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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1answer
14 views

Probability Notation “Hierarchy”

I was wondering if there's any "hierarchy" in probability notation, such as the basic "multiplication comes before summation" so that if you have: A * B + C you know that this means (A * B) + C and ...
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1answer
13 views

Covariance of max and min of two uniformly distributed random variables

Let X, Y be two independent random variables following a uniform distribution in the interval (0,1). Let U=Min(X,Y), and V=Max(X,Y). How do I find the Covariance(V,U)? Since X and Y are i.i.d., I ...
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0answers
17 views

Total possible ways + ensuring order

I was thinking a problem(concurrency related problem in operating system but involve basic maths). There are k programs each with $m_1, m_2,...m_k$ steps. Programs execute concurrently. But i need to ...
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2answers
36 views

Borel Cantelli Lemma

I started this year to study probability and i'm using Probability Oath of Resnik, i'm trying to solve this exercise but i can't see the link between the Borel Cantelli's Lemma than i studyied and the ...
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0answers
31 views

Show a function X is B-measurable [closed]

I'm a new student of probability and I'm using the book "A "robability Path"- Resnick. I've studied the concept of measurability but I don't know hot to start and solve the follow exercise. Thank you ...
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0answers
25 views

Is my proof showing that this Markov chain is irreducible correct?

I'm trying to show the irreducibility of the Markov chain of the simple random walk on $\Bbb Z$, defined as: $$P_{i,j}=\begin{cases} \ p , & \text{if $j=i+1$} \\ (1-p)=q , & \text{if $j=i-1$}...
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1answer
15 views

Number of parameters in joint distribution of n dependent Bernoulli r.v.s

I am trying to understand the result in this thread (Dependent Bernoulli trials), which says the number of parameters in a joint distribution of n dependent Bernoulli random variables is $2^n - 1$. ...
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0answers
23 views

Probability and Combinations of outcome of separate events [closed]

Our team is a recruitment firm, we are trying to figure out a formula for the following If we submit 10 resumes to 10 different employers what are the odds of each resume being rejected or ...
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1answer
23 views

$L^p$ norm inequality between random variable and conditional expectation (hypercontractivity)

Suppose we have two random variable $X$ and $Y$ and $1\leq q\leq p$ such that for any $Y$-measurable set $A$, \begin{align*} &\| \mathbb E[1_A(Y)|X] \|_p \leq \| 1_A(Y) \|_q\\ \Leftrightarrow&\...
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2answers
25 views

In a survey, how many people did not like either orange or apple juice (simple sets and probability question)

I came across the following sets and probability question and I don't particularly agree with the book's answer. I am interpreting the or as a union $\cup$. The problem The book's answer: $25$ My ...
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1answer
38 views

Find the law of a random variable

Let $X$ be a discrete random value taking values in $\mathbb{N}^* = \left\{1, 2, 3, \ldots \right\}$, and such that $\exists p \in (0,1) \forall n \geq 1$: $$ \mathbb{P}[X=n] = p \mathbb{P}[X \geq ...
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0answers
18 views

How to evaluate the log posterior and find the new parameters $\bar \mu, \bar \sigma ^2$

Given that the model distributon is Gaussian with known variance $\sigma^2 = 1$ $$ p(x_i | \mu, \sigma^2=1) = \frac{1}{\sqrt{2\pi}}\exp{-\frac{(x_i - \mu)^2}{2}} ,$$ the log of the data likelihood $\...
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1answer
15 views

Explanation of proof of theorem of hitting probabililties

I dont quite understand the second part of the proof of this theorem: The vector of hitting probabilities $h_A = (h_A(x) : x ∈ S)$ is the smallest nonnegative solution to the system of equations $f(...
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1answer
19 views

Percentile of positive values of max(0,X-500) given the CDF of X

$X$ follows a distribution with distribution function $$F(x)=\begin{cases}1-\left(\frac{2000}{2000+x}\right): &x\ge0\\ 0: &\text{otherwise}\end{cases}$$ Let $Y=max(0,X-500)$. Calculate the $60^...
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1answer
38 views

Different answers on a Conditional Density problem

Here's a joint density function for you: $f(x,y) = \begin{cases} \frac23(x+2y): &0\le x \le 1, 0 \le y \le 1 \\ 0: &\text{otherwise} \end{cases}$ I want to calculate the probability that $X +...
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3answers
37 views

Can we write Probability of Binomial distribution $p(x)=n_{c_x}{p^x}(1-p)^{(n-x)}$ As $p(x)=\frac{no. of successes}{no. of outcomes}$ [closed]

can anyone suggest how to calculate the binomial probability using $p(4\text{ redballs})=\frac{\text{no. of successes}}{\text{no. of outcomes}}=0.02835$; Example: There are 6 red balls and 14 yellow ...
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1answer
27 views

Branching Process - Extinction probability geometric

Consider a branching process with offspring distribution Geometric($\alpha$); that is, $p_{k} =α(1−α)^k$ for $k≥0$. a) For what values of $α ∈ (0, 1)$ is the extinction probability $q = 1$. b) Use ...
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1answer
43 views

Could someone point to flaws in my logic: How many people does a capricorn need to meet at random in order to find AT LEAST one more capricorns? [closed]

with 50% certainty full certainty I have computed the answer myself. I don't understand too much about the complexities of probability but the basics, and multiplication principle. My calculations ...
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3answers
46 views

Is there any advantage in using odds in probability ad statistics?

