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

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11 views

Dynamics of birth-death process with discouraged arrivals (alternatively, M/M/1 queue with balking customers)

Take a continuous-time birth-death process, where $k \in \{0,1,\ldots\}$ is the count and the arrival rate of death is $\mu \geq 0$ for $k = 1, 2, \ldots$ the arrival rate of births is $\alpha_k ...
1
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34 views

How to compute the probability of $P(a|b,e)$ given $P(a|b), P(a|e)$ and $A$ independent of $B$?

The following is given: $$ \begin{align} P(b) &= P(e) = 0.1\\ P(a|e) &= 0.2\\ P(a|b) &= 0.95\\ P(a|\neg b, \neg e) &= 0\\ B &{\perp\!\!\!\perp} E \end{align} $$ Is there any way ...
0
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0answers
11 views

Finding out expected number in penalty shootouts [on hold]

There are 3 expert players (Alex,Bob and Charlie) in a penalty shootout in a football team. The coach often has difficulty selecting an expert penalty shooter from the three expert players. Therefore, ...
0
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0answers
12 views

Stuck in how to take the Taylor Series for Expectation analysis: Doubt in how to take the Expectation

I don't know how to solve the following expression: $ = nm^2 E \bigg[\frac{ \exp(\theta) {(\log(R))}^2}{N(R,x)}\bigg] \hskip 5 pt Eq(4) $ which I have explained below. $R$ follows Poisson ...
0
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0answers
8 views

Distribution of a linear combination of the components of an elliptically distributed random vector

What is the distribution of a linear combination of the components of an elliptically distributed random vector? Is the linear combination an elliptically distributed random variable?
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0answers
17 views

Closed form Pocock Rejection Region solution for joint MVN correlated RVs?

This is related to my previous question (link), but I have a simpler way of expressing it. Suppose $ \left( \begin{array}{ccc} \ Z_1 \\ Z_2 \end{array} \right)$ follows a Bivariate standard normal ...
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0answers
23 views

How to derive the conjugate prior of an exponential family distribution

I am trying to derive the conjugate prior of the univariate Gaussian distribution over both the mean and the precision. I know that the prior I'm looking for is the normal-gamma distribution, but the ...
3
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0answers
27 views

Difference between pointwise and uniform confidence sets

According to Wasserman's All of Nonparametric Statistics p.6, let $\mathfrak{F}$ be a set of distribution functions $F$ and $\theta$ some quantity of interest. A uniform asymptotic confidence set ...
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1answer
64 views

Question on a probability question

I have this question for a test: A judge is 35% sure that Tim commited burglary. The witness would lie at a probability of 0.25 if Tim is guilty but would tell the truth if Tim is innocent What ...
1
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1answer
75 views

Three players with their own coin flip untill they have heads. The first one with heads wins

I have been stuck with this question for some time and I'm not sure if I'm doing it correctly. There are three players who each have their own coin—each coin having a different probability for it to ...
1
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1answer
48 views

Specify conditional probability of a continuous node given a continuous node as its parent

This question is essentially same as this one. The question is: How do you calculate conditional probability of a node in Bayesian network when it has a continuous node as a parent? However, I cannot ...
0
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1answer
47 views

$H_0=250g$ and $H_1\neq 250g$" [on hold]

We have a sample of size $100$ with a standard deviation of $5g$ It was decided that if the sample mean is between $245g$ and $255g$ while the sample average is $250g$ if $\mu=250g$ or $\mu\neq250g$ ...
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1answer
46 views

Coupon collecting problem

Suppose there are 8 different types of coupons, each coupon has probability 1/8 of being selected. Suppose a total of 6 coupons are collected. Let: $$ X_i=\begin{cases} 1, \textrm{if a type-i coupon ...
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0answers
13 views

A Hypothesis Testing Question and Verification

Suppose we have two p.d.f.'s: $f_0(x)=1,$ if $0\leq x\leq1$, and $f_0(x)=0$ otherwise $f_1 (x)=2x$ if $0\leq x\leq1$, and $f_1(x)=0$ otherwise We want to test the hypotheses ...
2
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1answer
17 views

Developing a heuristic for maximizing the “covering” of a distribution

Context There's a board-game called Settlers of Catan in which players compete to be the first to gain 10 victory points by trading various resources in exchange for pieces (or cards) worth victory ...
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1answer
22 views

How to estimate rating of a customer?

