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Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions.

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21 views

The probability of two independent random variables?

Let $X, X'$ be independent with $X \sim p(x)$, $X' \sim r(x)$ for $x, x' \in X$. I don't understand this equation: $\sum p(x)r(x)=Pr(X=X')$ What is intuitive to me is if $X \sim p(x)$, $X' \sim ...
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1answer
22 views

Is there a class of probability density functions includes the laplacian and the normal pdfs?

Is there a class of probability density functions that includes the Laplacian and the Normal pdfs? $$f(x\mid\mu,b) = \frac{1}{2b} \exp \left( -\frac{|x-\mu|}{b} \right)$$ and $$f(x \mid \mu, \sigma^2)...
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0answers
30 views

Probability Distribution for answering 2 exam questions correctly in succession

What is probability distribution can I use 'answering exam questions until you get two answers right in succession' Is there some adapted way of using geometric?
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0answers
30 views

Finding the probability that a sum of normal variables does not exceed a certain treshold

I have a question about a relatively simple probability exercise involving a sum of components. It goes as follows: A cargo ship has a maximum payload of $50,000$ kilograms. The ship is loaded ...
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0answers
15 views

Find the distribution of $X_t|Y_t,X_{t-1}$

Find the distribution of $X_t|Y_t,X_{t-1}$ given $$X_t=\alpha X_{t-1}+V_t$$ $$Y_t=X_t+\sigma_wW_t$$ where $X_1 \sim N(0,1)$, $V_t \stackrel{iid}{\sim} N(0,1)$, and $W_t \stackrel{iid}{\sim} N(0,1)$. ...
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2answers
24 views

Asymptotic behavior of combinations: approximating Hypergeometric by Binomial

A form of the hypergeometric distribution is $$P(X=x)=\frac{\binom{Np}{x}\binom{Nq}{n-x}}{\binom{N}{n}}$$ where $N\equiv$ total number of elements of the sample space $p\equiv$ probability of ...
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0answers
32 views

Gaussian process properties

I am reading Gaussian Processes (GP) for Machine Learning (http://www.gaussianprocess.org/gpml/). The GP definition is usually like this: "A Gaussian process is a collection of random variables, any ...
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0answers
60 views

Showing $E(S^2\mid \bar X)=\bar X$ for i.i.d Poisson random variables $X_i$

Let $X_1,X_2,\ldots,X_n$ be i.i.d $\text{P}(\lambda)$ random variables where $\lambda(>0)$ is unknown. Define $$\bar X=\frac{1}{n}\sum_{i=1}^n X_i\qquad,\qquad S^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\...
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0answers
10 views

order statistic, distribution of a partial sum

Please let me know if I can clarify my question in any way. I want to figure out the distribution of a partial sum of k largest observations in a n sample from a non-central Chi-square distribution. I ...
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1answer
44 views

What is the probability of 50% of the people dying out of a group of M people selected from N people?

After watching avengers infinity war this thought came to my mind. If there were N number of people and we know exactly half of them will vanish, then what is the probability that if we select a group ...
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2answers
24 views

Confidence Interval Intuition Conflict

I am trying to understand confidence interval CI, from simplest article I could find about it. I got to an extent and then stuck at crucial moment. Suppose if we decide confidence level we want is 95%,...
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0answers
14 views

Is this probability function continuous?

I have a probability function shown below, and I was wondering if it is possible to set the right side equal to 0, then take the derivative of the right side with respect to $P_j{_r}$ and find the ...
2
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0answers
22 views

Is there a category for distributions with curved boundary?

I wonder if there is a mathematical term or way to describe a continuous distribution with a "curved" boundary. By "curved" I mean the distribution has compact support whose boundary cannot be ...
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3answers
39 views

Sampling from product of exponential distributions

I have a distribution who's moment generating function is the product of two exponentially distributed variables moment generating functions. If I wanted to generate samples from the distribution, ...
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0answers
24 views

Why Sampling distribution not skewed when np < 10 and nq < 10?

My aim is to study "Sample distributions" via simulations convincing myself of the outcomes. After struggling with questions here, here and here, finally I hope am somewhat getting my head around it....
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1answer
22 views

Finding the conditional entropy on the sum of independent random variables

I have two independent random variables $X_1$ and $X_2$. I want to find the differential entropy defined as $$H(X_1+X_2\mid X_1)=\int_{X_1} \int_{X_2} p_{X_1,X_2}(x_1,x_2)\log\left(\frac{1}{p_{X_1+X_2\...
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2answers
31 views

Lottery ticket sales and ‘coverage’

Take a small 6/40 lottery with 3,838,380 possible combinations. Prior to the draw, the operator sells only 3 million randomly drawn tickets (for simplicity’s sake, I’m completely ignoring the fact ...
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0answers
44 views

How to understand the normal distribution? [on hold]

