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

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Visualize Sensitivity Results Using Combinations of Means and Standard Deviations of Two Normally Distributed Variables

I ran a sensitivity of my model, sampling the response space using two normally distributed variables. I used four nested repetition loops to generate this data, recording the average output of the ...
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3answers
420 views

Probability that the minimum of a normal random sample will exceed the maximum of another?

I sample independently $n$ data points following normal distribution with $\mu = 0$ and $\sigma = 1$. Then I divide the sample into two groups $G_1$ and $G_2$ of sizes $g_1$ and $g_2$ respectively ...
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26 views

Variation of the Coupon Collecting Problem

Suppose you are collecting coupons from a pool of 50 distinct types. These types can be divided into 10 sets of 5 types each. What is the expected number of coupons you need to collect in order to ...
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1answer
34 views

Test to determine observed vs. expected (not Chi-square)

I will work through how and what type of data I am getting before I ask my question: I made some code in R that flips n fair coins x times, and sums the number of successes (heads = 1, tails = 0) I ...
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35 views

Probability in the game Risk? [duplicate]

I've been playing a board game called risk lately, and I'm very interested in the probability behind it. Sorry if this question is it the wrong category, but I assumed that it was largely statistical. ...
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26 views

probability distribution of complex Gaussian column vector and conditional probability of complex Gaussian column vector

I have column vector $\vec r=[r_1\ r_2]^T$. $$\vec r =hA\vec s +\vec n$$ where $h$ is a complex number , $\vec n \sim \mathcal{C} \mathcal{N}(\begin{bmatrix} 0 \\ 0 ...
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1answer
53 views

Why don't log-likelihoods lead to log(0)?

I seem to oscillate between understanding this and not understanding this (which most likely means I've never understood it), but how do log likelihoods function in practice? I mean, when you take ...
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1answer
35 views

What does “asymptotically proportional” mean?

I am reviewing a paper that uses the term "asymptotically proportional" in a definition of a PDF. Unfortunately I can't find what this term precisely means. Can anyone fill me in?
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42 views

Minimum number of people such that 2 can be expected to sit next to each other

We are given a large, round table with $n$ seats. It is easy to see that whenever $p\geq \text{int}(\frac{n}{2}) + 1$ people are seated, at least $2$ people will sit next to each other (here ...
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23 views

Lorentz and Gaussian

I would like to ask the difference between Lorentz and Gaussian. In laser application, the intensity of Natural and Doppler broadening is different in a mathematical way simply due to the differences ...
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1answer
24 views

Equivalence of the dt and pt function in R

in my own curiosity I decided to question the equivalence of dt and pt in R. pt denotes the cumulative density function of the t-distribution. So this gives us $P(X \le x)$. So if I want to find the ...
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2answers
68 views

Particle filter for estimation of static parameters

I am considering particle filtering methods for the estimation of static and dynamic parameters. For the static parameters $\theta$, Liu and West (page 7, equation 3.1) describe an "artificial" ...
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1answer
46 views

The squared-norm of the projection of a Gaussian vector onto an independent $d$-dimensional subspace is a $\chi^2_{2d}$

How we can prove that: The squared-norm of the projection of a $N$-dimensional complex vector with i.i.d. unit-variance and zero mean Gaussian components onto an independent $d$-dimensional subspace ...
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1answer
152 views

Calculate mean and variance of a distribution of a distribution

I am using hierarchical distributions, of the following form: $\theta\sim N(\mu,\sigma)$ $\mu\sim N(a,b)$ I can calculate the mean, and variance using Mathematica, and find: $\mathrm{E}(\theta)=a$ ...
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0answers
23 views

Confidence intervals for estimates generated from a non-probability sample?

