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Questions tagged [probability]

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

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Can a probability distribution value exceeding 1 be OK?

On the Wikipedia page about naive Bayes classifiers, there is this line: $p(\mathrm{height}|\mathrm{male}) = 1.5789$ (A probability distribution over 1 is OK. It is the area under the bell curve ...
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11answers
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Why does a 95% Confidence Interval (CI) not imply a 95% chance of containing the mean?

It seems that through various related questions here, there is consensus that the "95%" part of what we call a "95% confidence interval" refers to the fact that if we were to exactly replicate our ...
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9answers
244k views

What is the difference between “likelihood” and “probability”?

The wikipedia page claims that likelihood and probability are distinct concepts. In non-technical parlance, "likelihood" is usually a synonym for "probability," but in statistical usage there is a ...
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4answers
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Generic sum of Gamma random variables

I have read that the sum of Gamma random variables with the same scale parameter is another Gamma random variable. I've also seen the paper by Moschopoulos describing a method for the summation of a ...
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3answers
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How to generate correlated random numbers (given means, variances and degree of correlation)?

I'm sorry if this seems a bit too basic, but I guess I'm just looking to confirm understanding here. I get the sense I'd have to do this in two steps, and I've started trying to grok correlation ...
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3answers
5k views

Functions of Independent Random Variables

Is the claim that functions of independent random variables are themselves independent, true? I have seen that result often used implicitly in some proofs, for example in the proof of independence ...
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7answers
10k views

How to tell the probability of failure if there were no failures?

I was wondering if there is a way to tell the probability of something failing (a product) if we have 100,000 products in the field for 1 year and with no failures? What is the probability that one of ...
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3answers
2k views

$P[X=x]=0$ when $X$ is continuous variable

I know that for continuous variable $P[X=x]=0$. But i can't visualize that if $P[X=x]=0$, there is infinite number of possible $x$'s. And also why do their probabilities get infinitely small ?
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5answers
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Wikipedia entry on likelihood seems ambiguous

I have a simple question regarding "conditional probability" and "Likelihood". (I have already surveyed this question here but to no avail.) It starts from the Wikipedia page on likelihood. They say ...
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14answers
7k views

What is the most surprising characterization of the Gaussian (normal) distribution?

A standardized Gaussian distribution on $\mathbb{R}$ can be defined by giving explicitly its density: $$ \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$ or its characteristic function. As recalled in this ...
24
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5answers
8k views

Time taken to hit a pattern of heads and tails in a series of coin-tosses

Inspired by Peter Donnelly's talk at TED, in which he discusses how long it would take for a certain pattern to appear in a series of coin tosses, I created the following script in R. Given two ...
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7answers
6k views

Combining probabilities/information from different sources

Lets say I have three independent sources and each of them make predictions for the weather tomorrow. The first one says that the probability of rain tomorrow is 0, then the second one says that the ...
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5answers
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Can anyone clarify the concept of a “sum of random variables”

In my probability class the terms "sums of random variables" is constantly used. However, I'm stuck on what exactly that means? Are we talking about the sum of a bunch of realizations from a random ...
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11answers
26k views

How to easily determine the results distribution for multiple dice?

I want to calculate the probability distribution for the total of a combination of dice. I remember that the probability of is the number of combinations that total that number over the total number ...
19
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3answers
16k views

Comparing and contrasting, p-values, significance levels and type I error

I was wondering if anybody could give a concise rundown as to the definitions and uses of p-values, significance level and type I error. I understand that p-values are defined as "the probability of ...
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7answers
16k views

How often do you have to roll a 6-sided die to obtain every number at least once?

I've just played a game with my kids that basically boils down to: whoever rolls every number at least once on a 6-sided die wins. I won, eventually, and the others finished 1-2 turns later. Now I'm ...
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3answers
659 views

How exactly do Bayesians define (or interpret?) probability?

Part of a series of trying to understand Bayesian vs frequentist: 1 2 3 4 5 6 7 I think I get the difference of how Bayesians and frequentists approach choosing between hypotheses, but I'm not quite ...
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5answers
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How does linear regression use the normal distribution?