I have data about customers transactions contains: ratings of customer $r_{ci}$ to service $s_m$ which has quality $q_m$. Quality is assumed to be a real value from [0, 1]. Rating is discrete values ...
0
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1answer
22 views

Does martingale model work for betting football matches?

Imagine I have 1 million USD and will be betting 1.000 USD on the win of FC Barcelona each time they play a match in La Liga (Spanish Tier 1 football league). If FC Barcelona loses or ties their last ...
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1answer
21 views

how to random sample proportional to probabilities

I have a ten-sided die, where I know the probabilities of rolling some, but not all of the values on the die: ...
2
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0answers
39 views

how to determine if two dice are fair using pymc and roll data

My scenario is that I have two six-sided dice (D1 and D2), either of which may be fair or loaded (biased). I have samples of combined roll data (i.e. D1 + D2). I would like to view the posterior ...
2
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1answer
35 views

What is the distribution family where one side is with light tail but the other side with heavy tail?

For example the distribution of weights of human. There are not many adults under 40 kg, but a lot more people heavier than 100 kg, although the average of an adult's weight is, let's say, 70 kg. ...
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0answers
24 views

Probability distrubution of 3 products

A lightbulb factory produces bulbs in 3 qualities. 50% of production is Low quality, 30% average quality, 20% good quality. A randomly selected lot of 4 bulbs gets tested. Low quality lights have a ...
3
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1answer
20 views

Variance of a linear combination of vectors

Let $A$ and $B$ be two constant matrices and let $x$ and$ y$ be two random vectors, what is the general formula for $Var(Ax+By)$? I know the formula for when $x$ and $y$ are scalar random variables ...
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0answers
12 views

Describing a distribution of probabilities/percentages

Is there a better way to describe this probability? \begin{equation} Pr(Y_{sij}|R_n) = \frac{Y_{sij|R_n} \sum_{ij}Y_{sij}}{\sum_{ij|R_n} Y_{ij}\sum_{sij}Y_{sij}} \end{equation} The $R_n$ describes ...
0
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1answer
27 views

What is a good model for revenue managment / price optimization problem

I am trying to create a dynamic pricing model to optimize revenue for a hypothetical business. Lets say I have an application that connects dog walkers to people who want to pay for their dog to get ...
3
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27 views

Regarding variance of “total number of records”

Here is my problem: Let $\{X_i\}_{i=1}^n$ be a sequence of continuous random variables. A record is said to have occured at time $k$ if $X_k > X_i$ $\forall i=1,...,k-1$. Let $N$ denote the ...
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0answers
30 views

What are the odds of this happening?

There is a list of exactly 94 names on it. One person randomly picks 5 names on the list and doesn't show them to anyone. Another person picks just 2 names on the list. What are the odds that one of ...
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16 views

Probability of paired combinations

There are 8 friends: Alfred, Alex, Billy, Barbara, Connie, Chris, David and Donna. The friends are paired up at random. What is the probability that at least two pairs will share the same first ...
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0answers
28 views

Threshold for Pearson correlation for hypothesis testing [closed]

How to determine threshold for Pearson correlation coefficient which reduces type 1 and 2 error and and can be used whether the sample is correlated or not?
0
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0answers
41 views

statistical analysis problems [closed]

Two persons, A and B, are to play a series of independent games, with A’s winning probability in each game = $p$ and B’s winning probability = $q = 1 – p$. The game ends when the total number of ...
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0answers
30 views

Estimate Probability of being in a time interval

​You arrive at a bus stop in an unfamiliar part of town. Assume that buses arrive at the stop with an unknown (to you) distribution and wait in the bus stop for a few ​minutes. The wait time ...
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3answers
379 views

Returning unlikely results from a probability distribution

The Law of large numbers addresses the issue of the average result of a large number of random trials. Is there a similar law which addresses the probability of obtaining an unusual result for a large ...
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30 views

Confidence intervals sum of dependent variables

How to construct confidence intervals for sum of dependent random variables. Specifically, I want a high probability claim regarding the difference between the empirical mean and the true mean of the ...
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0answers
4 views

Find the probability of a new data point using “density” function in R [migrated]

I am trying to find the best PDF of a data that has Gaussian distribution, using the "density" function in R. Now, given a new data point, I want to find the probability of this data point based on ...
32
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6answers
3k views

How to tell the probability of failure if there were no failures?

