\ 5\ 6$ Die B: $3\ 3\ 3\ 4\ 4\ 5$ How can I calculate the probability that Die A will roll higher than Die B for this pair or any other pair ...
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1answer
17 views

1st Yr Statistics Question: Create an approximate $\alpha$ level test of $H_0 : p_1 = p_2$

Let $X_1$ and $X_2$ be binomial random variables with respective parameters $n_1, p_1$ and $n_2, p_2$. Show that when $n_1$ and $n_2$ are large, an approximate level $\alpha$ test of $H_0 : p_1 = p_2$ ...
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2answers
38 views

What is the formalism that allows Random Variables to be treated algebraically like real or complex numbers?

We all know that if we have a variable x, then there is a meaning to - for example - $$y=e^x$$. And we all know how to manipulate that algebraically and to do calculus. For example, if $$y_1=e^{...
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2answers
22 views

Probability of stacked medication working

Say we're talking about contraception, and the probability of one contraceptive, $A$, working is $99\%$, and the other, $B$ is also $99\%$. What is the probability of them working using both at the ...
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1answer
12 views

Expectation of infimum of records

I'm trying to prove the next: Let $$L_{1}=\inf\{j\geq 2: X_j\space\text{is a record}\}.$$ Prove that $E(L_{1})=\infty.$ Here, we say $X_n$ is a record if $X_n>\max\{X_2,\ldots,X_{n-1}\}$ and $\{...
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1answer
18 views

Interpretation Weak Law of Large Numbers as Corollary of a convergence theorem in Hilbertspaces

I'm trying to solve an excerise in my probability book that states the following Show that for bounded orthogonal vectors ${ x }_{ 1 }+...+{ x }_{ i }$ in a Hilbert Space H the sequence $\frac { { x }...
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1answer
44 views

$Y = \frac{X_1 X_2}{X_3}$ where $X_i$ is a uniform random variable

$Y = \frac{X_1 X_2}{X_3}$ where $X_i\sim U(0,1)$ and $X_1,X_2,X_3$ are i.i.d I need to calculate $Var(Y)$ and $Var[Y|X_3=1.7]$ I know that for each $X_i$, $E[X_i]=\frac{1}{2}$ $Var[X_i]=\frac{1}{...
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2answers
25 views

Find pmf given a probability function

I'm learning probability theory and I am quite new in the concept. I'm stuck with the following problem: Consider a situation where people often get bitten by dogs (just as an example). Let $p_A(n)$ ...
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28 views

How big are the exponential moments of a truncated normal distribution?

Given a random variable $X$ valued on $[-1,1]$ say we can use Hoeffding's Lemma to get $$ \mathbb E[e^{\lambda X}] \le e^{\lambda^2/2}$$ I believe this bound cannot be improved much if for example $X$...
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1answer
20 views

Expected value of general diffusion

lets take measurable $b,\sigma:\mathbb{R}^+\times \mathbb{R}\to\mathbb{R}$ and consider the SDE $$dX_t=b(t,X_t)dt+\sigma(t,X_t)dB_t$$ with $X_0=x$. How can i use Itô's Lemma to show $$E_x[X_t-x]=t\...
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24 views

Rank-aware probability of meeting in knockout tournament

In a knockout format tournament of $2^n$ players, assume that the bracketing is chosen randomly. That is, the bottom row of the usual binary tree pairing diagram is selected as a permutation of the ...
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0answers
15 views

The probability of the substring entering the string for each possible value

Suppose we have random generated string of 4 characters $\{A, C, G, T\}$ with probabilities for each letter $p_A, p_C, p_G, p_T$ respectively, with maximal length of $1 \leq k \leq 100$ and $0 \leq ...
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1answer
27 views

Is the sum of two (non-real) random variables necessarily a random variable?

