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

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When do we have that $E[X_{T+1} | X_T] = X_T$?

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_n\}_{n \in \mathbb{N}}, \mathbb{P})$: Let $X = \{X_n\}_{n \in \mathbb N}$ be a $(\{\mathscr{F}_n\}_{n \in \mathbb{N}}, ...
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22 views

The Second Hearts Problem

Intro: According to the last part of these lecture notes, if we have a standard deck of playing cards and turn cards until the first heart appears, the probability that the next card is a heart is ...
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3 views

How to calibrate scored probabilities to counter over prediction of events in ranked deciles?

I have a dataset that has 3 class probabilites w.r.t. a reference class. Using neural network. I scored the dataset and getting over prediction of events when I rank order the scored probabilities, ...
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18 views

MLE for Bivariate Normal with Missing at Random data

I am trying to work the example on pg 16 of these notes from U Chicago. The setting is that we have $Y_i = (Y_{i1}, Y_{i2}) \sim N(\mu, \Sigma)$, all $Y_i$ iid, with mean $\mu = (\mu_1, \mu_2)^T$ ...
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17 views

Sum of two different probability functions

I'm have some problems estimating the marginal distribution from data. I found three different distributions that explain the data quite well: two log-normal (black) and one normal distribution (red). ...
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17 views

Does picking randomly from an imbalanced group balance the group?

If I had a bag of marbles with 75% blue and 25% red (ratio is what matters not raw number, so this applies to 100 marbles, 1000 marbles, 100000 marbles) So if I had this imbalance of 75% blue and 25% ...
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16 views

What can I do with NA values in my second-order Markov chain?

I have states A, B, C, I have developed both a 1st and 2nd Order Markov Chain for them. When developing my 2nd-order, the transition probability BA is 0: ($P(x_t = A | x_{t-1} = B) = 0 $) ...
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2answers
50 views

What's the problem with my sports streaks problem solution?

The problem is from the book "Think stats". It's formulated as follows: If there are 10 players in a basketball game and each one takes 15 shots during the course of the game, and each shot has ...
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23 views

How can we prove this equation using marginalization and conditioning? [on hold]

I want to prove $$P(A|C) = \sum_{B} P(AB|C) $$ How can we prove this using marginalization and conditioning?
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1answer
24 views

Compute likelihood of two events happening at the same millisecond [on hold]

I would like to write a python (or R, etc) function that computes the likelihood of two events happening at the same millisecond. The events are independent an can happen at any time of the day. My ...
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1answer
16 views

Probability of mutation at the same site

In two identical, but genetically isolated DNA sequences (which are essentially strings of so-called nucleotides - A, T, G, C), for example: ...
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18 views

Combining probabilities [duplicate]

I apologize in advance - I'm asking as a relative novice and may be using incorrect terminology, or may be missing a similar question already posed on this site. Let's assume I have a classification ...
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1answer
49 views

Some basic probability dilemma

So I started studying probability and while it seemed all ok I'm having trouble wrapping my head around this one. So the question is: We have 100 balls , 10 black and 90 white. In 50 tries what is the ...
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2answers
61 views

Student t-distribution parameter/s and MLE

So I always thought of the Student t-distribution as having only 1 parameter, v, the degrees of freedom (as described by wikipedia). When I searched however on how to find the MLE of v I keep coming ...
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1answer
20 views

convergence of geometric mean/harmonic mean

Does any one know papers regarding the convergence of geometric mean or harmonic mean in probability, parallel to central limit theorem?
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2answers
70 views

Fourier transform in Machine Learning

I want to know what are the specific areas in which Fourier methods are used in machine learning. Apart from feature extraction and spectral analysis, I want to know if there are any learning ...
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0answers
24 views

How to determine if a set of data is Student-t distributed

So I know how to use goodness of fit tests generally. For example if I wanted to check whether my data came from a Poisson distribution I would first estimate the parameter using MLE (so just the ...
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0answers
15 views

Balls in bins with minimum spacing between filled bins

Say we have n identical balls and a row of m identical bins (with n<m). Say that the balls must be placed in the bins such that: No bin has more than one ball, and There are at least j empty bins ...
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3answers
245 views

Monty hall problem, getting different probabilities using different formulas?

