Questions tagged [probability]

A probability provides a quantitative description of the likely occurrence of a particular event.

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0answers
10 views

Marginal distribution of a Poisson rate

I have a random sample of Yi~Poisson(θ) with i=1,..,n. Where I assume a gamma(a,b) prior for θ. Where I'm able to find the marginal distribution of Y to be a NegativeBionomial(a,p) where p=(1/(1+b), ...
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1answer
600 views

Probability distribution of functions of random variables?

I have a doubt: consider the real valued random variables $X$ and $Z$ both defined on the probability space $(\Omega, \mathcal{F},\mathbb{P})$. Let $Y:= g(X,Z)$, where $g(\cdot)$ is a real-valued ...
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1answer
10 views

How the variance of a potential loss X has been derived

I'm studying Insurance and I have a question about how the variance has been computed in this example. Imagine a case where an "agent" may suffer a loss, because of an event (an accident) occurring ...
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1answer
42 views

Understanding three prisoners (Statistical Inference - Cassella and Berger)

There are quite a few questions regarding three prisoners riddle, but my question relating to a particular line of reasoning from Cassella and Berger, as highlighted in the image below. Could you ...
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6answers
4k views

Expectation of 500 coin flips after 500 realizations

I was hoping someone could provide clarity surrounding the following scenario. You are asked "What is the expected number of observed heads and tails if you flip a fair coin 1000 times". Knowing that ...
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Percents of Random Samples

I have three sets $A,B,C$ of sizes $N_A=2508$, $N_B=36211$ and $N_C=2296$ respectively, containing binary values. I took 200 samples of each set to produce point estimates of the averages: $\hat p_A=0....
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3answers
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Does a sample version of the one-sided Chebyshev inequality exist?

I am interested in the following one-sided Cantelli's version of the Chebyshev inequality: $$ \mathbb P(X - \mathbb E (X) \geq t) \leq \frac{\mathrm{Var}(X)}{\mathrm{Var}(X) + t^2} \,. $$ Basically, ...
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Basic probability question in microeconomics [closed]

There are two players in a gamer N={1,2}, their actions lay between a=[1,infty) and their type lays between [0,200]. given that x is within [0,200] and P(t_2 < x|t_1)=x/200 (the type of each player ...
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Factorization of the Characteristic function for sum of dependent data [closed]

We know that for $S_n=X_1+X_2+...+X_n$ the characteristic function is $\mathbb{E}_X\left\{\exp(iuS_n)\right\}$, where if $X_1 ,X_2 ,..., X_n$ are independent, it can be expressed as follows \begin{...
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1answer
536 views

Proof of posterior consistency

Given the model $x_1,...,x_n$ ~ $Poi(\theta)$ (iid), and the prior $\theta$~$ Gamma(a,b)$. How do we prove the consistency of posterior distribution? Thanks in advance. Modification: The definition ...
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Extendability of finite sequences of exchangeable random variables?

The old paper Binomial Mixtures and Finite Exchangeability by G. R. Wood makes reference (see Table 2) to a 1971 conjecture of Crisma that gives a formula for the probability that an exchangeable ...
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56 views

Logit - probit regression

I was running regression of - determinants of acceptance into a social science college. I found this unrelated paper (Screenshot of relevant page attached herewith). Here, they have computed logit and ...
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22 views

Combinatorics and probability

I have 7 interviewees and 14 interviewers. Each interviewee is interviewed by 4 of those 14 interviewees. Is it possible to have every interviewee to be shared by at least 1 interviewer, as to allow ...
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1answer
64 views

Obtaining valid win probabilities from contest data using a binomial model

I conducted an enclosure experiment on lizards where I recorded contest outcome for every male-male combat. We had three morphs of lizards (o, w, y) in each enclosure. I am interested in obtaining ...
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25 views

Given competing risks, what is the probability of an event within d steps from time t?

