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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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### Distribution given sum

I'm stuck on an exercise (it's not homework, but preparation for finals). It goes like this: $X_1, \dots, X_n$ are iid Exponential($\lambda$) (with parametrization $f(x)=\lambda e^{-\lambda x}$). What ...
18k views

### If I have a 58% chance of winning a point, what's the chance of me winning a ping pong game to 21, win by 2?

I have a bet with a co-worker that out of 50 ping pong games (first to win 21 points, win by 2), I will win all 50. So far we've played 15 games and on average I win 58% of the points, plus I've won ...
54k 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 ...
13k 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 ...
7k views

### How to compute conditional expectations with respect to a sigma field?

Example: Toss a coin twice. Letting $\mathbb P$ be a probability measure, suppose $\mathbb P(HH)=p^2,\mathbb P(HT)=\mathbb P(TH)=p(1-p), \mathbb P(TT)=(1-p)^2.$ I would like to answer the following ...
8k views

### 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 ...
2k views

### 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 ...
796 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 ...
6k views

### Book recommendations for probability

I am looking for a book (English only) that I can treat as a reference text (more colloquially as a bible) about probability and is as complete - with respect to an undergraduate/graduate education in ...
13k views

### Brain teaser: How to generate 7 integers with equal probability using a biased coin that has a pr(head) = p?

This is a question I found on Glassdoor: How does one generate 7 integers with equal probability using a coin that has a $\mathbb{Pr}(\text{Head}) = p\in(0,1)$? Basically, you have a coin that may or ...
7k views

### 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 ...
21k views

### Why are probability distributions denoted with a tilde?

What is the meaning of the tilde when specifying probability distributions? For example: $$Z \sim \mbox{Normal}(0,1).$$
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### What is probabilistic inference?

I am reading Chris Bishop's Pattern Recognition and Machine Learning textbook. I came across the term probabilistic inference several times. I have a couple of questions. Is probabilistic inference ...
10k views

### If $X$ is normally distributed, can $\log(X)$ also be normally distributed?

Suppose $X$ is distributed $N(\mu, \sigma^2)$ where $\mu \neq 0$. Can I use the Delta Method to say that $log(X)$ ~ $N(log(\mu), \sigma^2/\mu^2)$?
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### 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 ...
364 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 ...
8k views

### Would a Bayesian admit that there is one fixed parameter value?

In Bayesian data analysis, parameters are treated as random variables. This stems from the Bayesian subjective conceptualization of probability. But do Bayesians theoretically acknowledge that there ...
5k views

### 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, ...
58k views

### What's the difference between a probability and a proportion?

Say I have eaten hamburgers every Tuesday for years. You could say that I eat hamburgers 14% of the time, or that the probability of me eating a hamburger in a given week is 14%. What are the main ...
8k 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 ...
14k views

### Converting (normalizing) very small likelihood values to probability

I am writing an algorithm where, given a model, I compute likelihoods for a list of datasets and then need to normalize (to probability) each one of the likelihood. So something like [0.00043, 0.00004,...
20k 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?
8k views

### How to choose significance level for a large data set?

I am working with a data set having N around 200,000. In regressions, I am seeing very small significance values << 0.001 associated with very small effect sizes, e.g. r=0.028. What I'd like to ...
10k views

### Success of Bernoulli trials with different probabilities

If 20 independent Bernoulli trials are carried out each with a different probability of success and therefore failure. What is the probability that exactly n of the 20 trials was successful? Is there ...
5k views

### How to define a distribution such that draws from it correlate with a draw from another pre-specified distribution?

How do I define the distribution of a random variable $Y$ such that a draw from $Y$ has correlation $\rho$ with $x_1$, where $x_1$ is a single draw from a distribution with cumulative distribution ...
2k views

### Decomposing the normal distribution

Does there exist a positive-only distribution such that the difference of two independent samples from this distribution is normally distributed? If so, does it have a simple form?
2k views

