Questions tagged [probability]
A probability provides a quantitative description of the likely occurrence of a particular event.
11,186
questions
0
votes
0
answers
8
views
Computing conditional distribution of hidden state given observed states?
I am interested in the following Gaussian linear system that describes a Hidden Markov Model (HMM):
$$x_{k+1}=Ax_k + u_k + \xi_k, \xi_k \sim N_2((0,0), 0.01I_2)\\
y_{k+1}=C^tx_{k+1}+\eta_k, \eta_k \...
- 157
0
votes
0
answers
6
views
Betting on a sample from a known distribution
This was an interview question. Given a known distribution, sample a value from it with replacement for many times. Two people A and B bet on the sample with their own guesses, and the one closer to ...
- 1
-2
votes
0
answers
9
views
"Consensus" on Analyzing Mixed Continuous and Categorical Data in the field of Statistics? [closed]
I have been trying to determine the popular "consensus" as to how mixed continuous and categorical data (e.g. a dataset that has variables on income and gender) is generally analyzed in the ...
- 5,750
1
vote
1
answer
19
views
MNIST with a TWIST, no labels given, only probabilities
Let's say we have basic MNIST dataset, and we have the same goal to predict the digit, BUT we're swapping all the labels by RED ...
- 121
1
vote
0
answers
13
views
Generate a random covariance matrix with specified eigenspectra and diagonal elements?
I want to generate a random covariance matrix ($c \in \mathcal{R}^{n \times n} $) whose eigenspectra, i.e., $n$ eigenvalues $e_0 \in \mathcal{R}^{n\times 1}$ and diagonal elements $c_{ii} \,\, i=1 \,\,...
- 31
2
votes
1
answer
47
views
What is Galton's paradox?
From One Thousand Exercises in Probability I found the exercise:
Galton's paradox. You flip three fair coins. At least two are alike, and it is an even chance that the third is a head or a tail. ...
- 3,403
1
vote
0
answers
37
views
Probability of seeing sun rise tomorrow using Bayes Theorem
It is a stormy night and an alien prisoner is abandoned on earth during the first ice age. Scared, he finds a cave and only comes out for a few hours at night to hunt some food. He has no luck though. ...
- 277
0
votes
0
answers
19
views
What is the correct definition of the correlation of two random variables?
In the book Signal Processing for Communications by Prandoni and Vetterli, correlation of two variables is defined as the expected value of their product, while the covariance is refered to as the &...
- 247
0
votes
0
answers
14
views
Expectation values [closed]
What is the expectation value of the expectation value. How to calculate:
<∆j> where, ∆j=spread of data= j-<j.> ;for any variable j
- 1
3
votes
1
answer
38
views
What is meaning of such notations in general?
I read a notation on a paper about statistics and machine learning ("High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models" by
Tengyuan Liang,
Subhabrata Sen, and Pragya ...
- 262
1
vote
2
answers
37
views
Probability of selecting 3 students being in different classes
I have come across this problem and I can't quite figure out the solution.
A class of 60 students is divided into 3 groups of 20 randomly. Three friends are in this year group. Find the probability ...
1
vote
0
answers
26
views
Do the variables used in computing the mean and variance have to be i.i.d.?
For example if you have a theoretical server that processes incoming requests where in this case there are so many requests that the callback delays depend upon each other, can you just take the ...
- 111
2
votes
1
answer
427
views
In a game with 0.01 chance of survival, there are 100 participants, a specific player survives twice what are the odds?
Sorry ahead if my question is very beginner-ish but I'm confused with this example. Your help is appreciated
Let's say we have a survival game where 100 players participate each time and only one of ...
- 133
0
votes
0
answers
18
views
Probability of bringing a card that is already in the collection
Let's say there is a card collection that is made of 1,000 different cards. I start calling my friends and each of them brings 15 random cards to the collection. The first friend will bring 100% new ...
1
vote
0
answers
17
views
Uniform distribution of skewed dice and the central limit theorem
I am reading MacKay's Information Theory, Inference, and Learning Algorithms. I am not a student.
On pg. 36, ex 2.16, c and d:
C. How can two cubical dice be labelled using the numbers
{0,1,2,3,4,5,6}...
- 373
0
votes
0
answers
48
views
If Y is the sum of normalized multivariate pareto random variables then Y is a Feller-Pareto random variable
If we let $\underset{k\times 1}{\boldsymbol{X}}=(X_1, \dots, X_k)' \sim MP^{(k)}(\boldsymbol{0},\boldsymbol{1}, \alpha)$ where MP denotes a Multivariate-Pareto distribution, with joint survival ...
- 11
1
vote
0
answers
34
views
Issue with bounded in probability
I have tried to prove the following problem that I read in the lecture and it seems not transparent to me.
Suppose that $Y_{i}$ be independent random variables (with $i=1,2,3, \dotsc$). Each has the ...