In competing risk analysis, we can only identify the cause-specific hazard and functions of that hazard. The cumulative incidence function (CIF) is a function of the cause-specific hazards. Because ...
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2answers
150 views

Statistical confidence in voting samples

I'm currently watching the USA elections and I'm wondering how the confidence is calculated. So let us say we have a state of 500,000 inhabitants; 5,000 votes have been counted and 80% (4,000 votes) ...
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Which come first? (random walks)

Suppose I have a continuous time random walk in one (non-time) dimension, based on not-necessarily-Gaussian white noise. I know its value at the beginning and end of an interval, and from inside of ...
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1answer
179 views

Elo ranking for attack and defence seperately

I want to create a simple statistical model for the bowler versus batsman contest in a game of cricket (it's similar to pitcher versus batter in baseball). Let's say the simple the model is this: One ...
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0answers
31 views

random walk with probability [closed]

I just have no idea how to do this. Can anyone give a hint? Thanks for your time!
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1answer
16 views

Subset of the event {$i$ aces in $i$ cards} and its probability

As far as I'm aware, $A$ is a subset of $B$ if $x\in A$, then $x\in B$ for all $x\in A$. If so, how could this statement (please see the image below) say the event {4 aces in 4 cards} was a subset of ...
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1answer
44 views

Why Pareto distribution has pointy head? How its symmetrical version looks like?

I want to use Pareto distribution for the stock price probability distribution. They say (B. Mandelbrot and N. Taleb Mild vs Wild Randomness) it represents the price changes, especially the tail ...
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3answers
2k views

Estimating population size from the frequency of sampled duplicates and uniques

There is a web service where I can request information about a random item. For every request each item has an equal chance of being returned. I can keep requesting items and record the number of ...
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2answers
3k views

Understanding Add-1/Laplace smoothing with bigrams

I am working through an example of Add-1 smoothing in the context of NLP Say that there is the following corpus (start and end tokens included) ...
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Distribution of X/(X+Y) when X and Y are independent, but non-identical Binomial distributions [closed]

$X \sim Bin(n, p)$, and $Y \sim Bin(m, q)$, then what is the distribution of $\frac{X}{X+Y}$?
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How to compute the probability for a firm to move from on group to another in a panel database?

I have a panel database for firms i (90 firms) across year t (from 2013 to 2019). Some firms witnessed a cross border investment in a year that is between 2013 and 2019. I created a dummy=1, when the ...
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1answer
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Bayesian Question [closed]

A person belongs to 5 different groups. Group A consists has 10 people, 6 of them male. Group B consists has 20 people, 11 of them male. Group C consists has 30 people, 16 of them male. What is the ...
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2answers
197 views

Why convergence in probability is defined as convergence to R.V.?

Wikipedia defines convergence in probability as A sequence ${X_n}$ of random variables converges in probability towards the random variable $X$ if for all ε > 0 $$\lim_{n\rightarrow\infty} P(|X_n-X|&...
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Approximate posterior distribution [closed]

Let $\mathbf{a_k} \sim \mathcal{GP}(\mathbf{m}_k,\mathbf{C_k})$ where $\mathbf{a} \in \mathbb{R}^T$ is modeled as Gaussian process with mean $\mathbf{m} \in \mathbb{R}^T$ and prior covariance $\mathbf{...
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1answer
134 views

Showing independence between two functions of a set of random variables

I've been working on the following problem and I'm confused about how to get started: Let $X_1, X_2,\dots, X_n$ denote i.i.d. real valued random variables, each absolutely continuous with an ...
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1answer
25 views

How to Change prior probabilities for predicted variable in neural networks and other methods in SPSS Statistics

i am trying to find right model for predicting categorical variable with two values. Problem is that ratio of cases in group 1 and group 2 is not equal but rather in ratio of 2:1. When i try to find a ...
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30 views

What information in general is necessary to fully specify a multivariate distribution?

