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Questions tagged [probability]

This tag is for basic questions about probability and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using measure theory), please ask under (...

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system reliability, probabilities of rocket launch suspended

A rocket has 3 computers (a,b,c). If a fails than b will take over and so on. Each computer has 0.01 failure rate. These 3 computers are linked to an expensive sensor. If this sensor fails than all 3 ...
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Function of limit inf and limit inf of a function

Consider a Lipschitz continuous function $f : L^{\infty} \rightarrow \mathbb{R}$. Let, $\{X_{n}\}_{n=1}^{\infty}$ be a sequence on $L^{\infty}$, such that $\lim_{n \rightarrow \infty}{X_n} = X$, $X \...
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Help solving Bigram Model with the following probabilities

I came across the following problem involving bigram models which I am struggling to solve. Following this tutorial I have a basic understanding of how bigram possibilities are calculated. Problem: ...
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1answer
28 views

the definition of normal vector

I got confused with definitions of a normal vector. Assume, that $X \in \mathcal N(\mu, \sigma^2)$ and let $Y = 1 - X$. Is this true that $(X, Y)$ is multivariate normal? The problem is that ...
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A random sample of 100 adults from a large district was taken

A random sample of 100 adults from a large district was taken, of whom it was found that 38 had visited a dentist during the previous year and 62 had not. a) Obtain a 99% confidence interval for the ...
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1answer
12 views

Probability conditioned on a zero chance event.

Conditional probability $P(A|B)$ is defined as: $P(A|B) = \frac{P(A\cap B )}{P(B)}$ when $P(B) > 0$, where $A$ and $B$ are events in the sample space. Is $P(A|B)$ not defined for $P(B)=0$? ...
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34 views

Changing expectations into probability.

Going through a proof (related to probability) I found on a paper(Comp sci.) i could not undertstand the steps he followed. Basically the author did the following steps: $$E(T_{Fail}) = E(t_{f_{\...
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1answer
25 views

Find unordered cycle

In an undirected random graph of 11 vertices the probability of an edge being present between a pair of vertices is 2/5. What is the expected number of unordered cycle of length 3. I think the answer ...
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21 views

Flip the coin. Variant of calculation. [on hold]

Flip the coin. The easiest way to accept {+1,-1} for the outcome of each game , then the process of winning/losing can be represented by {+1,-1}-sequence. Step-by-step {+1, -1} - process( win/lose) ...
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31 views

Question about probability theory.

A forest has a population of $b$ animals. A uniformly random sample of $a$ of them are picked, marked and then released back into the forest. After that, we keep on capturing the animals in a ...
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How to calculate the odds of someone gambling or keeping the money

Alright, not too sure whether the title makes sense or not. basically, I'm trying to come up with a formula that will give a rough idea of whether a player is a gambler or not. The figures i have to ...
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10 views

Relationships between almost sure convergence, convergence in density, and convergence in moments

I've been trying to map out the relationships between the modes of convergence associated with random variables. However, I've had a hard time finding out online whether the statements below are true ...
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1answer
14 views

Variance of squared $l_2$ distance ratio

Problem: Assume we have an i.i.d data set $\{\bar{x} \} \subset \mathbb{R}^n$, sampled from an $n-$dimensional normal distribution where each dimension is independent, with zero mean and $\sigma^2$ ...
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26 views

Moment Generating function of sum of random number of random variables

I have a question on this problem https://i.stack.imgur.com/1qwiF.png , It states that E[(${\phi_x}(t))^N$]= ${\phi_y}(t)$ such that $\phi$ is a moment generating function. He then goes on to take the ...
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3answers
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Basic Multiplication Rule vs Conditional Probability

I'm looking at old probability problems and trying to interpret them in different ways to test my understanding. Given information in this problem: probability of being windy = .2 Probability rain ...
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1answer
35 views

Continuous random variable $X$ with p.d.f. $f$ given [on hold]

Consider a continuous random variable $X$ with p.d.f $f$ given by : $$f(x) = \frac{c}{(1 + x)^{f+1}} \quad \quad (x \geqslant 1)$$ Prove that if $f > 4$ then : $$ \text{Var}(X) = \frac{f}{(f − 1)...
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1answer
19 views

Proving that expectation of stopping time of a random walk is finite

Suppose we have a random walk, $S_n = X_1 + \cdots + X_n$ where $X_i = \pm 1$. I have proved that $M_n = S_n^2-n$ is a martingale. For fixed $N$, we define the stopping time \begin{align} T &= \...
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1answer
36 views

Probability of 'top six' match-ups on a Premier League match-day

There are 20 teams in the English Premier League. Of these 20, there is generally considered to be a clearly defined 'top six' - Arsenal, Manchester United, Manchester City, Liverpool, Chelsea and ...
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1answer
31 views

Find probability distribution given constraints?