- 23
3
votes
1
answer
32
views
Probability for a bin in a binned histogram
This question is very basic, but I cannot figure the error in my thinking. According to the author of the book "Pattern Recognition and Machine Learning", we can get the Probability ...
- 33
1
vote
1
answer
42
views
Fitting discrete data to continuous distributions
I'm creating a simulation model, in which some stochastic factors are included. On of my stochastic factors is the amount of containers arriving daily for a specific delivery location. A plot of this ...
- 113
0
votes
0
answers
22
views
Given two sequences of random variables whose moments match, does their difference tends to 0?
Suppose you have two sequences of random variables $(X_n)_{n \in \mathbb{N}}$ and $(Y_n)_{n \in \mathbb{N}}$ and you know that for every $n$
$$ \mathbb{E}[X_n^r] = \mathbb{E}[Y_n^r] \quad \forall \, r ...
- 1
5
votes
1
answer
68
views
Show that no two sets in the probability space with $\mathbb{P}(\{k\})=2^{-k!}$ are independent
Let $\mathcal{P}(\mathbb{N})$ denote the power set of $\mathbb{N}$.
Show that no two non-trivial sets in the probability space $(\mathbb{N},\mathcal{P}(\mathbb{N}),\mathbb{P})$ with $\mathbb{P}(\{k\})=...
- 151
0
votes
0
answers
49
views
How do I calculate the constant K that makes the area under the pdf equal to 1 [closed]
You are analyzing the probability of defective machine parts coming out of an assembly line.
Data collected over 60 consecutive days, revealed 2 days where defects were observed in machine parts.
To ...
- 1
1
vote
0
answers
26
views
How many random draws needed to sample a uniform space?
I am looking for a way to estimate how many random draws took place to produce a produce a known number of unique results. I'm sampling a uniform distribution.
For example, if there were a lottery ...
- 11
2
votes
2
answers
46
views
Logistic regression simulation with respect to event occurrence (prevalence)
I am trying to simulate logistic regression data, but under the constraints of prevalence.
$$\text{logit}(y_i) = \beta_0 + \beta_1 X_1 + \beta_2X_2$$
For example, I want to create a dataset that has ...
- 111
1
vote
1
answer
67
views
Consecutive coin flips, what is the appropriate statistical test for this word problem? [closed]
I was listening to a podcast by NDGT (Neil deGrasse Tyson, a prominent scientist) and he posed a simple thought experiment to illustrate the susceptibilities to cognitive bias. What I've come here to ...
- 111
2
votes
1
answer
46
views
Prove $P(X_1+X_2> 2C) \leq P(X_1>C)$ if $X_1,X_2$ are identical, but dependent?
If $X_1,X_2$ are dependent but identically distributed, it seems obvious that $P(X_1+X_2\geq2C) \leq P(X_1\geq C)=P(X_2\geq C)$. At least if we additionally assume that the joint distribution is ...
- 23
0
votes
0
answers
11
views
Multivariate analysis of sample mean and sample variance
$\{X_n\}$ Let be a sequence of iid probability vectors with mean vector$ \mu$ and variance-covariance matrix$ Σ$.
In this case, sample variance and sample covariance are defined as follows
$S_{n,j}^2=\...
0
votes
1
answer
30
views
Distribution function of the linear combination of standardized student-t quantiles [duplicate]
Assume to observe 2 quantiles, x and y, associated with the z% probability. These quantiles are generated by 2 non-independent standardized student-t distributions X and Y.
In case of linear ...
- 1
0
votes
0
answers
11
views
When I have population data do I need probability and can I infere? [duplicate]
Say I have data about a population and I want to analyze them, then in my understanding the "sample" is the population. And then do I need probability, like p-values, confident intervalls ...
- 11
1
vote
0
answers
25
views
Calculating Probability Using Bayes? [closed]
A recent survey of residents in Texas concluded that 55% of Austin city residents and 46% of Houston city residents broke a bone at some point during their childhood.
Let’s say Austin has 5200 ...
- 743
2
votes
0
answers
25
views
Fitting Sparsed Constrained regression with non-negative coefficients adding to 1
I see a similar problem in How do I fit a constrained regression in R so that coefficients total = 1?
Specifically, my model is
$Y_i= \pi_1 X_1+\pi_2 X_2 +...+ \pi_K X_K +\epsilon_i$ with $\pi_k \ge 0$...
2
votes
1
answer
26
views
probability that the players will exchange their initially drawn number
Consider the following two-player game. The players simultaneously draw one sample
each from a continuous random variable X, which follows $Uniform\ [0, 100]$. After observing the value of her own ...
1
vote
2
answers
18
views
Notation of expectation with conditional in subscript
Inside the book "The Elements of statistical learning", I stumbled upon the following notation (Ex. 2.7)
$$E_{\mathcal{Y|X}}(f(x_0) - \hat{f}(x_0))^2$$
where $\mathcal{X, Y}$ are two random ...
