# Tagged Questions

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### Probability and Sampling distribution

Would you please explain me the difference between Probability distribution and Sampling distribution easily ? Is that the difference : in probability distribution we have probability for every ...
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### Calculating the probability of an inequality with two random variables

I am analyzing a timing circuit I designed, and I need to calculate the probability of a certain event (bit error). For example, I have derived this equation: $(1 + x) / d < 1 / M$, where x is a ...
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### Independent gamma and normal distribution

Let $Z = X + Y$ where $X \sim N\left(\mu, \sigma^2 \right)$ and $Y \sim \Gamma\left(k, \theta \right)$ using this parametrization of the Gamma distribution. Also assume $X$ and $Y$ are independent. ...
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### Dependence of distribution standard deviation with subtraction of its mode from mean of normal distribution

The following question might have relations to this question: Given $\mu$ and $\sigma$ (mean and standard deviation) of normal distribution, find the set of all distributions with their $\mu_1$ and ...
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### Correlation of distribution standard deviation with subtraction of its mean from mean of normal distribution

The following question might have relations to this question: Given $\mu$ and $\sigma$ (mean and standard deviation) of normal distribution, find the set of all distributions with their $\mu_1$ and ...
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### Sum of sample mean and sample variance sampling distribution

Let $X_1, X_2, \cdots, X_n$ be an identical and independently distributed sample from $N(\mu, \sigma^2)$, define: $$D = \frac{1}{t}\left[\overline{X} + \frac{1-\rho}{2} S^2\right]$$ where: $t$ and ...
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### How to calculate the binomial probability when expected frequency is a random variable

I am trying to write a simple likelihood function to calculate the binomial probability of $X$ successes from $N$ trials. The problem is that the expected proportion of successes ($p_X$) is itself a ...
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### Probabilities from Logistic Regression

I have built a logistic regression model in R and though the result appears to be satisfactory to some degree, there is one question I have not be able to address. I am not sure if my approach is at ...
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### Modeling time: Probability distribution over time?

I'm trying to model users' posting behavior during a day. Say we have a bunch of users, with the time they post tweets. Now, for each user, I would like to estimate the likelihood of he post a new ...
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### For which distributions does uncorrelatedness implies independence?

A time-honored reminder in statistics is "uncorrelatedness does not imply independence". Usually this reminder is supplemented with the psychologically soothing (and scientifically correct) statement ...
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### kurtosis, positive skewed and negative skewed for probability distribution

When discussing probability distribution, I always read something such as excess kurtosis, positive kurtosis, positive skewed and negative skewed. What exactly do these concepts indicate? In practical ...
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### How to compute expectations from a probability density function?

How to find a tax/subsidy in an income probability density function situation? I am asked the following question: Suppose all families with $Y \lt 20$ are given transfer payments equal to ...
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### Independent but not identically distributed

Let $X_1, X_2,\ldots ,X_n$ be discrete random variables. I'm looking for a way to prove the random variables are independent but not identically distributed. Can anyone suggest some ideas ?
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### Minimum Sample Size Required to Estimate the Probability $P(X \le c)$ for a Constant $c$ (Given a Confidence Level & Confidence Interval)

I have a large population of size $n$ from an unknown continuous random variable $X$, and I do not know the underlying distribution of $X$. Given a constant number $c$, I want to determine the minimum ...
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### Interpretation of a PDF squared [duplicate]

I have a problem where the crucial variable is the integral of the squared PDF of a random variable, i.e. $\int f(x)^2dx$ How should I interpret this property of a distribution? If $f(x)$ is ...
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### Probability of pairwise difference of samples from distribution with finite support

I'd appreciate any help on the following problem: Let $X_1, X_2, \dots, X_N$ be i.i.d. continuous random variables with support $[0, 1]$. What is a reasonable bound on the probability that some pair ...
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### How to combine the chi-square distributions (with one model parameter) of many items in a sample?

I have a sample of items, for each of which I have fitted models to obtain the best-fitting ($\chi^2$-minimising) value of a parameter $\alpha$. So for each, I have the values of $\chi^2_i(\alpha)$ ...
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### What distribution would be expected for number rapists ( vs number of victims)?