I am looking at the following problem. I have a function $f(x)$ with support on $[0, \infty)$. Furthermore, $f(x)$ is bounded between 0 and 1, monotonically increasing and concave everywhere. ...
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1answer
22 views

Probability Question dice [on hold]

for the first part, I think its £10.50? Can anyone help me with the second part? thank you.
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31 views

The expected value of $C$ is equal to $\frac{a}{b}$ for coprime positive integers $a$ and $b.$ What is $a+b?$

A fair, $6$-sided die is rolled $20$ times, and the sequence of the rolls is recorded. $C$ is the number of times in the 20-number sequence that a subsequence (of any length from one to six) of rolls ...
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Bayesian Updating and Projection Theorem with Multi-Variables

I am interested in mathematical techniques frequently used in finance literature. If $\tilde{x}$ and $\tilde{y}$ are normally distributed, we can use the so-called projection theorem. $E[\tilde{x}\...
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20 views

Hazard and Intensity Function Revisited in the light of Hawkes Process

I am looking at the following post in the Statistics Stack Exchange: Difference between hazard function and intensity function? where there was a question about the hazard function in survival ...
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34 views

Confidence Intervals - Inconsistent Statistical Results

After my last SE question on confidence Intervals here, which clarified the intuition, I tried then to verify statistical results if they are convincingly compliant with theory. I started with CI for ...
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1answer
32 views

What would be $P(f(A) \leq x) $ for a function $f$ and random number $A \in [0,1]$?

Say that we generate a random number $A \in [0,1]$. We then square that number and graph the probability $P$ of the result being $\leq x$ (a constant). A small simulation gives us some approximate ...
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4answers
62 views

probability of at least 1 girl

I am having trouble understanding how the probability of at least one girl is 3/4 in this question below. Given a family of two children (assume boys and girls equally likely, that is, probability ...
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8 views

Conditional density in hierarchical model

Let $Y_1\sim Ber(k(X_1)), Y_2\sim Ber(k(X_1)), \ldots ,Y_n\sim Ber(k(X_n))$, where $$k(X_i)=\frac{\exp(X_i)}{1+\exp(X_i)}, \ \ \ i=1,2,\ldots, n$$ $X_1,X_2,\ldots ,X_n\sim N(0,\sigma^2)$, where $\...
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3answers
48 views

Circular Definition of Experiment in probability

I was trying to understand what an experiment was in the theory of probability. I found several definitions. Definition by Wikipedia Any procedure that can be infinitely repeated and whose ...
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1answer
41 views

Let $X$ and $Y$ be independent random variables. Find $ \mathbb P \{ \max(X, Y) - \min(X, Y) \gt 0.2 \} $.

Let $X$ and $Y$ be independent random variables for which $X \sim > \text{exp}(2)$ and $Y \sim \text{exp}(3)$. Find $P(\max(X, Y) - \min(X, Y) \gt 0.2) $. 1) I know this answer has already been ...
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1answer
30 views

How come the counting theorem isn't working here

I was doing a seemingly trivial question, and I though it was a simple application of the counting theorem but it turns out it doesn't work. Here's the question From a deck of 52 cards, how many ...
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22 views

Covariance of 2 conditionally independent gaussian random variables

Consider three identically distributed Gaussian Random Variables $X; Y;Z \thicksim N(0; 1)$. The RV's $X;Z$ are conditionally independent given $Y$ . If the covariance between $X$ and $Y$ is $\rho_1$ ...
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$X\sim U[0,1]$, given $X=x$, $Y\sim U[0,x],$ what is $\rho(X,Y)?$

I want to compute $$\rho(X,Y)=\frac{\text{Cov}[X,Y]}{\sqrt{\text{Var}[X]\text{Var}[Y]}}=\frac{E[XY]-E[X]E[Y]}{\sqrt{\text{Var}[X]\text{Var}[Y]}}.$$ I know that $E[X]=1/2$, $\text{Var}[X]=1/12$. To ...
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1answer
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Conditional probability (false positives and negatives) question. What Benefit-cost ratio is required to continue with testing program

The full question can be viewed here - https://imgur.com/a/Vrd47ng I understand how to answer all but the very last part, (D). I'll include a summary of the question below. The problem is a very ...
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1answer
26 views

Poisson process. Probability that next vehicle is a motorcycle.