I was wondering if there is a way to tell the probability of something failing (a product) if we have 100,000 products in the field for 1 year and with no failures? What is the probability that one of ...
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0answers
27 views

Calculate the probability of a string

I have a classification problem. I am trying to classify strings using I normalize the data before training. So far I get the best results with addition, but I cannot understand why. The ...
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0answers
41 views

Bayesian credible intervals: “superiority” even if 1 is included?

In a recent medical publication comparing a cardiac device to anticoagulation ("blood thinners") using a Bayesian statistical model to evaluate the efficacy of preventing strokes and cardiovascular ...
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0answers
18 views

rate of convergence of sample mean

Law of large number ensures the convergence of sample mean to population mean. But it does not tell about the rate of convergence. Now CLT tells about the rate of convergence. Then what is the ...
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14 views

survival mixture model of gamma distribution via EM algorithm

I generated survival data for a three component survival mixture of gamma distribution with random right censored observations in R with the code; ...
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2answers
187 views

Independence and orthogonality

I know what it means to say two variables are independent, but can't understand what does it mean to say two variables are orthogonal.Can anyone help me?
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1answer
38 views

Why $p(y_n|y_{0:n-1})$ is a constant in Kalman filter derivation?

In a derivation of Kalman filter, It says that in Equation: $p(x_n|y_{0:n})=\frac{p(y_n|x_n)p(x_b|y_{0:n-1})}{p(y_n|y_{0:n-1})}$ the denominator $p(y_n|y_{0:n-1})$ is a constant. (See the ...
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0answers
8 views

Dempster-Shafer sampling

If we consider classical probability on finite support, we assign some probability to each elementary event. The sampling is then easy. In Dempster-Shafer, we assign a value to all subsets of the ...
0
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1answer
35 views

Is summing posterior probabilities valid for classification problems?

A classification for two mutually exclusive problem can be formulated by having a decision hinge on whether $P_0(x) > P_1(x)$ or $P_0(x) < P_1(x)$ where $P_0(x)$ and $P_1(x)$ are posterior ...
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0answers
18 views

Laplace transform of function of more than one random variable

I have tried to be very clear and precise with this question. Please let me know if I can write it in better way or give more explanations. I don't need a solution for this problem, only opinion if my ...
1
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1answer
32 views

How to calculate probability that another player has a card

In a poker game, how would we calculate the probability that another player has at least one certain card? Lets say that there are five other player, there are four cards in middle of the table, and ...
0
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1answer
46 views

Graphical dependence in the DAG X->Z<-Y

In Barber's book pp. 40-41 he says that the belief network X->Z<-Y: is "graphically dependent" since: $$p(x,y|z) \propto p(z|x,y)p(x)p(y)$$ I don't understand why graphical dependence follows ...
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0answers
28 views

How is the probability of Weibull Distribution exceed $1$ for some value? [duplicate]

Here is R code which show that probability of Weibull distribution is $1.7$ for some value $t$. But as far i know probability lies between $0$ and $1$. How is it possible ? R code: ...
2
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0answers
78 views

Birthday paradox, two generations, different continents

I have been reading about the birthday paradox & was trying to take it further & apply it to an example from my own life:- I am a Brit married to an American. My British sister-in-law shares ...
3
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0answers
28 views

If X is independent of Y given Z, is X/Z independent of Y?

When is the following true? If $X$ is independent of $Y$ given $Z$, $\frac{X}{Z}$ is independent of Y.
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1answer
35 views

Variation on the urn problem and frequency distribution

I have $6$ machines each producing different coloured balls. The balls are mixed together in a large vessel. Groups of $6$ balls are extracted at random for packing. Each pack will therefore have a ...
2
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0answers
29 views

parameter and prediction confidence intervals

We have a correctly specified linear model $z = x\beta + \varepsilon$ with 100 independent observations. Given: $y = e^z$, $\sigma^2 = 4$ (assumed to be known), sample means $\bar{x} = -3$ and ...