The question title says it all. I have always wondered why in some areas, mostly gambling, people use odds instead of probability. But I have never seen a probability/statistics book (at least one ...
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2answers
33 views

Joint entropy of 2 independent random variables

Say we have two independent random variables $X$ and $Y$. What is their joint entropy $H(X,Y)$? I worked this out, but I am not sure if the result I reached is correct. The definitions of entropy ...
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1answer
18 views

Implications of convergence in quadratic mean over convergence in the 4-th mean

I am trying to understand whether the convergence in quadratic mean of a sequence $X_n$ to some $X$ has implications over its own convergence in the $4$-th mean to the same $X$. Are there any ...
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1answer
34 views

A basic question on random variables

Consider the following: $x$ is an instantiation of a Bernoulli random variable over $\pm1$. $y: y^2-1=0$, so $y=\pm1$. How do we explain the difference between $x$ and $y$ to a high school student? ...
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0answers
33 views

Geometric intuition behind Monte Carlo integration

I found 2 seemingly different explanations of the geometric intuition behind Monte Carlo integration. Watch from 4:51 of this video by Jared Niemi. https://youtu.be/MKnjsqYVG4Y There is a set of 4 ...
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2answers
53 views

probability of the $n$-th ball drawn is white is same as probability of first ball drawn

Consider an urn which initially has w white balls and b black balls. Draw one of the balls in the urn at random, then put this ball back into the urn and add another ball to the urn of the same color ...
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2answers
24 views

Integers and probability

Started my first Probability class and already struggling. Any help/pointers with these questions? Choose one number from {1.. 1000} randomly. What is the probability that the chosen number is: a) ...
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1answer
33 views

How to prove that pairs of probability are independent?

Given the following probabilities: p(A|C) = 0.5, p(A|C') = 0, p(B|C') = 0.6, p(B'|C) = 0.4, p(C) = 0.3 How do I prove which of the pairs containing A,B and A,C are independent, without having any ...
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1answer
23 views

What is the probability that the smallest of the two results is 3 given that the sum of the two results is 8?

Two fair six sided dice are rolled. (i) What is the probability that the smallest of the two results is 3 given that the sum of the two results is 8? (ii) What is the probability that the sum of the ...
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2answers
38 views

Absolute Value of a normally distributed random variable.

Below is a problem I did. My answer matches the back of the book, but some how, I do not have confidence in my answer. I am hoping somebody here can confirm that my solution is right. Problem: Let $X$ ...
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1answer
30 views

Is a coupling defined on the Cartesian product of two sample spaces or a single sample space?

A book I'm reading presents two definitions of a coupling which seem to me to be contradictory. First it says A coupling of two probability distributions $μ$ and $ν$ is a pair of random variables $(...
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1answer
23 views

Show that the discrete random variable converges to zero

Given a discrete random variable $X= \begin{pmatrix} -1 & 0 & 1 \end{pmatrix}$ with $p = \begin{pmatrix} \frac{1}{2n} & 1-\frac{1}{n} & \frac{1}{2n} \end{pmatrix}$ How would I show ...
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0answers
17 views

Probability generating function for total number of individuals who are ever part of the population. [duplicate]

For the first part, it's just $1+\mu +\mu^2 + \mu^3 +\mu^4...... $ and that will give is $\frac{1}{1-\mu}$ where $\mu <1$ Need help with the second one - how can I condition it on $Z_{0}$?
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1answer
17 views

The mean remaining operation lifetime of a parallel system

I have a system with components: A and B. The operating times until failure of two are independent and exponentially distributed with $A \sim \mathrm{Exp}(2)$ and $B \sim \mathrm{Exp}(3)$. Assumption ...
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0answers
15 views

Variance of a truncated multivariate gaussian

Let $z$ be a gaussian vector of mean $m$ and covariance matrix $V$, where the components will be noted $f_1$, $f_2$ and $f_3$. I'm interested in the probability distribution defined by: $p(f_1,f_2) = ...
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0answers
26 views

Calculating large exponential shares / probabilities

Let there be an event space ES. Let there be some sets of objects OS[]. The probabilities of selecting any object are mutually disjoint. Now, assume that the size of each set is based on a number X[...
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2answers
46 views

Prove $\sum\limits_{m=0}^\infty {{2m+1}\choose m} p^{m+1}(1-p)^{m+1}(2p-1) = 1-p$

How can one prove that: $$\sum_{m=0}^\infty {{2m+1}\choose m} p^{m+1}(1-p)^{m+1}(2p-1) = 1-p$$ ?
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0answers
46 views

Exchange the limit with the expectation: why can I do it?

