Questions tagged [probability]
A probability provides a quantitative description of the likely occurrence of a particular event.
1,418
questions
5
votes
2
answers
841
views
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 ...
- 195
91
votes
6
answers
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 ...
- 881
61
votes
13
answers
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 ...
- 611
16
votes
6
answers
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 ...
- 161
13
votes
2
answers
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 ...
- 799
11
votes
1
answer
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 ...
- 121
8
votes
4
answers
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 ...
7
votes
2
answers
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 ...
- 213
7
votes
7
answers
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 ...
61
votes
11
answers
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 ...
- 1,514
43
votes
3
answers
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 ...
- 34.2k
31
votes
2
answers
21k
views
Why are probability distributions denoted with a tilde?
What is the meaning of the tilde when specifying probability distributions? For example:
$$Z \sim \mbox{Normal}(0,1).$$
- 403
14
votes
2
answers
11k
views
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 ...
12
votes
5
answers
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 $log(X)$ ~ $N(log(\mu), \sigma^2/\mu^2)$?
- 1,692
10
votes
1
answer
505
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 ...
- 344
5
votes
1
answer
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 ...
- 81
49
votes
8
answers
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 ...
- 1,831
35
votes
3
answers
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, ...
- 603
31
votes
8
answers
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 ...
- 9,702
28
votes
3
answers
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 ...
- 381
27
votes
1
answer
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,...
- 453
19
votes
2
answers
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?
- 191
15
votes
2
answers
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 ...
- 343
14
votes
2
answers
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 ...
- 151
14
votes
1
answer
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 ...
- 1,049
13
votes
1
answer
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?
- 3,805
13
votes
1
answer
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 ...
- 131
12
votes
1
answer
29k
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
$$\Pr(Y=1) = \frac{1}{1 + \exp\left(-[-3.92 + 0....
- 2,445
12
votes
1
answer
5k
views
What is the expected norm $\mathbb E \lVert X \rVert$ for a multivariate normal $X \sim \mathcal N(\mu, \Sigma)$? [duplicate]
$\DeclareMathOperator\E{\mathbb E}
\DeclareMathOperator\Var{\mathrm{Var}}
\newcommand\R{\mathbb R}
\DeclareMathOperator\N{\mathcal N}
\DeclareMathOperator\tr{\mathrm{tr}}$Suppose
$X \sim \N(\mu, \...
- 23.4k
9
votes
11
answers
1k
views
How do Bayesians interpret $P(X=x|\theta=c)$, and does this pose a challenge when interpreting the posterior?
I have seen the post Bayesian vs frequentist interpretations of probability and others like it but this does not address the question I am posing. These other posts provide interpretations related to ...
- 2,632
8
votes
1
answer
527
views
Mars attack (probability to destroy $n$ spaceships with $k \cdot n$ missiles)
Suppose Earth has been attacked by $n$ Martian spaceships and suppose that we have $m=k \cdot n$ missiles to release against the $n$ spaceships. The probability to hit and destroy each spaceship by ...
- 669
7
votes
2
answers
1k
views
How much of neural network overconfidence in predictions can be attributed to modelers optimizing threshold-based metrics?
Neural network "classifiers" output probability scores, and when they are optimized via crossentropy loss (common) or another proper scoring rule, they are optimized in expectation by the ...
- 51.2k
7
votes
1
answer
4k
views
PDF/CDF of max-min type random variable
For i.i.d. random variables, we may write the CDF of $t=\max(t_1,\cdots,t_N)$ as
$$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$$
When we have $X=\...
- 223
6
votes
1
answer
5k
views
Determining beta distribution parameters $\alpha$ and $\beta$ from two arbitrary points (quantiles)
Suppose I have two points $(p_1,x_1)$ and $(p_2,x_2)$ where $p_i$ is a probability on the beta CDF and $x_i$ is a value on that same CDF. How would I go about determining the beta distribution shape ...
- 163
4
votes
2
answers
2k
views
Probability of getting the exact same letters in Scrabble 2 turns in a row?
I would guess that the odds are astronomical that when you play 7 letters (as I did with LAUHTER added to the G on the board to play the bingo: LAUGHTER) and then get 7 new letters that are exactly ...
2
votes
2
answers
359
views
How to include the observed values, not just their probabilities, in information entropy?
Shannon entropy measures the unpredictability in a random variable's outcome as the weighted average of the probabilities of that variable's outcomes or observed values. However, it discards the ...
- 3,401
146
votes
21
answers
109k
views
What's the difference between probability and statistics?
What's the difference between probability and statistics, and why are they studied together?
- 1,597
39
votes
1
answer
32k
views
Derivation of change of variables of a probability density function?
In the book pattern recognition and machine learning (formula 1.27), it gives
$$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 ...
- 15.3k
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 ...
- 433
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.
...
- 311
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 \...
- 1,441
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, ...
- 465
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 ...
- 3,125
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 ...
- 517
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 ...
- 333
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 ...
- 86.4k
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 - \...
- 487
17
votes
2
answers
3k
views
Which is largest, of a bunch of normally distributed random variables?
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 $1$. ...
- 6,443
17
votes
2
answers
670
views
Interpretation of confidence interval
Note: apologies in advance if this is a duplicate, I didn't find a similar q in my search
Say we have a true parameter p. A confidence interval C(X) is a RV that contains p, say 95% of the time. Now ...
- 171