On a particular highway cars pass as a possionprocess with intensity 3 cars per minute and motorcycles with intensity 1 motorcycles per minute. Other types of vehicles are not counted. a) ...
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1answer
33 views

Probability in a fun game show

On a game show, the - contestants each day can win $1, 000, 000 by correctly guessing an integer between 1 and 100 inclusive (which is chosen randomly each day). Before guessing the contestant can ...
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23 views

Harmonic functions and Markov processes

I was reading some lecture notes, in which it is stated that, given a discrete time Markov process $X_t$, for its Markov semigroup $P_t$ it holds that: $$ P_tf(x) = E_x[f(X_t)]=E[f(X_t)|X_0=x] $$ For ...
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relation between standard deviation and singular value

Let $L:E\rightarrow F$ be linear map between d-dimesional inner product, then singular value $\sigma_{1}(L)\geq...\geq \sigma_{d}(L)$ are eignvalues symmetric operator$(L*L)^{1/2}$.So $\sigma_{1}(L)$ ...
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2answers
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Why am I wrong? - Probability that at least two consecutive knights are selected when three knights are selected from a round table?

Can someone explain to me why my solution to this problem is wrong? Where have a made a logical error? The problem is Twenty five of King Arthur's knights are seated at their customary round ...
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44 views

A probability inequality applying Feller-Chung lemma

Let $\{X_i,i\geq 1\}$ be independent r.v.'s with $E[X_i]=0$. Then prove that for any $\epsilon>0$ there is \begin{equation*} P\{\max_{1\leq j\leq n}S_j>\epsilon\}\leq 2P\{S_n\geq\epsilon-E|S_n|\}...
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37 views

Why is the expression $\frac{I(x>\theta)}{I(x>\theta)}$ independent of $\theta$?

This question has been asked before but I can't really grasp the explanations. Let $f(x\mid \theta)=\frac{I(x>\theta)}{I(x>\theta)}$ for some particular value of $x$. I want to show that $f$ is ...
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Conditional expectation given RV and an event [on hold]

I am having hard time finding out what is $E(E(X|Y, Z = z)|Y)$ equal to. Somewhat intuitively makes sense the following $$E(E(X|Y, Z = z)|Y) = E(X|Y, Z = z)$$ but I am having hard time in finding good ...
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3answers
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Two six-sided dice are rolled (associated with the random variables $X$ and $Y$) . Find the probability distribution of $X | \max\{X,Y\} = z$.

By definition, $$ P( X = x \mid \max\{X,Y\} = z ) = \frac{P(X = x, \max\{X,Y\} = z )}{P(\max\{ X,Y \} = z)}.$$ (am I right?) I've found that $$P(\max\{X,Y \} = z) = \frac{2z - 1}{36},$$ but I can'...
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1answer
32 views

Calculate probability of an infinite geometric series

Suppose in a single round, I win with probability p, and you win with probability 1 − p. We play repeatedly and keep score until one of us is two ahead (for example a score of 2 − 0, 3 − 1, 2 − 4, ...
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1answer
37 views

Explanation of Bayesian probability

I'm using the MultiNomial Naive Bayes algorithm provided by NLTK to predict pos/neg sentences, with words as features. Where F is feature and V is the pos/neg class, a features informativeness is ...
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1answer
21 views

Computing the expected number of overlapping committee members

Here is the problem, which comes from the probabilistic method section of Introduction to Probabilty. A group of 100 people are assigned to 15 committees of size 20,such that each person serves on ...
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18 views

Probability space with pairwise disjoint sets

I asked myself, if there is a probability space $(\Omega,\mathcal{F},P)$ such we can find pairwise disjoint sets $(A_n)_{n\in \mathbb{N}}\in\mathcal{F}$ with $P(A_n)=1$ for all $n\in \mathbb{N}$, but ...
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1answer
20 views

Bound on tail of beta distribution

Let $X$ be a random variable with a beta distribution $\beta(j,k)$. Is there a convenient upper bound for the left tail when $j$ and $k$ are large: $$ \mathbb{P}(X \leq \varepsilon) \leq ?? $$
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1answer
46 views

Formula to generate “Score” that ranges between 0 and 100 based on many inputs [on hold]

Suppose you have a set of data looking like this: views: 1500 likes: 23 avgRating: 7 ratings: 2 purchases: 6 Note, that those numbers may vary dramatically. Now ...
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24 views

Find distribution of random variable which is a function of two other random variables

Let $X$ and $Y$ be two independent random variables whose densities are $\rho(x)=\frac{e^{-|x|}}{2}, x \in \mathbb{R}$ and $\phi(y)=e^{-y} , y > 0$ Find distribution of random variable $Z=min(X^...
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2answers
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Difference between constant and random variable always equal to constans

I am interested in the difference between constant (let's call it $c$) and random variable which is always a constant: $P(X = c) = 1$. Is there any trap in thinking that it is the same? What about ...