Hie everyone, I need a check on the following passage I found in a proof: Let $X$ be an a.s. positive random variable and let $h,t>0$. Then $$\lim_{h \rightarrow 0^{+}} E[\frac{e^{-hX} - 1}{h}]...
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0answers
21 views

Geometry distribution not sure [closed]

I am not sure how can I work on this exercise. I have the geometry type is f(x)=p(1-p)^(x-1). The exercise says : A company has a phone line the 40% of all ...
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0answers
16 views

Expectation of exp(-ax) where x is a random variable following a gamma distribution [closed]

I need to find the expectation of $\exp(-ax)$ where $P(x) = \Gamma(x;1,\beta)$. Here $\Gamma(x;1,\beta)$ is the gamma distribution. In other words, what is $\mathbb{E}_{P(x)}[\exp(-ax)]$?
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0answers
23 views

Random bijections with finite expectation are bounded?

Consider a random bijection $F:\mathbb{Z}\to\mathbb{Z}$ such that: $\mathbb{P}(F(i)=i)=0$, for all $i\in\mathbb{Z};$ For all $i,j\in\mathbb Z$, the probability $\mathbb{P}(F(i)=i+j)$ is independent ...
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1answer
23 views

$E[Z_{n}^2]$ in terms of $E[Z_{n-1}^2]$

After calculating part i), I am getting $E[Z_{n}^2]$ = $(\sigma^2 + \mu^2)$ $E[Z_{n-1}]$ which I think is incorrect. $E[Z_{n}^2]$ = $\sum_{k=0}^{inf}$ $E(Z_{n}^2/ Z_{n-1}=k) * P(Z_{n-1}=k)$ = $\...
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1answer
19 views

Calculate density of Z = X - Y when X, Y are i.i.d with $N(2, 3^2)$

Joint density of them seems to be: $$ f(x,y) = \frac{1}{2 \pi 3^2}e^{-\frac{(x-2)^2 + (y-2)^2}{2 \cdot 3^2}} $$ Should I calculate $$ F_Z(t) = P(X \leq Y + t) $$ and then just $\frac{\partial}{\...
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1answer
24 views

How is the probability sign taken out of equation

I'm having trouble understanding the following text: $$\lim_{n \to \infty} \mathbb P\left( \left|\frac{\hat I_n - I}{\sigma/\sqrt{n}} \right| \leq a \right) = \int_{-a}^a \gamma(x) \, \mathrm{d} x = \...
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0answers
24 views

What is the probability that the average value of die rolls will be within p of the average value of the die?

Basically, if I roll a die $x$ times, what is the probability that the average of all the results will be within $p$ of the actual average value of the die? For example, say I roll a six-sided die $6$...
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1answer
17 views

Proving an inspection was statistically likely or unlikely [closed]

My total number of customers is 1,250.  The total number of boxes shipped to these 1,250 customers was 13,000.   Customer X purchased 36 boxes. During inspection, I will pick completely at random ...
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1answer
34 views

probability problem (rolling a die) [closed]

Can someone help please? Brian rolls a fair die 10 times. What is the probability that he rolls exactly 6 fours?
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0answers
23 views

Expectation value from 2d gaussian

Let $$ G({R_q},L)= \exp\left[-\frac{3}{2}\int_{-\infty}^{\infty} \frac{dq}{2 \pi}\frac{q^2 R_{q}^2}{l_{1}(q)}\right] $$ from this the expected value is calculated as $$\langle R_{q}^2\rangle=q^{-2}...
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1answer
35 views

N keys and N locks at once

There are $N$ keys and $N$ locks. We test all keys at once. What is the probability that $k$ keys are correctly matched to $k$ locks ($k \leq n$)? Thanks a lot!
2
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1answer
37 views

Sufficiency for AR(1) model

Consider the following AR(1) model: $X_1=\epsilon_0$,$X_t=\rho X_{t-1} + \epsilon_t$ where $t=2,3,..,n$ where $\epsilon_t \sim N(0 , \sigma^2)$ independently. It is given that $|\rho| <1$. Let $...
0
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1answer
33 views

joint probability x +y < 4

This is rather simple but I cannot seem to figure out the proper integral ranges. Take x ~ uniform (0,2) and y ~ uniform (1,3) what is the joint probability that $x + y < 4$? I've computed it ...
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1answer
52 views

What are the chances of (not) getting drafted to army

My country has new law, which requires every male from 18 to 23 to be added to possible recruit list. The computer algorithm chooses randomly from 10% of those young men. That also means, that from 18 ...