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

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Which topic does this question fall under?

One percent of population cannot drive even if they try very very hard, but everyone applies for the driving license. The driving test fails those who cannot drive with a chance of 97%, but because ...
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27 views

Actuarial question [on hold]

A family buys two policies from the same insurance company. Losses under the two policies are independent and have continuous uniform distributions on the interval from 0 to 10. One policy has the ...
4
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1answer
33 views

Correlation multinomial distribution

Problem 1.14 from Categorical Data Analysis 2nd. For the multinomial distribution, show that $$\operatorname{corr}(n_j,n_k)=\frac{-\pi_j\pi_k}{\sqrt{\pi_j(1-\pi_j)\pi_k(1-\pi_k)}}$$ Show ...
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26 views

Predictions from betting quotes adjusted for money-flows

Recently at least two predictions were made based on (pools) of bookmaker quotes one for the UEFA Euro 2016 (see here and Zeileis A, Leitner C, Hornik K (2016). “Predictive Bookmaker Consensus Model ...
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2answers
878 views

Blackwell's bet

I have read about Blackwell's bet paradox on Futility closet. Here is the summary: you are presented with two envelopes, $E_x$ and $E_y$. The envelopes contain a random amount of money, but you don't ...
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16 views

Calculate probability of a measurement assuming a uniform distribution around a mean

Assuming a uniform distribution and given it's mean value, can I calculate the probability of a particular value occurring? This is a real world problem. The expected value is 3.35, so I guess that ...
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0answers
14 views

Statistics Question [on hold]

Researchers conducted a study in which they invited 85 nutrition experts to an ice cream social (Wansink et al., 2006). These experts were randomly given either a 17- or a 34-ounce bowl, and they were ...
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0answers
47 views

Probability of stacked ensemble classifiers using a simple threshold function

The problem There are N classifiers which each give a probability of some event occurring. We want to combine these probabilities in an ensemble classifier to make a final prediction. We also want to ...
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0answers
20 views

How can I cluster data sets? [on hold]

I have some data sets generated by different probability distributions. Each data set concludes a number of item IDs(not have to be continuously) and the frequency of each item. I'm wondering which ...
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0answers
18 views

Variance of an unbiased estimator is 0 when the sample size goes to infinity

So I would like a proof for the following but I can't seem to do it myself. I have a random variable $X$ and I draw $n$ samples($\{X_1, \ldots, X_n\}$) from it and I have $$ Z_n = \frac{\sum_{i = ...
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1answer
19 views

Methods and Demographics

Specific question: How do statisticians determine the likelihood that such-and-such type of person will commit a crime? What is their methodology? To illustrate what I'm talking about, people will ...
3
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1answer
42 views

Variance of the modulus of a random variable

Let $X$ be a random variable with mean $\mu$ and variance $\sigma^2$. What is the upper-bound on the variance of $Y=\left|X\right|$? My gut feeling says that $\operatorname{Var}(Y) \leq \operatorname{...
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37 views

Find the density of a function of a random variable with known distribution

For example, if $X$ is a normal r.v than the distribution of $Z=X^{2}$ is (F is the CDF): $$F_{Z}(z)=P(Z<z)=P(X^{2}<z)=P(-\sqrt{z}<X<\sqrt{z})=F_{X}(\sqrt{z})-F_{X}(-\sqrt{z}) −P(X=\sqrt{z}...
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1answer
40 views

Multinomial Probability Question

A lift starts with 5 passengers and stops at 8 floors. Find the probability that no two or more passengers leave at the same floor. Assume that all arrangements of discharging passengers have the same ...
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0answers
19 views

CLT and Wald-Wolfowitz runs test asymptotic distribution

I need help finding a theorem which could be used to prove that the Wald-Wolfowitz runs test is asymptotically normal. Let me formalize my question. We have a random sample $\{X_0,X_1,...,X_n\}$ (if ...
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0answers
13 views

