A probability provides a quantitative description of the likely occurrence of a particular event.
25
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3answers
3k views
A Probability distribution value exceeding 1 is OK?
On the Wikipedia page about naive bayes classifiers here there is this line "P(height|male) = 1.5789 (A probability distribution over 1 is OK. It is the area under the bell curve that is equal to 1.)" ...
8
votes
3answers
1k 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 ...
29
votes
16answers
6k views
What's the difference between probability and statistics?
What's the difference between probability and statistics, and why are they studied together?
22
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12answers
2k 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 ...
3
votes
1answer
130 views
Probability of getting between
...2 to 5 questions answered correctly, out of 20 of them? Each question has 5 choices. Probability of getting one right is 1/5. Probability of getting exactly 1 right is ${20 \choose 1} p^1 q^{19}$, ...
49
votes
6answers
3k views
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 ...
24
votes
12answers
1k 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, ...
5
votes
2answers
212 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 ...
5
votes
1answer
496 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 ...
4
votes
1answer
578 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 + ...
24
votes
7answers
1k views
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 ...
2
votes
1answer
375 views
Normal distribution and weight of babies problem
X is the weight of a baby when born (in gram). If the distribution of X is N(3315, 575) and Y is the number of babies with weight lower than 2719 in a random sample of 25, then P[Y≤4] is about...?
...
4
votes
2answers
413 views
What will be the correct answer, if we modify the “Best statistics question ever”?
There is a popular question, called "Best statistics question ever".
If you choose an answer to this question at random, what is the chance you will be correct?
A) 25%
B) 50%
C) 60%
D) 25%
This ...
2
votes
1answer
69 views
Probability associated with no of fleet calculation
Assume last year I commuted to work in a taxi and suppose if there are 'n' taxis in th fleet I used and if I took one of these taxis every trip at random and with replacement then what is the number ...
7
votes
3answers
461 views
Measure the uniformity of distribution of points in a 2D square
I have a 2D square, and I have a set of points inside it, say, 1000 points. I need a way to see if the distribution of points inside the square are spread out (or more or less uniformly distributed) ...
3
votes
2answers
172 views
What is the difference between using the multiplication rule or using Venn diagram subtraction for probability?
Take two problems:
Andrew is 35, and the probability he will be alive in 10 years is .72. Ellen is 35, for her, .92. Assuming these are independent, what is the probability they both will be alive ...
3
votes
2answers
179 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 ...
2
votes
4answers
2k views
How to create a dataset with conditional probability?
Suppose that a certain disease (D) has a prevalence of 3/1000. Also suppose that a certain symptom (S) has a prevalence (in the general population = people with that disease D and people without that ...
1
vote
3answers
135 views
Derive househould weights from a uniformly distributed person sample
The Swiss Public-Use Sample of the national census is a 5% sample drawn from the entire census survey. According to the documentation, the persons are sampled uniformly without replacement. Persons ...
65
votes
7answers
18k 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 ...
29
votes
6answers
2k views
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 ...
12
votes
3answers
654 views
What is a tight lower bound on the coupon collector time?
In the classic Coupon Collector's problem, it is well known that the time $T$ necessary to complete a set of $n$ randomly-picked coupons satisfies $E[T] \sim n \ln n $,$Var(T) \sim n^2$, and $\Pr(T ...
9
votes
5answers
937 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 ...
4
votes
1answer
419 views
Understanding proof of a lemma used in Hoeffding inequality
I am studying Larry Wasserman's lecture notes on Statistics which uses Casella and Berger as its primary text. I am working through his lecture notes set 2 and got stuck in the derivation of lemma ...
13
votes
5answers
213 views
Does this quantity related to independence have a name?
Obviously events A and B are independent iff Pr$(A\cap B)$ = Pr$(A)$Pr$(B)$. Let's define a related quantity Q:
$Q\equiv\frac{\mathrm{Pr}(A\cap B)}{\mathrm{Pr}(A)\mathrm{Pr}(B)}$
So A and B are ...
11
votes
1answer
447 views
Probability inequalities
I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts.
My problem is to find an exponential upper ...
9
votes
3answers
923 views
How to compute the probability associated with absurdly large Z-scores?
Software packages for network motif detection can return enormously high Z-scores (the highest I've seen is 600,000+, but Z-scores of more than 100 are quite common). I plan to show that these ...
4
votes
1answer
629 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 - ...
3
votes
1answer
220 views
How to interpret the divergence of Fisher information expectation?
Consider translated Weibull distribution with probability density function:
$$
f(x ; k, \lambda, \theta) = \frac{k}{\lambda} \left( \frac{x-\theta}{\lambda} \right)^{k-1} \exp\left( - ...