Given some multivariate probability distribution, we can fully describe it with its density or mass function -- we can associate each point in the space with either a probability density or mass, ...
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1answer
135 views

Time series forecast with probability

I have historical data for a particular metric for each month for the last 3 years for different categories. The metric is a percentage and its heavily skewed towards 1 with more than 75% of values ...
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0answers
14 views

Finite Mean and Finite Variance [duplicate]

Let $X$ be a random variable. Suppose there exists a constant $c ∈ R$ such that $E(|X − c|^2) < ∞$. Show that the random variable $X$ has finite mean and variance. And I'm quite confused about the ...
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12 views

One-way ANOVA method validation

Here is my case: Let's say we have 50 polygons(looks like this:and a point set distributed within these 50 polygons. So that for each polygon, there is an associated point density. What I want to ...
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1answer
173 views

Realization of a conditional expectation wrt a random variable

Let $Z$ and $X$ be a real-valued random variables. Is it true that $\mathbb{E}[Z|X](\omega)=\mathbb{E}[Z|X=X(\omega)]$ ? If so, why ? P.S : I'm looking for a rigorous mathematical proof, not ...
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1answer
28 views

How many losing runs of length r playing x numbers in roulette in a sample of n bets?

A losing run is defined as a number of consecutive losing bets. I've written a program to simulate this but the results I'm getting are different from what the formula gives, so one or another is ...
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0answers
19 views

How many times would you need to watch a movie to have seen every moment of it (i.e. not blinking)?

Say we blink stochastically for .1 seconds every 4 seconds on average. How many times would we need to watch a 120 minute movie (assuming your eyes are glued to the screen the whole time), to have a ...
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1answer
37 views

Where is the measure theoretic probability theory actually applied?

Where is measure theoretic probability theory actually applied? I've done quite a bit of graduate work in machine learning, Bayesian machine learning, information theory, and statistics (both ...
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1answer
124 views

Probability Question - Mutual Exlusive/Independent (To the power)

A bridge has to be designed for a region which is subject to the influence of both floods and earthquakes. Its expected design life is 120 years. At the particular site, the probability of a flood ...
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0answers
33 views

How to change the distribution of classes?

I have two classes [0,1] and I want to evaluate algorithm on different distributions of classes. I did label flipping incrementally such as 0%, 10%,20%,....,90%,100%. Does label flipping change the ...
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1answer
160 views

The effect size of difference

I have this interesting data where I would like to estimate possibly a parameter of the difference (between $A+B$ and $A+C$, inference using both) that would allow me to infer the development of $A$ (...
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29 views

Expectation of the k-th order statistic of a standard Gaussian sample

Let $(X_1,\dots,X_n)$ be independent random variables with common distribution $\mathcal{N}(0,1)$. The order statistics satisfy $X_{(1)} \leq X_{(2)} \leq \dots \leq X_{(n)}$. I am interested in the ...
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0answers
14 views

Calculate conditional probability using sample from 3variate normal distribution

I am working in R and have a matrix (200X3) named X where I have simulated data from a 3variate normal distribution given its parameters (variance-covariance matrix & and mu). My task is to ...
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0answers
22 views

How high is the probability that copy N bytes of data from /dev/random to a file will result in a working application? [closed]

After talking with some colleagues, a question came up about the probability of copying N bytes from /dev/random to a given file on disk (either creating a new file, or truncating and overwriting an ...
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1answer
16 views

How to rank data based on cross-validation

I had this problem from a long time. I have small dataset with about 1000 data points. The data is labeled as 1 or 0 (i.e. ...
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2answers
98 views

Multinomial distribution: probability that one outcome is greater than another

Consider a multinomial distribution with three outcomes. Let $x_i$ denote the number of occurences of the $i^{th}$ outcome, and the $i^{th}$ outcome occurs with probability $p_i$, $i=1,2,3$. Let $n$ ...
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1answer
16 views

Range of probability for non-independent events

Suppose the probability of rain on day 1 is $p_1$ and probability of rain on day 2 is $p_2$. Then the probability of rain for the entire two-day period is $1-(1-p_1)(1-p_2)$, under the assumption that ...
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0answers
11 views

Estimation of generalized gamma convolutions

How can i estimate on a data sample parameters of a generalised gamma convolution ? To be more specific, if my estimation gives me only a gamma convolution and not a generalised gamma convolution i'll ...
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1answer
22 views

Understanding t bounds on MGF

I'm having trouble understanding what the bounds of the $t$ variable are for an mgf. My questions are bolded. Here's an example from a textbook: Suppose X is a random variable for which the pdf is: ...