KL Divergence between an i.i.d sequence and non i.i.d sequence

I have two sequences of probability density functions $Q=\prod_{i=1}^{n} q(x_i)$ (independent and identically distributed, i.i.d )$P=p(x_1,\cdots,x_n)$ (non i.i.d). How can I calculate the KL ...
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1answer
40 views

Problem on convergence of sequence of random variables

Given any sequence of random variables $\{ X_n \}$; how do I show that there exists a sequence of real numbers $\{ \alpha_n \}$, such that $\{ \alpha_n X_n \}$ converges in probability to 0 ?
4
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1answer
62 views

Variance computed using Taylor series does not agree with numerical experiment

I would like to estimate an angle $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ given the noisy observations of its sine and cosine (this is related to my earlier question). My estimator is ...
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0answers
29 views

Randomly generating transition probabilities for Markov chains

I'm trying to simulate a person moving through a household using a Markov chain. Each state would be a room in the house. The issue I'm running into is that I have no existing data telling me what a ...
3
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1answer
35 views

Continuous uniform random variables convergence question

Let $X_1, X_2, \ldots$ be independent $U(0,2)$ random variables and let $$Y_n = \prod_{i=1}^n \, X_i \;.$$ How do I prove or disprove that that $Y_n$ converges to $0$ almost surely ?
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2answers
75 views

Permutations and probabilities for indistinguishable balls into indistinguishable fixed-sized buckets

I'm neither a statistician nor a mathematician so apologies if this question is asked and answered elsewhere or if I've misdescribed my problem. I am trying to figure out in a thread in stackoverflow ...
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32 views

How would a bayesian and a frequentist answer this question? [closed]

I have a site which which in which there are lets say 1000 visitors every month and out of 1000, 10 people actually sign up on my site which leads to a 1% conversion rate. Now I invest in the ...
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18 views

What do attendance figures tell me about regularity? What does the average tell me about individual attendance?

Suppose I have a group of $N$ people, attending a series of $M$ events, and (for simplicity) let's assume the overall attendance happens to be the same at each event, say $A$ people (ranging between 1 ...
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1answer
29 views

What is the best way to approximate probability from a PDF of a Gaussian distribution? [closed]

In a program I am writing, I have a Gaussian Distribution function that returns the PDF given a specific vector. The issue is, this is obviously not the actual probability. To further complicate ...
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43 views

How can I check the quality of my probabilities?

I have a data set where the number of accidents are given together with the probability (assumed to be correct) that the number of accidents is less than 4. The task is to calculate the probabilites ...
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21 views

Question about probability modelling only for Google Analytic users [closed]

I've used this free super quick and easy Regression Analysis tool for Google Analytics called SuperBrain Modeler. It seems to calculate the r-value correctly, but for me it seems the probability ...
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2answers
49 views

mean of the product of n dependent indicator random variables

How can I simplify $E( I_1*I_2*…*I_n )$, which is a mean of the product of n dependent indicator random variables. Can I write down something like $E( I_1*I_2*…*I_n )$= $E( I_1)*E(I_2)*…*E(I_n)+Cov$ ...
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95 views

Probability of Birthdays using Python [closed]

I am very new to coding using python and seem to have hit a wall. This is the full problem I am attempting to answer: Birthdays Suppose you are teaching a freshman class populated with forty ...
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1answer
69 views

How to correctly calculate the type I error in a two-step clinical trial when stopping after the first step?