Please note that I'm working with the following definition of random variable, which allows for a codomain other than $\mathbb{R}$. Definition: Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a ...
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1answer
46 views

Tossing Coin Expected Value Against Intuition

I have the following two coin toss games: Game 1: A and B tosses a coin. At first the coin is unbiased. Through the game, if heads comes A wins and game stops. If tail comes, the coin is swapped with ...
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1answer
27 views

Properties of the probability generating function

My understanding of PGF is that it is just an efficient way to find the properties of the distribution (mean variance etc) and represents the whole distribution, and there isn’t really any meaning to ...
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2answers
27 views

The ratio between positive/negative numbers

I am creating a model that compares multiple data points of two attributes and compares them vs one another as a percentage of 100. So for instance, comparing the values A=1.24 v B=0.44 would be: ...
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1answer
38 views

How can I prove that these events are independent?

I have a pair of sets: $A=\{n\in\mathbb{N}\mid p\cdot n\}$ $B=\{n\in\mathbb{N}\mid q\cdot n\}$ Where $p$ and $q$ are two different prime numbers. And the following event definitions: $X_n$: $n\in{...
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0answers
18 views

Conditional Joint probability of three random variables

I have encountered a problem in my work where x,y,z are my independent exponential random variables. I need to find out the probability given I have the limit for random variable z which is $(z \lt \...
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1answer
30 views

Finding probability distribution of $X$

In the box, there are 3 red balls and 3 blue balls, and from this box, the extraction of the ball is continued until the blue ball comes out. $X$ denotes extracted balls until blue ball comes out. ...
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1answer
22 views

Compute the posterior on a Gamma distribution with a gaussian random variable

i have the following problem: given a variable $x \in \mathbb{R}$ drawn from a Gaussian distribution with known mean $\mu$ and unknown precision $\tau$ (the inverse of the variance). So: $$p(x \...
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0answers
20 views

Proving expectation and variance of a function of a random variable tends to a fix point

Given $f:\mathcal{X} \rightarrow \mathbb{R}$ is a continuous function and $\mathbb{E}_{Q(X)}[X] \rightarrow x^\star$ ($x^\star$ is a fix number), $\mathbb{V}\text{ar}_{Q(X)}[X] \rightarrow 0$. How can ...
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0answers
23 views

Bound for the expectation of a function of a Gaussian random vector

A paper I am currently reading seem to be using the bound $$ E_h\left[\inf_{\|z\|_{\infty}\leq t} \|z-h\|^2\right] \leq n\frac{1}{\pi}\frac1t e^{-t^2/2},$$ where $n$ is the dimension of the vector, ...
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2answers
42 views

60 students sitting at 5 tables randomly

There are $60$ students from 12 different grade years, $5$ students from each year. Also, there are $5$ tables A,B,C,D,E. Tables A-D have $14$ seats and E has $4$ . Students sit randomly to the ...
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13 views

Proving convergence of expectation and variance given Rényi's $\alpha$-divergence tends to 0

I denote $p, q$ as density function of $P, Q$. Given $Y, X$ are random variables and \begin{align} \int q(x)\mathbb{D}_{\alpha}[p(Y\mid X=x)\,||\,p(Y\mid x^\star)] \,dx \rightarrow 0 \end{align} ...
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1answer
33 views

Probability Puzzle: 9 People and 9 Seats (Seeking a Better Method)

Q: There are 3 universities- University A, B & C. Each university will send 3 students, where the 3 students from each university represent the faculty of Health, Engineering and Law to ...
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12 views

Queuing Theory: How to calculate probability of a customer joining a queue via Erlang C Curve

I am attempting to calculate the the probability of a customer joining a queue by using the Erlang C curve in an M/M/m queue, but I am struggling to get to the final answer. In the question I am ...
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32 views

Right inverse of the Poisson CDF

For an exercise, I have to find the right inverse of the Poisson CDF $$F_X = e^{-\lambda} \sum_{i=1}^{\lfloor k\rfloor }\frac{\lambda^i}{i!}$$ where the right inverse is: $$F_X^{-1}(p):=\text{inf}\{x ...
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29 views

Showing uniform distribution on (0,1) of a random variable

Let $X,Y$ be two random variables and $Z:=(X,Y)$ and $Z\sim \mathcal{U}([0,1]^2)$. Show: $X\sim \mathcal{U}([0,1])$. The density of $Z$ is $1_{[0,1]\times[0,1]}(x,y)$ (indicatorfunction) and I want ...
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2answers
20 views

Can Conditional Expected Value be negative in normal distribution?

