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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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184 votes
6 answers
125k views

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 ...
babelproofreader's user avatar
110 votes
10 answers
26k views

What is meant by a "random variable"?

What do they mean when they say "random variable"?
Baltimark's user avatar
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29 votes
2 answers
9k 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 ...
Zyzzva's user avatar
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307 votes
15 answers
112k views

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 ...
Mike Lawrence's user avatar
59 votes
5 answers
36k views

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 ...
OSE's user avatar
  • 1,217
674 votes
12 answers
496k 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 ...
Douglas S. Stones's user avatar
34 votes
3 answers
4k 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 ...
kjetil b halvorsen's user avatar
47 votes
4 answers
22k 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 ...
JohnK's user avatar
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22 votes
6 answers
16k views

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

I know that for any continuous variable $P[X=x]=0$. But I can't visualize that if $P[X=x]=0$, there is an infinite number of possible $x$'s. And also why do their probabilities get infinitely small ?
time's user avatar
  • 1,607
32 votes
5 answers
14k 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 ...
lafrasu's user avatar
  • 423
15 votes
3 answers
2k 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 ...
BCLC's user avatar
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67 votes
14 answers
12k 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 ...
62 votes
4 answers
68k views

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 ...
Joseph Weissman's user avatar
61 votes
4 answers
95k views

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 ...
luciano's user avatar
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58 votes
9 answers
32k 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 ...
melonfresh's user avatar
39 votes
4 answers
12k views

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 ...
Gosset's user avatar
  • 555
49 votes
7 answers
23k 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 ...
Biela Diela's user avatar
56 votes
9 answers
44k views

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 ...
BYS2's user avatar
  • 1,505
36 votes
5 answers
3k views

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 ...
Creatron's user avatar
  • 1,665
30 votes
3 answers
17k 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(...
emcor's user avatar
  • 1,231
29 votes
4 answers
17k views

Why is a normalizing factor 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, ...
Sreejith Ramakrishnan's user avatar
27 votes
4 answers
19k 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 ...
BYS2's user avatar
  • 1,505
27 votes
11 answers
51k 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 ...
C. Ross's user avatar
  • 695
199 votes
5 answers
56k views

Why do we need sigma-algebras to define probability spaces?

We have a random experiment with different outcomes forming the sample space $\Omega,$ on which we look with interest at certain patterns, called events $\mathscr{F}.$ Sigma-algebras (or sigma-fields) ...
Antoni Parellada's user avatar
43 votes
7 answers
21k 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 ...
Jonas's user avatar
  • 1,678
25 votes
6 answers
30k 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{...
qed's user avatar
  • 2,828
11 votes
2 answers
4k 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 ...
rudivonstaden's user avatar
34 votes
2 answers
7k views

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 ...
Vidar S. Ramdal's user avatar
15 votes
3 answers
35k 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 ...
Monolite's user avatar
  • 1,443
37 votes
1 answer
18k views

"Absolutely continuous random variable" vs. "Continuous random variable"?

In the book Limit Theorems of Probability Theory by Valentin V. Petrov, I saw a distinction between the definitions of a distribution being "continuous" and "absolutely continuous",...
Tian's user avatar
  • 619
55 votes
2 answers
75k views

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?
kay's user avatar
  • 681
71 votes
6 answers
233k views

Find expected value using CDF

I'm going to start out by saying this is a homework problem straight out of the book. I have spent a couple hours looking up how to find expected values, and have determined I understand nothing. ...
styfle's user avatar
  • 937
38 votes
5 answers
63k views

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 ...
Barry's user avatar
  • 533
59 votes
5 answers
42k 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 ...
Greg's user avatar
  • 793
42 votes
15 answers
7k views

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 ...
Rizwan Kassim's user avatar
23 votes
3 answers
27k views

Normalizing constant in Bayes theorem

I read that in Bayes rule, the denominator $\Pr(\textrm{data})$ of $$\Pr(\text{parameters} \mid \text{data}) = \frac{\Pr(\textrm{data} \mid \textrm{parameters}) \Pr(\text{parameters})}{\Pr(\text{...
amateur's user avatar
  • 384
18 votes
2 answers
3k 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 $X+Y$? ...
Dilip Sarwate's user avatar
8 votes
3 answers
1k views

$X_i, X_j$ independent when $i≠j$, but $X_1, X_2, X_3$ dependent?

I've seen the statement: It's possible that random variables $X_i, X_j$ are independent for $i≠j$, but $X_1, X_2, X_3$ are dependent. I haven't been able to find examples of this though. Any ...
mavavilj's user avatar
  • 4,109
36 votes
5 answers
16k views

How to derive the probabilistic interpretation of the AUC?

Why is the area under the ROC curve the probability that a classifier will rank a randomly chosen "positive" instance (from the retrieved predictions) higher than a randomly chosen "positive" one (...
mff's user avatar
  • 361
17 votes
6 answers
5k views

Probability theory books for self-study

Are there any good books that explain important concepts of probability theory like probability distribution functions and cumulative distribution functions? Please, avoid referring books like "...
12 votes
1 answer
677 views

Can we think of a probability in both the classical and subjective sense simultaneously?

I'm a statistics student. I am trying to understand the classical and objective definitions of probability and how they are related to frequentist and Bayesian inference. It's not obvious to me why ...
Kareem Carr's user avatar
78 votes
11 answers
12k 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 disallow ...
Chill2Macht's user avatar
  • 6,309
18 votes
1 answer
2k views

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 ...
Davd's user avatar
  • 181
11 votes
2 answers
2k views

What is $P(X_1>X_2 , X_1>X_3,... , X_1>X_n)$?

All $X$ are mutually independent and from normal distributions, each with its own mean and variance. If it's easier, $P(X_1 \geq X_i \forall i \in \{1, ..., n\})$ is fine although I suspect it's the ...
Chris Idzerda's user avatar
130 votes
9 answers
102k 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 ...
arjsgh21's user avatar
  • 2,653
72 votes
5 answers
22k views

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 ...
gui11aume's user avatar
  • 14.8k
27 votes
4 answers
30k 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 ...
TheChymera's user avatar
11 votes
2 answers
16k 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 ...
user1516425's user avatar
61 votes
13 answers
56k 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 ...
user68492's user avatar
  • 611
44 votes
4 answers
14k views

Intuition for Conditional Expectation of $\sigma$-algebra

Let $(\Omega,\mathscr{F},\mu)$ be a probability space, given a random variable $\xi:\Omega \to \mathbb{R}$ and a $\sigma$-algebra $\mathscr{G}\subseteq \mathscr{F}$ we can construct a new random ...
Nicolas Bourbaki's user avatar

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