I just finished reading the following article by Berger & Berry (1988) in which they explain how subjectivity enters statistical analyses. One of their examples concerns a clinical trial with ...
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38 views

Fast markov chain sampling of multiple initial states in R [migrated]

Assuming there are $K$ states. I would like to draw the next state for every initial state in a state vector 'initial_states' of length $N$ given a transition matrix 'trans_matrix' of dimension $K \...
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1answer
28 views

What does the height represent in a probability density function? [duplicate]

I know the integral represents the probability, but what does the height represent? It can’t be the number of times that a feature appears, because the function is continuous. I understand that it is ...
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2answers
125 views

How to determine if proportions are significantly different

I have a database of purchase events. Each event has a field green (a boolean field that determines whether the item was green) and a field ...
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Linear probability model and multinomial logit inconsistent signs

I have a dependent variable with three potential outcomes. When I run linear probability models independently on the dependent variables and compare the results to that of a multinomial logit model ...
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3answers
85 views

Sampling without Replacement and Non-uniform Distribution

There are $N$ items, numbered $1 \ldots N$. The probability of selecting item $i$ in one draw is $p_i$. Items are drawn without replacement, so after each draw we need to do a re-normalization. Now, ...
3
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1answer
63 views

Convergence to zero of a sum involving the median

Can anyone help me with this? Let $\{X_i:i=1,...,n\}$ be a sequence of i.i.d random variables with a continuous density function. If $m$ is the true median and $\hat m _n$ is the sample median, is it ...
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How to prepare for a Theory of Probability class?

I have taken statistical theory and applied statistics classes at the graduate level, and I may take Theory of Probability next semester. I have heard from PhD students that this class was the hardest ...
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36 views

How to integrate lognormal equation?

The problem comes from the personal research $h \sim \log N(\mu,\sigma^2)$ then $$f_H(h) = \frac{1}{h\sigma\sqrt{2\pi}}\exp\left[-\frac{1}{2}\left(\frac{\log h-\mu}{\sigma}\right)^2\right].$$ Here ...
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Probabilistic Information Retrieval Question

I thought up an interesting information retrieval question that I am having a difficult time answering. Consider a semantic search engine i.e. a search engine that matches on semantically similar ...
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34 views

Coin flipping probability taking turns between players

I've worked out the result to a basic coin flipping problem and want to generalize it. The basic problem is: there are N players, and they take turns of flipping a coin (in the same cyclical order) ...
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1answer
52 views

Proof for probability of union of n events

I'm trying to refresh my knowledge of probability so I'm working my way through Haigh's Probability Models 2e. I'm looking at one of the corollaries presented (1.3) and I don't understand it. The ...
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1answer
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Is it possible to add up the accuracy rates of 2 predictors?

Weather channel 1 has a 65% accuracy rate of predicting tomorrows weather Weather channel 2 has a 59% accuracy rate of predicting tomorrows weather Is it somehow possible to take into account ...
3
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0answers
32 views

Sufficient conditions for variance convergence in CLT

More generally, if $\{X_n\}_{n\in\mathbb{N}}, X$ are real random variables with finite variance such that $X_n\xrightarrow{d}X$, what are some sufficient conditions to assure that $\operatorname{Var}(...
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How to calculate the expected loss of a credit card transaction?

I have used an algorithm to calculate the probability of a credit card transaction to be fraudulent. The algorithm outputs a classification (fraud/no fraud) and the probability of each, such that $P(\...
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2answers
30 views

Expected number of points in k turns?

I'm not too familiar with expected values so I'm wondering if people could verify if I'm on the right track with my thought process here. If the probability of winning one point in a turn is 1/9, ...
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0answers
52 views

How to Combine Multiple Probabilities

I have a list of many products, each with a given probability of the owner's gender and I'd like to compute the gender probability for users who have multiple products. I am forced to assume that the ...
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PDF from difference of two Weibull distributions

How should I derive a PDF from the difference of a couple of given two parameter Weibull distributions ? The convolution integrals become very difficult to evaluate. PS : The parameters are not ...
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Confusion regarding conditional distirbution of the product of gaussian process with a normal

I am getting confusedregarding a simplae problem. say $Y = WX$ where $X$ is a q X n matrix with each row a Gaussian process denoted as $\mathcal{GP}(M(\mathbf{X_i}),C(\mathbf{X_i},\mathbf{X_i}))$, in ...