In linear regression, each predicted value is assumed to have been picked from a normal distribution of possible values. See below. But why is each predicted value assumed to have come from a normal ...
22
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2answers
653 views

I've heard that ratios or inverses of random variables often are problematic, in not having expectations. Why is that?

The title is the question. I am told that ratios and inverses of random variables often are problematic. What is meant is that expectation often do not exist. Is there a simple, general explication of ...
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2answers
7k views

Extreme Value Theory - Show: Normal to Gumbel

The Maximum of $X_1,\dots,X_n. \sim$ i.i.d. Standardnormals converges to the Standard Gumbel Distribution according to Extreme Value Theory. How can we show that? We have $$P(\max X_i \leq x) = P(...
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2answers
2k views

Risk of extinction of Schrödinger's cats

I am interested in how uncertainty can be accounted for when considering the risk of extinction of a species. Forgive me for extending a rather tired thought experiment, but at least it's familiar ...
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6answers
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Bayesian vs frequentist Interpretations of Probability

Can someone give a good rundown of the differences between the Bayesian and the frequentist approach to probability? From what I understand: The frequentists view is that the data is a repeatable ...
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5answers
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How to sample from a discrete distribution? [duplicate]

Assume I have a distribution governing the possible outcome from a single random variable X. This is something like [0.1, 0.4, 0.2, 0.3] for X being a value of either 1, 2, 3, 4. Is it possible to ...
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2answers
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How many stickers do I need to complete my FIFA Panini album?

I'm playing the FIFA Panini Online Sticker Album, which is an Internet adaption of the classic Panini albums that are usually published for the soccer world cup, European championship, and possibly ...
16
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3answers
8k views

Why Normalizing Factor is Required in Bayes Theorem?

Bayes theorem goes $$ P(\textrm{model}|\textrm{data}) = \frac{P(\textrm{model}) \times P(\textrm{data}|\textrm{model})}{P(\textrm{data})} $$ This is all fine. But, I've read somewhere: Basically, ...
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1answer
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Odds made simple

I am having some trouble in understanding odds, and I would like just a basic explanation for how to interpret them. I have found various posts related to odds but most of them are more complex than ...
6
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1answer
305 views

Is accuracy an improper scoring rule in a binary classification setting?

I have recently been learning about proper scoring rules for probabilistic classifiers. Several threads on this website have made a point of emphasizing that accuracy is an improper scoring rule and ...
64
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10answers
7k views

Is there any *mathematical* basis for the Bayesian vs frequentist debate?

It says on Wikipedia that: the mathematics [of probability] is largely independent of any interpretation of probability. Question: Then if we want to be mathematically correct, shouldn't we ...
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2answers
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PP-plots vs. QQ-plots

What is the difference between probability plots, PP-plots and QQ-plots when trying to analyse a fitted distribution to data?
21
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2answers
6k views

Meaning of probability notations $P(z;d,w)$ and $P(z|d,w)$

What is the difference in meaning between the notation $P(z;d,w)$ and $P(z|d,w)$ which are commonly used in many books and papers?
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2answers
6k views

How does a uniform prior lead to the same estimates from maximum likelihood and mode of posterior?

I am studying different point estimate methods and read that when using MAP vs ML estimates, when we use a "uniform prior", the estimates are identical. Can somebody explain what a "uniform" prior is ...
34
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5answers
17k views

Probability distribution for different probabilities

If I wanted to get the probability of 9 successes in 16 trials with each trial having a probability of 0.6 I could use a binomial distribution. What could I use if each of the 16 trials has a ...
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5answers
9k views

Distribution of ratio between two independent uniform random variables

Supppse $X$ and $Y$ are standard uniformly distributed in $[0, 1]$, and they are independent, what is the PDF of $Z = Y / X$? The answer from some probability theory textbook is $$ f_Z(z) = \begin{...
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1answer
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If random variables are drawn from an identical distribution, why doesn't this guarantee they are independent?