- 13
3
votes
1
answer
47
views
What is the intuition behind the odds scale?
What is an intuitive explanation of the odds scale?
In a logistic regression such as $$logit(p) = \beta_0 + \beta_1 x$$
we often interpret $\beta_1$ by looking at the odds ratio, $e^{\beta_1}$, which ...
- 327
0
votes
1
answer
20
views
How to calculate real probability with multiple scenarios?
I would like to know how to calculate the probability of being profitable on an event where there are multiple scenarios. For example paying 5 dollars to grab a bill from a box box full of bills where ...
- 1
1
vote
2
answers
43
views
Conditioning of join gaussian over a line
I need to compute the conditional probability of bivariate normal distribution over a line.
Let's suppose that X and Y both are normal distributions and that they are independent.
Let's suppose that ...
- 117
1
vote
0
answers
59
views
What is the probability mass function of Rock, Paper, Scissors?
I was curious about the statistics behind the game of rock, paper, scissors. Let's say n people are playing, where n is greater than or equal to 2. If when all n people reveal their play and only 1 ...
- 743
1
vote
0
answers
9
views
General way to describe dot product of N different random variables [duplicate]
I'm a novice to DS, so feel free to correct me.
Imagine we have $N$ biased coins each with different probability of getting heads (which are known to us prior). What's the probability of getting $k$ ...
- 111
0
votes
0
answers
14
views
Conditional probability problem (Bayes Theorem ?) [closed]
Hello, basically, I can't find 1/2 for the very last question.
I tried to use the Baye's Theorem, but it wasn't successful
Could someone help me ?
0
votes
0
answers
27
views
Given 15 random birthdays in a year, what are the odds that there are at least 120 consecutive days in a year without a birthday? [closed]
The 120 days can be shared between consecutive years too, such as roughly November through February.
- 1
1
vote
1
answer
30
views
MLE of the Uniform Distribution
In a uniform distribution where $0\leq X \leq \theta$, the pdf is represented as $f(X|\theta) = \frac{1}{\theta}I(0\leq X \leq \theta)$, and the likelihood is $L(\theta) = \prod\frac{1}{\theta}I(0\leq ...
- 61
0
votes
0
answers
35
views
MLE for the Uniform distribution [duplicate]
I understand how a random sample $x_1, ..., x_n$ following the Uniform Distribution with $0 \leq x \leq \theta$ has a log-likelihood proportional to $\frac{1}{\theta^2}$. I am told that the MLE for $\...
- 61
1
vote
0
answers
32
views
Do Mixture Models "Defy" Entropy?
Recently, I have learned about the principle of Maximum Entropy with regards to Probability Distribution (https://www.youtube.com/watch?v=2gTrsLVnp9c) - in particular, when certain "information&...
- 5,750
0
votes
0
answers
30
views
Expected value of a random variable with truncation
Let $f:[0,\infty)\to \mathbb R_+$ denote the PDF of a random variable $X$ and $c>0$ a constant. I want to evaluate the following integral:
$$I(c)=\int_0^\infty{\min(x,c)f(x)dx}.$$
This can be ...
- 113
4
votes
2
answers
407
views
0
votes
0
answers
12
views
How to actually apply martingales when conditioning on a random variable (not filtration)?
For a Galton-Watson process, I've shown that $\frac{Z_n}{\mu^n}$ is a martingale i.e. $E[\frac{Z_{n+1}}{\mu^{n+1}}|\mathcal{F}_n]=\frac{Z_n}{\mu^n}$.
However, I want to show that, for $n>m$,
$$E[Z_{...
- 157
0
votes
1
answer
20
views
Finding probability of a team winning the title given probability of future matches
Say, we have two teams: Manchester City and Liverpool. Both of them have 3 games remaining with the following probabilities of winning/drawing:
...
0
votes
0
answers
52
views
What is the conditional expectation of two random Poisson variables?
Say I have two random variables $Z^{p}_{i} \mid Y^{p}_{i} \sim \text{Poisson} \left(t^{p}_{i}Y^{p}_{i}\right)$ and $Z^{q}_{i} \mid Y^{q}_{i} \sim \text{Poisson} \left(t^{q}_{i}Y^{q}_{i}\right)$ for ...
0
votes
0
answers
19
views
T-testing on sparse data
Problem: I have two stochastic processes, $S_1$ and $S_2$, that frequently are zero, but occasionally have positive values with unknown probabilities $q_1$ and $q_2$. e.g.
$$
S_1 = \{0,0,0,0,0,21,0,0,...
- 1
3
votes
2
answers
98
views
Calculating the probability of guessing 3 card draws out of 6
My friend guessed the correct card drawn from a full deck of 52, 3 times out of 6 guesses (card not replaced each time).
I'm trying to work out the probability. My thinking is that if you guessed ...
- 1,047