In my Monty hall problem, I am computing what is the probability that P(H=1|D=3) i.e. price is behind door 1 and the 3rd door is opened. $P(H=1|D=3) = p(H=1) * \frac{p(D=3|H=1)} {p(D=3)} = 1/3 * ...
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1answer
45 views

Find 'p' in geometric distribution when 'P(X)' and 'k' are known

This may be a bit more of an advanced algebra question, but here goes: I'm trying to use the equation for a geometric distribution P(X) = p(1-p)^k to find the ...
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10 views

how to deal non response in simple random sampling if sample saize is about 380? [on hold]

How to deal non-response in simple random sampling when sample size is 380 from about 20000 population
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24 views

Probability of getting a certain correlation by random chance

Say I have two different binary vectors (containing only 0s and 1s) with 20,000 observations in each and find the correlation between them. I'm interested in finding out if a very high correlation ...
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39 views

How to properly initialize a stochastic vector?

I'd like create a stochastic vector $$\mathbf{v} = (v_1, \dots, v_n)$$ of length $n$, so that its elements are assigned weights according to a given parameter ("entropy"): the weights are either ...
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1answer
97 views

How many times will a 12 letter word appear in a string of 1 quintillion random letters?

Let's say we think of a specific 12 letter word. How many times on average will that particular word appear in a string of 1 quintillion (10^18) completely random letters (i.e., uniform/equal ...
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1answer
43 views

How to show $\int_{-\infty}^{\infty}f_{X}\left(x\right)\left(P\left(X<x\right)-P\left(X>x\right)\right)dx =0$

For a continuous random variable X , intuition tells me that $$\int_{-\infty}^{\infty}f_{X}\left(x\right)P\left(X<x\right)dx=\frac{1}{2}$$ and more weakly that ...
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1answer
34 views

Variance Estimation of MA(1) with known autocovariance function

I haven't worked with time-series in a while now and stumbled upon them in a different setting. Given $X_t\sim\mathcal{N}(0,\sigma^2)$ for $t=1,\ldots,n$ and the process $Y_t$ for $t=1,\ldots,n-1$ ...
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25 views

Joint probability and normal random variables

Say that A and B are independent and continuous random variables, and the normal distribution of A is N~(10,1), B is N~(5,2). Is the normal distribution of A and B jointly simply N~(15,3)? If no, ...
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1answer
38 views

Conjugate prior for parameter W when the likelihood is normal with mean and variance both functions of W

Suppose that $x$ is an observable scalar variable and $$ x \thicksim N(W\mu_0,W^2\sigma_0^2) $$ Where $W$ is a parameter that must be estimated from data, and $\mu_0$ , $\sigma_0$ are known ...
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28 views

Find the most frequent timespan of event occurrence

I'm working on bike-robbery events dataset. I'm trying to figure out the most frequent timespan of a bike robbery. Most of time, the robbery does not occur when the owner is watching => the declared ...
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1answer
54 views

how can I calculate $P(X_1\leq X_2\leq X_3)$ [closed]

suppose $X_1,X_2,X_3$ are random sample of distribution with probability function $P(X=k)=(1-\theta)\theta^k , k=0,1,2,... ,\theta \in (0,1)$ how can calculate $P(X_1\leq X_2\leq X_3)$
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1answer
14 views

Ranking submissions based on multiple judges

I have a situation in which we have 20 proposals and five judges. Each proposal is reviewed by three of the five judges, assigned to the proposal at random. Each judge scores the proposal out of ...
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1answer
29 views

Understanding statistical independence of events using a relative frequency interpretation

This is what I've read in my textbook: "If $n_A$ and $n_B$ are the number of times the independent events $A$ and $B$ have occurred, then we expect that the ratio $\frac{n_{AB}}{n_A}$ (num. of times ...
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16 views