### 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 ...
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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 $$\Pr(Y=1) = \frac{1}{1 + \exp\left(-[-3.92 + 0.... 12 votes 1 answer 5k views ### What is the expected norm \mathbb E \lVert X \rVert for a multivariate normal X \sim \mathcal N(\mu, \Sigma)? [duplicate] \DeclareMathOperator\E{\mathbb E} \DeclareMathOperator\Var{\mathrm{Var}} \newcommand\R{\mathbb R} \DeclareMathOperator\N{\mathcal N} \DeclareMathOperator\tr{\mathrm{tr}}Suppose X \sim \N(\mu, \... 9 votes 11 answers 1k views ### How do Bayesians interpret P(X=x|\theta=c), and does this pose a challenge when interpreting the posterior? I have seen the post Bayesian vs frequentist interpretations of probability and others like it but this does not address the question I am posing. These other posts provide interpretations related to ... 8 votes 1 answer 527 views ### Mars attack (probability to destroy n spaceships with k \cdot n missiles) Suppose Earth has been attacked by n Martian spaceships and suppose that we have m=k \cdot n missiles to release against the n spaceships. The probability to hit and destroy each spaceship by ... 7 votes 2 answers 1k views ### How much of neural network overconfidence in predictions can be attributed to modelers optimizing threshold-based metrics? Neural network "classifiers" output probability scores, and when they are optimized via crossentropy loss (common) or another proper scoring rule, they are optimized in expectation by the ... 7 votes 1 answer 4k views ### PDF/CDF of max-min type random variable For i.i.d. random variables, we may write the CDF of t=\max(t_1,\cdots,t_N) as$$F_t(t)=F_{t_i}(x)^n$$and the CDF of x=\min(x_1,\cdots,x_N) as$$F_x(x)=1-(1-F_{x_i}(x))^n$$When we have X=\... 6 votes 1 answer 5k views ### Determining beta distribution parameters \alpha and \beta from two arbitrary points (quantiles) Suppose I have two points (p_1,x_1) and (p_2,x_2) where p_i is a probability on the beta CDF and x_i is a value on that same CDF. How would I go about determining the beta distribution shape ... 4 votes 2 answers 2k views ### Probability of getting the exact same letters in Scrabble 2 turns in a row? I would guess that the odds are astronomical that when you play 7 letters (as I did with LAUHTER added to the G on the board to play the bingo: LAUGHTER) and then get 7 new letters that are exactly ... 2 votes 2 answers 359 views ### How to include the observed values, not just their probabilities, in information entropy? Shannon entropy measures the unpredictability in a random variable's outcome as the weighted average of the probabilities of that variable's outcomes or observed values. However, it discards the ... 146 votes 21 answers 109k views ### What's the difference between probability and statistics? What's the difference between probability and statistics, and why are they studied together? 39 votes 1 answer 32k views ### Derivation of change of variables of a probability density function? In the book pattern recognition and machine learning (formula 1.27), it gives$$p_y(y)=p_x(x) \left | \frac{d x}{d y} \right |=p_x(g(y)) | g'(y) |$$where x=g(y), p_x(x) is the pdf that ... 33 votes 7 answers 9k views ### What is the name of the statistical fallacy whereby outcomes of previous coin flips influence beliefs about subsequent coin flips? As we all know, if you flip a coin that has an equal chance of landing heads as it does tails, then if you flip the coin many times, half the time you will get heads and half the time you will get ... 31 votes 3 answers 12k views ### Extending the birthday paradox to more than 2 people In the traditional Birthday Paradox the question is "what are the chances that two or more people in a group of n people share a birthday". I'm stuck on a problem which is an extension of this. ... 28 votes 1 answer 5k views ### Central limit theorem and the law of large numbers I have a very beginner's question regarding the Central Limit Theorem (CLT): I am aware that the CLT states that a mean of i.i.d. random variables is approximately normal distributed (for n \to \... 27 votes 4 answers 35k views ### Intuition for cumulative hazard function (survival analysis) I'm trying to get intuition for each of the main functions in actuarial science (specifically for the Cox Proportional Hazards Model). Here's what I have so far: f(x): starting at the start time, ... 24 votes 4 answers 2k views ### The magic money tree problem I thought of this problem in the shower, it was inspired by investment strategies. Let's say there was a magic money tree. Every day, you can offer an amount of money to the money tree and it will ... 23 votes 3 answers 6k views ### Uniform random variable as sum of two random variables Taken from Grimmet and Stirzaker: Show that it cannot be the case that U=X+Y where U is uniformly distributed on [0,1] and X and Y are independent and identically distributed. You should not ... 20 votes 3 answers 10k views ### To maximize the chance of correctly guessing the result of a coin flip, should I always choose the most probable outcome? This is not homework. I am interested in understanding if my logic is correct with this simple stats problem. Let's say I have a 2 sided coin where the probability of flipping a head is P(H) and ... 20 votes 2 answers 37k views ### Why is the probability zero for any given value of a normal distribution? I noticed that in the Normal distribution, the probability P(x=c) equals zero, while for the Poisson distribution, it will not equal zero when c is a non-negative integer. My question is: Does ... 19 votes 4 answers 985 views ### Draw integers independently & uniformly at random from 1 to N using fair d6? I wish to draw integers from 1 to some specific N by rolling some number of fair six-sided dice (d6). A good answer will explain why its method produces uniform and independent integers. As an ... 19 votes 1 answer 14k views ### What are the sharpest known tail bounds for \chi_k^2 distributed variables? Let X \sim \chi^2_k be a chi-squared distributed random variable with k degrees of freedom. What are the sharpest known bounds for the following probabilities$$ \mathbb{P}[X > t] \leq 1 - \...
I have random variables $X_0,X_1,\dots,X_n$. $X_0$ has a normal distribution with mean $\mu>0$ and variance $1$. The $X_1,\dots,X_n$ rvs are normally distributed with mean $0$ and variance $1$. ...