From what I understand, to generate a margin of error to have confidence intervals for a given estimate one needs the standard error of the estimate (SE). For the SE one needs information about the ...
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22 views

Calculating probability of sale from auction data

I have some data representing the last 6 months of closed auction data from a particular website. The data I have includes market value of product, actual sale amount, and date sold. I have about 600 ...
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2answers
588 views

Can someone tell me how to read this equation in words?

$$ P\left[\frac{\text{MTBF}\cdot2r}{\chi^2_{1-\alpha/2,2(r+1)}} \le \text{True MTBF} \le \frac{\text{MTBF}\cdot2r}{\chi^2_{\alpha/2,2r}} \right] \ge 1-\alpha $$ Can someone please help me to ...
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1answer
30 views

How to prove the identifiability of a likelihood

Consider the likelihood function for parameter vector ...
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1answer
97 views

Proof of Simplification of Conditional Expectation of Product of Random Variables

Could someone please provide detailed steps to prove or disprove the following? $E[XY\mid XY>k] = E[XE[Y\mid XY>k]]$ Here, $X,Y$ are independent random variables that could be discrete or ...
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19 views

Stata's xttrans - revolving transition probability for each year? [closed]

Edit: Crossposted at Statalist (see here) I am using Stata 13 to estimate a few models with some panel data I have. Among others, I am interested in reporting transition probabilities fot the change ...
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27 views

Probability to be the millionth customer (What Would You Do?)

I saw this episode of "What Would You Do?" a few months ago, and I keep wondering what would statistically be the best thing to do in this situation. Here is the problem formulation: You are waiting ...
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1answer
21 views

Proof that Conditional Expectation of Sum is Sum of Conditional Expectations

\begin{eqnarray*} E\left[\left.\left(X+k\right)\right|\left(X+k\right)>0\right] & = & E\left[k\left|\left(X+k\right)>0\right.\right]+E\left[X\left|\left(X+k\right)>0\right.\right] ...
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9 views

Permutations or Hypogeometeric [closed]

Problem 7. a) In how many ways can 13 people be seated around a round table? b) In how many ways can 13 people be seated around a round table if one cares only for the relative positions of the people ...
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3answers
96 views

How well does a single data point fit a distribution?

I have to come up with a way to measure the 'quality' of a distribution for a research project. We collect data over a a period of time $t_0$ through $t_1$ and then estimate the distribution that ...
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15 views

is there any R package for calculating transition probabilities for multi order markov chain?

I need to estimate the transition probabilities of a user whose k recent selections were x1, . . . ,xk and will select the item x' next. I know to calculate first order transition probabilities but ...
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10 views

Probabilistic models - clarification on separation of model and solver

Almost all of the material I read on probabilistic models/ probabilistic programming mentions separating solver and the model, thus stating the benifit that the model can be changed by the user ...
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1answer
27 views

Independence of random variables after a transformation

We are given four random variables $S:=(S_1,S_2,S_3,S_4)$ defined on $(\Omega, \mathcal A,P)$. The random variables can be viewed as being extracted from a stochastic process. I assume that all ...
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1answer
78 views

Find standard deviation of arbitrary game with multiple payouts

This earlier question asked how to get a 5.76 standard deviation for a single number bet on Roulette. The answer gave the formula, but unfortunately, the formula doesn't easily generalize to more than ...
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1answer
186 views

How many distributions are in the GLM?

I've identified multiple places in textbooks where the GLM is described with 5 distributions (viz., Gamma, Gaussian, Binomial, Inverse Gaussian, & Poisson). This is also exemplified in the family ...
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1answer
26 views

How to generate probability function for uncertain data based on euclidean distance?

I am calculating pairwise distances between some points. The obtained distances can either be accurate, over-estimated or under-estimated. The respective probability is 80%, 5% and 15%. And the error ...
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2answers
161 views

relation between probability and probability density function

According to this link: PDF To translate the probability density $ρ(x)$ into a probability, imagine that $I_x$ is some small interval around the point $x$. Then, assuming $ρ$ is continuous, the ...
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2answers
65 views

Median of ratio of independent variates with Beta distributions

Let $X, Y$ be independent random variables where $X \sim Beta(\alpha_1,\beta_1)$, $Y \sim Beta(\alpha_2,\beta_2)$, and $Z = X/Y$. Recall $X, Y$ are supported on $(0,1)$, so $z > 0$. I've ...
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1answer
84 views

Proof for Simplifying Integral involving Gaussian and Error Function

How do we simplify this integral? \begin{eqnarray*} \int_{-\infty}^{\infty}\left\{ ...
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40 views

Comparing likelihoods from non-nested models

Short: I have a series of joint probabilities (likelihoods) for how likely sample $Q$ belongs to group $K$. I need to compute a p-value describing how "significant" the "top" group is compared to ...
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38 views

When does it make sense to fit a distribution to data?

