# Tagged Questions

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

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### 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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### Why does a 95% 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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### A Probability distribution value exceeding 1 is 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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### What's the difference between probability and statistics?

What's the difference between probability and statistics, and why are they studied together?
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### Motivation for Kolmogorov distance between distributions

There are many ways to measure how similar two probability distributions are. Among methods which are popular (in different circles) are: the Kolmogorov distance: the sup-distance between the ...
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### What does the hidden layer in a neural network compute?

I'm sure many people will respond with links to 'let me google that for you', so I want to say that I've tried to figure this out so please forgive my lack of understanding here, but I cannot figure ...
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### What is the probability that this person is female?

There is a person behind a curtain - I do not know whether the person is female or male. I know the person has long hair, and that 90% of all people with long hair are female I know the person has a ...
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### 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 ...
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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, ...
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### Are there any good movies involving mathematics or probability?

Can you suggest some good movies which involve math, probabilities etc? One example is 21. I would also be interested in movies that involve algorithms (e.g. text decryption). In general "geeky" ...
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### Why law of large numbers does not apply in the case of Apple share price?

Here is the article in NY times called "Apple confronts the law of large numbers". It tries to explain Apple share price rise using law of large numbers. What statistical (or mathematical) errors does ...
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### 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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### What is Bayes' theorem all about?

What are the main ideas, that is, concepts related to Bayes' theorem? I am not asking for any derivations of complex mathematical notation.
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### Can two random variables have the same distribution, yet be almost surely different?

Is it possible that two random variables have the same distribution and yet they are almost surely different?
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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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### The operation of chance in a deterministic world

In Steven Pinker's book Better Angels of Our Nature, he notes that Probability is a matter of perspective. Viewed at sufficiently close range, individual events have determinate causes. Even a ...
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### Combining classifiers by flipping a coin

I am studying a machine learning course and the lecture slides contain information what I find contradicting with the recommended book. The problem is the following: there are three classifiers: ...
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### Statistics, war stories, data intuition

I think it is fair to say statistics is an applied science so when averages and standard deviations are calculated it is because someone is looking to make some decisions based on those numbers. Part ...
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### 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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### 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 ...
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### 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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### 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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### Confidence interval and probability - where is the error in this statement?

If someone makes a statement like below: "Overall, nonsmokers exposed to environmental smoke had a relative risk of coronary heart disease of 1.25 (95 percent confidence interval, 1.17 to ...
### Closed form expression for the quantiles of $\alpha_1\sin(x)+\alpha_2\cos(x)$
I have two random variables, $\alpha_i\sim \text{iid }U(0,1),\;\;i=1,2$ where $U(0,1)$ is the uniform 0-1 distribution. Then, these yield a process, say: P(x)=\alpha_1\sin(x)+\alpha_2\cos(x), ...