This is a horrible question to ask. But it would be useful to know (rather than someone spouting an opinion that 99.999% are/are not.). It has been estimated that 18.3% of women will be raped at some ...
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### About variance mixture models and probability distributions

I was wondering if anyone knows a good resource to learn about variance mixture models ? My interest is in particular the normal variance mean mixture. I know what they mean with their definition of ...
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### What distribution does “arrival of people” at an event in time follow?

Suppose that there is a Cricket match scheduled on Sunday, this weekend. We know that people do not arrive at the stadium at constant rate. Few hours before scheduled start of the match, people start ...
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### Phase space distribution of heads and tails (coin toss)

So I have the equation $$h(t) = 1 + vt - \frac{1}2 gt^2 \pm sin(\omega t)$$ to describe the motion of a flipped coin. It is just a kinematics equation with an angular component added to it, where ...
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### Convergence in distribution and limiting distribution

I'm trying to solve the problem below: Let $X_1,...,X_n$ be independent with PDF $f(x)=e^{-x}$ if $x>0$ and zero otherwise and define $$X_{(n)} = \mathrm{max}\{X_1,..,X_n\}$$ Find the CDF of ...
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### Probability of a random outcome given a sample size

Say you gave an IQ test to a random sample of $n$ people, but instead of having the actual test results, you have only the normalized data for this sample (mean=0, $\sigma$=1). What is the ...
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### % chance, when the success rate goes up with every failure

In an online game the chance of succeeding at an action starts at 5% and goes up 5% every time the action fails. Upon success the chance resets back to 5%. (So you know how they say a die has no ...
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### Distribution of correlation coefficient between two discrete random variables and their collapsed form

I have two discrete random variables with PMF of the form \begin{align*} P(X) = \begin{cases} p_0, & \mbox{if } X=0 \\ p_1, & \mbox{if } X=1 \\ p_2, & ...
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### Distribution of the hyperbolic distance between random points in the Poincaré disc

Let two points at polar coordinates $(r_i, \theta_i)$ and $(r_j, \theta_j)$ be hyperbolic points in $\mathbb{H}^2_\zeta$ with curvature $K=-\zeta^2$. The radial coordinates of these points are ...
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### How to sample from a discrete distribution?

Assume I have a distribution governing the possible outcome from a single random variable X. This is something like [0.1, 0.4, 0.2, 0.3] for X being a value of either 1, 2, 3, 4. Is it possible to ...
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### How many people need to be screened if the disease has an R0 of 2, and starts with 1 infection in 1000?

I need to calculate how many people I would need to screen to catch a disease in its early stages. If the disease has an R0 of 2, and starts with 1 infection in 1000, and I screen one person in every ...
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### How to visualise “equal in distribution” in the context of stochastic dominance?

I am reading about stochastic dominance in wikipedia. I don't comprehend the meaning of "equal in distribution". $x_B \overset {d}{=} (x_A+y)$ What does $\overset {d}{=}$ mean? I understand that ...
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### Understanding multinomial distribution

There are K categories. Let x be a discrete random variable taking on values $1,2...,K$. Given data $\textbf{X}= \{x_1, x_2, ..., x_N\}$, and suppose $\textbf{p}$ is a vector probabilities where, ...
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### Expected value of a multinomial distribution

A multinomial distribution can be given as $M(m_1,\dots,m_K|N,P) = {N \choose m_1\dots m_K}\prod_k p_k^{m_k}$ The expected value is $Np_k$. How can I prove it?
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### The correct probability distribution / way to identify large deviations in a set of daily changes to portfolio value

I am working on a report which is being sent through to end users that should flag to them any "large changes" in the day-to-day values for the past 30 days. These values are day-to-day differences ...
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### Is there a smooth probability density function with finite moments and a closed form quantile function?

I am looking for a smooth probability density function with finite moments and closed form quantile function. As one knows, an example of smooth probability density function with finite moments is the ...
By redefining the energy function, $E(x)$, can any $p(x)$, be written as a boltzmann distribution, ie. $p(x) = \frac{e^{-E(x)}}{Z}$, where Z is the partition function?