More in general, if a fitted generative model is at best a biased approximation, and let's say we're interested in the predictive distribution, there is always an information loss associated with the ...
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1answer
34 views

How do I obtain the uncertainty of a value generated from a known distribution?

I wish to semi-randomly generate values from the known distribution of a particular quantity. The distribution is represented in my data as a list of values with a probability for each value, where ...
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28 views

click through rate stats

I have a group of 10 users. Each user receives a daily notification for one week. Ideally, each user will click on all notifications received. At the end of the week I collect the number of clicks per ...
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1answer
49 views

How to calculate the probability distribution F(X,Y) when the distributions of X and Y are known?

Suppose $X$ and $Y$ are normally distributed with known means and standard deviations. How do I calculate what is the probability distribution some function $f(X,Y)$? For example, say $f(X,Y)=2X+Y$. ...
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12 views

Kullback-Liebler's divergence on a conditioned function

Let $q$ be a conditioned pdf over $\mathbf{X}=X_1,\dots,X_n$ binary r.v.s in the form $$q(\mathbf{X})=\begin{cases}q_{0}(\mathbf{X}_{\setminus i}) \text{ if } X_{i}=0\\q_{1}(\mathbf{X}_{\setminus i}) ...
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15 views

generate multivariate cauchy random variable

How can we generate multivariate random variable in matlab? Will the set of t distribution with 1 degree of freedom will give multivairate cauchy noise, How to generate?
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24 views

Joint probability for non-mutually exclusive events

90% of people who like bananas eat a lot. 60% of those who like apples eat a lot. If person "A" likes banana and likes apples, what’s the probability that they eat a lot. Is this possible to solve?
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1answer
35 views

Latent Dirichlet Allocation - definitions

I am self-studying the article on LDA by Blei, Ng and Jordan (https://www.cs.princeton.edu/~blei/papers/BleiNgJordan2003.pdf). at the start of the second section - the following definitions are given: ...
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2answers
32 views

How do you reconcile the definitions of multiple mutually independent events and multiple mutually independent random variables?

1) If $A, B, C$ are events then to prove mutual independence it's not enough to show $P(A,B,C)=P(A)P(B)P(C)$. You also have to show $P(A,B)=P(A)P(B)$ etc. 2) For random variables, the definition of ...
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1answer
52 views

Conditions on transformation function in Monte Carlo expectation

If I have an i.i.d. set of samples $\theta_1, \ldots, \theta_n$ from my posterior $p(\theta | y)$ then: $ E(f(\theta | y)) = \int f(\theta) p(\theta | y)\, \mathrm{d}\theta \approx \frac{1}{n} ...
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11 views

Finding the distribution of a dependent function given the mean and coefficient of variation (COV) of the independent parameters

It is given that $A$ is log-normally distributed with a mean of 0.19 and coefficient of variation of 0.10, and $B$ is log-normally distributed with a mean 0.040 and COV equal to 0.10. How do I find ...
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1answer
35 views

Probabilities of conditional expectation values in uniform distribution

Let's consider a continuous random variable $X$ as follows: $f_X(x)=\left\{ \begin{array}{ll}\frac{1}{2}, &\mbox{if} \ x\in[0,1] \\ \frac{1}{4}, &\mbox{if}\ x\in(1,3]\end{array}\right.$ ...
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26 views

Bayesian decision theory - loss function

i am trying to learn bayesian decision theory. the book that i am using (Kevin Murphy: Machine Learning: A Probabilistic Perspective) - has the following introduction on the action a: however - ...
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1answer
22 views

Distribution parameterized by its median and value at risk (VaR)

I am trying to find if there is an appropriate distribution whose parameterized by the median as a measure of center and VaR (Value at Risk) or CVaR (Conditional VaR, Tail VaR) as a dispersion. Is ...
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30 views

How to calculate the probability of no show rate in airlines

At present I am analyzing the historical airlines data which has got the information of bookings and no shows. Now,to calculate the probability of no show rate,is it simple percentage or is there any ...
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65 views

Bernoullli distribution probabilities

Given these events I= {X=1} and J= {X > 3}, for the Bernouli distribution with p = 1/4, find the probability of each event. I am not sure how p can be 1/4 if there is only one event and it is X = ...