Estimating The Probability Of Success For Text Data

Imagine the phrase toy maker and by itself, it has 1 success and 887 trials for a success ...
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3answers
5k 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 ...
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27 views

Calculating Mutual Information Rate in the sense of Entropy Rate

My background is not in mathematics nor statistics so please excuse any errors in notation. Let's assume that we have two discreet random variables that are not independent $x_i$ and $y_i$ where ...
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Calculating Emission Probability values

I'm new to HMM and still learning. I'm currently using HMM to tag part-of-speech. To implement the viterbi algorithm I need transition probabilities ($ a_{i,j} \newcommand{\Count}{\text{Count}}$) and ...
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3answers
102 views

Probability that exactly y of n rolls of an r-sided die are unique

Consider an $r$-sided die that is rolled $n$ times. What is the probability that of the $n$ rolls exactly $y$ of the rolls are unique? For example, consider $n = 2$ and $r = 3$. The possibilities ...
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About the evaluation of covariance of linear Gaussian model in PRML

Section 8.1.4 of Pattern Recognition and Machine Learning introduces the linear Gaussian model where each node has distribution $$ p(x_i \big|pa_i) = \mathcal{N}\left ( x_i \Bigg| \sum_{j\in ...
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46 views

Probability example. “All of Statistics”

I'm going through the book "All Of Statistics" I've come across a part I don't quite fully understand this example about probability. Especially the part P(A2). Why does P(A2) have the probability ...
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Probabilities concerning incoming calls at a call center and a question of independence

I work at a call center in a department where we receive two grades of calls: A and B. 15% of our calls are grade A, and 85% are grade B. Only two people work in my specific department, me and my ...
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19 views

Probability of receiving each preference in an internship drawing

I have to write a 23 itens preference list of places where to perform my internship. There will be a lottery which I don't know how exactly they will weight the preferences to distribute us. The ...
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Service Level Caluclation with Double Truncated Normal Distribution

The purpose of this exercise is to determine how the number of employees working affects the service level of their job within the organization they are a part of. EX. A call center gets 100 calls a ...
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1answer
36 views

Quantile and inverse CDF? [closed]

Since the cdf $F$ is a monotonically increasing function, it has an inverse; let us denote this by $F^{−1}$. If $F$ is the cdf of $X$, then $F^{−1}(\alpha)$ is the value of $x_\alpha$ such that ...
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Finding discrepancies in distances between random variable samples

There are $N$ independent and unknown points $p_i$, $q_i \in \mathbb{R}^3$, $i \in [1, N]$. I want to find if the pairwise squared Euclidean distance of each $i$ pair is the same for all of them, or ...
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Effect of partitioning on graph density and edge cuts

When a graph is partitioned using partitioners like METIS, what are the expected effect on density (ratio of number of actual edges to the maximum total number of edges ) and edge cuts(comparison ...
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Use Markov chains to compute probability of rolling a 1 followed by a 2 before rolling two consecutive sixes

I was curious about how to apply the concept of Markov chains to the following problem: Rolling a die, what is the probability that 1 followed by 2 will happen before two sixes in a row? I have ...
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1answer
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Suggestions for simple model of who wins the league in a game of fantasy football

In a game of fantasy football players score points each week depending on the team they have selected, and at the end of a season the player with the most points wins. There are $W = 38$ weeks, and a ...
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22 views

conditional probability of birthdays

what is the probability that two people have the same birthday, their fathers have the same birthday and mothers have the same birthday? The two original people met at a bar in the US with 80 people ...
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36 views

Bayesian Inference - How do I derive the prior and posterior pdfs? [closed]

Brief: "A new bank manager is concerned that customers at the bank are waiting too long in queues for service" I need to plot the prior pdf for λ, π(λ). I have the following information: New ...
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2answers
131 views

Simple example of how “Bayesian Model Averaging” actually works

I'm trying to follow this tutorial on Bayesian Model Averaging by putting it in context of machine-learning and the notations that it generally uses (i.e.): ...