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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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Total Probability Theorem over multiple variables

Let $X$ and $Z$ be continuous random variables. Let a conditional PDF be defined as $f\left(X\big|Z\right)$. To use total probability theorem (TBT), we introduce a discrete random variable $\Theta$ as ...
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2answers
19 views

How to use the law of total variance

I know that the law of total variance states $$Var(X)=\Bbb E[Var(X|Y)]+Var(\Bbb E[X|Y])$$ But how does one treat $Var(X|Y)$ and $\Bbb E[X|Y]$ as random variables? For example, say we know that $$\Bbb ...
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1answer
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Madison has a choice of two entrees (soup or salad), three main courses (fish, chicken or steak) and three desserts (ice-cream, lemon tart or cheese).

Madison has a choice of two entrees (soup or salad), three main courses (fish, chicken or steak) and three desserts (ice-cream, lemon tart or cheese). c. Suppose that Madison has the choice to omit ...
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1answer
17 views

Picking marbles at random from one bin vs either bin

So here is a basic probability problem and its answer which I extracted from a MIT tutorial: Suppose that in front of you are two bowls, labeled A and B. Each bowl contains five marbles. Bowl A has 1 ...
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1answer
18 views

Finding expected value when conditional distribution is known

If the distribution of $Y$ conditional on $X=x$ is known, and the distribution of $X$ is known, what would be the general process for finding the expected value $\Bbb E[Y]$? IS there a general ...
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0answers
11 views

Upperbound on expectation of supremum

If $X$ is a supermartingale. I need an upper bound as: $\mathbb{E}[\sup_{-\tau\leq\theta\leq0}\|X(\theta)\|^k]\le K \sup_{-\tau\leq\theta\leq0}\mathbb{E}[\|X(\theta)\|^k], $ where $\tau>0$ and $...
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1answer
21 views

Moment generating function of two variables

I am able to do all the parts except the very last. I have been trying to coax the differential equation $\frac{M'}{M}=t$ or something to that effect but I don't see how I can achieve this. Hints ...
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5answers
226 views

Why probability of picking a random prime is 0? [duplicate]

"It's well known that there are infinitely many prime numbers, but they become rare, even by the time you get to the 100s," Ono explains. "In fact, out of the first 100,000 numbers, only 9,592 are ...
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How Abraham De Moivre prove $2pqn$ $b(n,p,np)$ [on hold]

$E(|x-np|)= ∑|i-np|b(n,p,np)=2p q n$ $b(n,p,np)$,why ?
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1answer
11 views

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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33 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
20 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
23 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
33 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
23 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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2answers
23 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
73 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
23 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
13 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
27 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
47 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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3answers
35 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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0answers
36 views

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

Given a random variable $X$ valued on $[-1,1]$ with mean zero. We can use say Hoeffding's Lemma to get $$ \mathbb E[e^{\lambda X}] \le e^{\lambda^2/2}$$ I believe this bound cannot be improved much ...
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1answer
31 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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0answers
26 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
18 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
28 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
47 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
28 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
40 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
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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
24 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
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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
43 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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0answers
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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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0answers
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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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0answers
39 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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0answers
31 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
26 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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0answers
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
83 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
53 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 ...