Having read a little about exchangeability, I went back to thinking about the iid condition required for the central limit theorem. It struck me that if two random variables are drawn from an ...
113
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9answers
79k views

Numerical example to understand Expectation-Maximization

I am trying to get a good grasp on the EM algorithm, to be able to implement and use it. I spent a full day reading the theory and a paper where EM is used to track an aircraft using the position ...
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3answers
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What is so cool about de Finetti's representation theorem?

From Theory of Statistics by Mark J. Schervish (page 12): Although DeFinetti's representation theorem 1.49 is central to motivating parametric models, it is not actually used in their ...
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13answers
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The Monty Hall Problem - where does our intuition fail us?

From Wikipedia : Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who ...
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5answers
7k views

Probability of a run of k successes in a sequence of n Bernoulli trials

I'm trying to find the probability of getting 8 trials in a row correct in a block of 25 trials, you have 8 total blocks (of 25 trials) to get 8 trials correct in a row. The probability of getting any ...
22
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2answers
3k views

Difference of two i.i.d. lognormal random variables

Let $X_1$ and $X_2$ be 2 i.i.d. r.v.'s where $\log(X_1),\log(X_2) \sim N(\mu,\sigma)$. I'd like to know the distribution for $X_1 - X_2$. The best I can do is to take the Taylor series of both and ...
5
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1answer
228 views

Probability associated with experiencing all outcomes

Assume last year I commuted to work in a taxi, and suppose if there are $n$ taxis in the fleet I used. If I took one of these taxis every trip at random and with replacement, then what is the number ...
8
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2answers
5k views

Transformation Chi-squared to Normal distribution

The relationship between the standard normal and the chi-squared distributions is well known. I was wondering though, is there a transformation that can lead from a $\chi^2 (1)$ back to a standard ...
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13answers
33k views

Does 10 heads in a row increase the chance of the next toss being a tail?

I assume the following is true: assuming a fair coin, getting 10 heads in a row whilst tossing a coin does not increase the chance of the next coin toss being a tail, no matter what amount of ...
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3answers
16k views

“The total area underneath a probability density function is 1” - relative to what?

Conceptually I grasp the meaning of the phrase "the total area underneath a PDF is 1". It should mean that the chances of the outcome being in the total interval of possibilities is 100%. But I ...
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3answers
9k views

A normal divided by the $\sqrt{\chi^2(s)/s}$ gives you a t-distribution — proof

let $Z \sim N(0,1)$ and $W \sim \chi^2(s)$. If $Z$ and $W$ are independently distributed then the variable $Y = \frac{Z}{\sqrt{W/s}}$ follows a $t$ distribution with degrees of freedom $s$. I am ...
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2answers
5k views

What is the difference between moment generating function and probability generating function?

I am confused between the two terms " probability generating function" and "moment generating function." How do those terms differ?
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2answers
899 views

Is joint normality a necessary condition for the sum of normal random variables to be normal?

In comments following this answer of mine to a related question, Users ssdecontrol and Glen_b asked whether joint normality of $X$ and $Y$ is necessary for asserting the normality of the sum $X+Y$? ...
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4answers
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Dungeons & Dragons Attack hit probability success percentage

In D&D players roll a 20 sided die trying to beat a set number to determine if an attack hits the target. Players often can add modifiers to this roll to help the odds in reaching this target ...
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2answers
372 views

How can I determine accuracy of past probability calculations?

I do not study statistics but engineering, but this is a statistics question, and I hope you can lead me to what I need to learn to solve this problem. I have this situation where I calculate ...
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1answer
25k views

Graphing a Probability Curve for a Logit Model With Multiple Predictors

I have the following probability function: $$\text{Prob} = \frac{1}{1 + e^{-z}}$$ where $$z = B_0 + B_1X_1 + \dots + B_nX_n.$$ My model looks like $$\Pr(Y=1) = \frac{1}{1 + \exp\left(-[-3.92 + 0....
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1answer
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Intuitive examples of importance sampling

My background is computer science. I am fairly new to monte carlo sampling methods and, although I understand the math, I have hard time coming up with intuitive examples for importance sampling. More ...