So, the problem gives me this facts (for a Normal bivariate distribution X,Y) $$Var(Y|X=x) = 5$$ $$E(Y|X=x) = 2 + x$$ It asks me to find $$E[Y^2|X=7]$$ I tried this: using the conditional variance ...
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1answer
25 views

Calculating Probability of an event and its expected value over a large number of independent events

So I am in a casino betting on Black in an American Roulette wheel. Scenario1: I bet for successive 1000 bets starting with a bet of 1 dollar. If I loose, I bet the double of amount that I bet ...
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2answers
36 views

Why does $\sum_{n=1}^\infty \frac{1}{n(\log(n))^{1+2\epsilon}}$ converge?

I am looking through examples on convergences of random series, and in one of the proofs the following result is used: If $\epsilon > 0$ then $$\sum_{n=1}^\infty \frac{1}{n(\log(n))^{1+2\epsilon}}&...
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1answer
19 views

Sigma algebra field for a toss of two coins [on hold]

What are the sigma fields over the sample space of a toss of a two coins
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1answer
13 views

Poisson process time of arrival proposition

This is taken from a stochastic processes text. Consider $N$ a Poisson process (number of arrivals at time $t$) with rate $\lambda$. Let $T_n$ denote an arrival time ($n$ a natural number). The ...
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33 views

About two events being mutually exclusive and independent give that one of the event's probability is zero.

I'm studying statistics and probability using an introductory textbook and it had this question: ...
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3answers
82 views

Why is $y$ separated into two intervals?

So, here's a question and a solution to part b). I do not understand why they make $y^{1/2}$ belong to interval $[0,1)$ and then separately to the interval $[1,3)$.
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2answers
52 views

The probability of meeting in a tournament, version 2: fixed ranking

This is a variation on this question $2^n, n\in\mathbf N$ tennis players compete in a tournament. In the first round, they partition into a set of $2^{n-1}$ disjoint pairs. The two players in each ...
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1answer
70 views

an exam has 50 multiple choice questions with 5 options

An exam has 50 multiple choice questions. Each question has five answer options and each question has 2 grades A-. Assuming that "a student" has no prior knowledge and randomly guess on all questions ...
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16 views

Stochastic population survival [on hold]

Suppose there exists a species of bug such that for each time discrete time period that passes, the bug has a probability $\frac{1}{3}$ of dying, and a probability $\frac{2}{3}$ of reproducing a new ...
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1answer
19 views

Notations question regarding the equation for sample mean and variance

Let $X$ be the outcome of a chance experiment with $E(X) = \mu$ and $V(X) = \sigma^2$. When $\mu$ and $\sigma^2$ can be estimated by repeating the experiment $n$ times with outcomes $x_1, x_2,...,x_n$,...
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3answers
37 views

Probability of 1 bucket being empty and one bucket having two balls with 7 buckets and 7 differently-colored balls overall

Idea is to get a string that is a permutation of $(0211111)$. I had two approaches, but both seem to miss by two orders of magnitude (expected answer is 32%). In the first one, I chose 2 balls out ...
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1answer
23 views

Random Variable Y following uniform distribution with parameter Random X that follows geometric.

Random variable X follows geometrical distribution with p=1/4. Random variable Y follows uniform distribution in [-X,X]. I'm looking for P(Y>3/2) and also P(X=2|Y>3/2).I know for a fact that Σ(from k=...
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0answers
10 views

calculate call by value of a bond

Matt purchased a 20 year par value bond with annual coupon rate of 8% payable semiannually at a price of $1722.25$. The bond can be called at par value $X$ on any coupon date starting at the end ...
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3answers
30 views

Probability of 6 people in a lift getting of at the same floor in a building with 10 floors

My reasoning is that there are 10 different events that match our desires (because there are 10 floors), and that each person chooses one of the floors to get of on, and hence $$P=\frac{10}{6*{{10}\...
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2answers
22 views

Let $T$ be a normal random variable that describes the temperature…

Let $T$ be a normal random variable that describes the temperature in Rome on the 2nd of June. It is known that on this date the average temperature is equal to $µ_T = 20$ centigrade degrees and that $...