1
vote
1answer
64 views
Determining the probability that two functions produce the same outputs, when not all outputs of one function are known
Let's assume I have a (one-dimensional) array $I$ with a finite number $N$ elements. I also have a function $A$ that maps each element $I_n$ (for $0\leq\ n \lt N$) to the corresponding element $O_n$ ...
8
votes
1answer
288 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 ...
7
votes
4answers
280 views
How can I estimate the density of a zero-inflated parameter in R?
I have a data set with lots of zeros that looks like this:
set.seed(1)
x <- c(rlnorm(100),rep(0,50))
hist(x,probability=TRUE,breaks = 25)
I would like to draw ...
6
votes
4answers
561 views
Derive P(C | A+B) from Cox's two rules
I am working my way (self-study) through E.T. Jaynes' book Probability Theory - The Logic of Science
Original Problem
Exercise 2.1 says: "Is it possible to find a general formula for $p(C|A+B)$ ...
5
votes
1answer
276 views
Minimum tickets required for specified probability of winning lottery
In a lottery, 1/10 of the 50 000 000 tickets give a prize.
What is the minimum amount of tickets one should buy to have at least a 50% chance to win?
Would be very glad if you could explain your ...
4
votes
1answer
494 views
Geometric distribution without replacement
On an attempt to solve this problem I've managed to reduce it to finding the expected number of white balls picked until one black ball is observed (let's call that value $v$). Except that, unlike the ...
4
votes
2answers
368 views
Meaning of probability notations $P(z;d,w)$ and $P(z|d,w)$
What is the meaning between the notation $P(z;d,w)$ and $P(z|d,w)$ which is commonly used in many books and papers?
3
votes
1answer
92 views
Probability of an order statistic
If I randomize time points $$T=\{266.14646, 107.28526, 108.89631, 593.03129,\\ 118.10284, 425.60470, 39.02817, 291.90210\}$$ into two groups of equal size, then what is the probability that $T_i$ (the ...
7
votes
3answers
282 views
What are the chances my wife has lupus?
*1.5 million people have lupus in America out of a population of 308 million. 90 percent are women. My wife is white, as is 63 percent of the country. Minorities are three times more likely to have ...
7
votes
4answers
246 views
Variance of resistors in parallel
Suppose you have a set of resistors R, all of which are distributed with mean μ and variance σ.
Consider a section of a circuit with the following layout: (r) || (r+r) || (r+r+r). The equivalent ...
7
votes
1answer
293 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?
7
votes
2answers
481 views
What is the distribution of maximum of a pair of iid draws, where the minimum is an order statistic of other minima?
Consider $n\cdot m$ independent draws from cdf $F(x)$, which is defined over 0-1, where $n$ and $m$ are integers. Arbitrarily group the draws into $n$ groups with m values in each group. Look at the ...
4
votes
1answer
512 views
Simple combination/probability question based on string-length and possible-characters
Assuming "complete randomness" and given a string with a length of 20 characters where each character may be one of 62 possible characters:
What are the total number of combinations possible? ...
3
votes
2answers
113 views
Properties of moment-generating functions
I am new to statistics and I happen to came across this property of MGF:
Let $X$ and $Y$ be independent random variables. Let $Z$ be equal to $X$, with probability $p$, and equal to $Y$, with ...
3
votes
1answer
276 views
Posterior distribution for multinomial parameter
(topic moved from maths.stackexchange.com)
I'm currently developing an application integrating a probabilistic inference engine for Bayesian Networks. The Bayesian Network integrates some form of ...
3
votes
2answers
244 views
What methods can be used to specify priors from data?
Background
I am generally interested in learning appropriate methods of using data to specify priors. A previous question asks how to elicit priors from experts and received some good ...
2
votes
1answer
327 views
probability of one random variable being greater than another
Using the normal distribution. Let $X \sim N(1, 2)$ and $Y \sim N(2, 3)$ where $N(\mu, \sigma^2)$ denotes the normal distribution with mean $\mu$ and variance $\sigma^2$. $X$ and $Y$ are independent.
...
2
votes
2answers
159 views
1
vote
0answers
88 views
Should the intersection of 2 event sets be on the same sample space?
I'm trying to understand a stochastic process and consecutive events in time.
Should the intersection of two events be on the same sample space, on a combination of simultaneous sample spaces, or a ...
1
vote
3answers
112 views
Confusion regarding a probability problem
A certain auditorium has 30 rows of seats. Row 1 has 11 seats, while Row 2 has 12 seats, Row 3 has 13 seats, and so on to the back of the auditorium where Row 30 has 40 seats. A door prize is to be ...
1
vote
2answers
256 views
Decomposition of the sum of two random variables
We observe the input samples as $Z = X+Y$, where distribution of $Z$ could be estimated by histogram of samples and $X,Y$ are two independent random variables. One of the variable was known following ...