My goal is to gain deeper insight into the expression for the Normal distribution. Question: What is the significance of each term in the expansion of $N(\mu, \sigma)$? $$f(x) = \frac{1}{\sqrt{2 \...
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1answer
21 views

Normal (and hypergeometric?) distribution probability problem [on hold]

An Airline Engineer knows from experience that a person's time of delay when taking a flight can be modeled as a normal random variable with average 10 minutes and deviation standard 1.93 minutes. If ...
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1answer
15 views

Determining the distribution for a sample data set

I have a sample data set of heights of students where X = (1.65,1.55,1.78,1.43,1.69) and I'm required to fit a normal distribution to the data X. The thing I'm ...
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0answers
23 views

Probability distribution type of the following variable

I'm not sure to which Probability distribution type the variable "Number of previous times defaulted on loan repayment" belongs to. I was thinking of Geometric distribution. Can someone please clarify ...
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1answer
26 views

Let X be the number of times you and your friend get the same outcome. Then X is distributed as

You and your friend have a fair coin each. Both of you toss the coins simultaneously, record the outcomes, and repeat the process, for a total of $n$ times. Let $X$ be the number of times you and your ...
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2answers
37 views

What is the cdf for a partially non-continuous pdf?

Suppose there is a pdf/pmf (?!) which places an atom of size 0.5 on x = 0 and randomizes uniformly with probability 0.5 over the interval [0.5,1]. Such that... \begin{equation} f(x)= \...
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1answer
25 views

Calculating expectation w.r.t. the empirical dist. fcn

I am trying to evaluate the plug-in estimator $$\hat{\theta} = T(F_n)$$ where $F_n (t) = \frac{1}{n} \sum_{i = 1}^{n}1\{t_i \leq t\}$ and $T : \eta \to \int_{}^{}x \ d\eta(x) $, (Riemann-Stieltjes ...
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1answer
25 views

Conditonal probability that person C won the game?

I had this problem on an exam a few weeks ago: The persons $A$, $B$ and $C$ are playing a game. What the game is about is of no concern, the only thing we need to know is that the person who obtains ...
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0answers
16 views

The CDF of the maximum of some function of the maximum two order statistics

Let the random variables $X_1,\,X_2,\,\ldots,\,X_K$ be i.i.d. exponential random variables with parameter 1. Also, let the random variables $Y_1,\,Y_2,\,\ldots,\,Y_K$ be defined similarly. Now let $...
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1answer
16 views

Two Simultaneous Cumulative Hypergeometric Distributions

Suppose we have a standard 52 card deck from which we draw five cards. What are the chances of drawing one or more Aces and one or more Kings? How do I calculate this? I know that we can calculate ...
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1answer
29 views

Polynomial function of random variable

let $x$ be a random variable with pdf $p(x)$, e.g., $ prob(A)=\int_A p(x)dx$. Define a random variable $y=f(x)$. If $f$ and $p$ are polynomial functions, what we can tell about pdf of random variable $...
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1answer
70 views

How Sample mean equals Population Mean?

I did try searching here before posting, but could not find a satisfactory answer. This is my attempt to prove mean of sample means equals to population mean where I am stuck. Note: Rewritten as ...
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0answers
42 views

Can we split a random variable into intervals on its domain of possible values and express it in terms of “simple” distributions on those intervals? [on hold]

So suppose we have a random variable $Z$ which can take values in $\left[-A, A \right]$. Suppose we do not know the exact distribution of $Z$. Now if we take $N$ disjoint intervals of $A$ such that $\...
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1answer
42 views

Solving the probability for sample size

I'm given a sample data of heights of customers for a coffee shop in previous week. y = (1.78,1.65,1.62,1.84,1.75,1.85,1.52,1.55) The mean of the dataset is 1....
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0answers
18 views

Sampling from matrix valued distributions

How does random sampling from matrix valued distribution work in general? In the univariate case we can draw from the uniform distribution over the interval from 0 to 1, then plug that value into ...
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1answer
81 views

Sufficient conditions for continuous functions of continuous random variables to themselves be continuous random variables

I've been trying to figure out nontrivial conditions for continuous functions of continuous random variables to themselves be continuous random variables without much success. Here's what I know so ...
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0answers
58 views

On the convergence in probability of $n^{-d}\sum\limits_{k=1}^n (X_{(k)} - Y_{(k)})^2$ to $0$, for every $d>0$

For $n \in \mathbb{N}$, let $(X_1, \dots, X_n)$ and $(Y_1, \dots, Y_n)$ be iid. samples from the same distribution. I write $X_{k:n}$ the $k$-th order statistic out of a sample of size $n$. I am ...
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0answers
32 views

Probability of repeated events with fixed delay times co-occuring

I am trying to estimate the probability of signal collisions for a research project I am working on. The project uses acoustic emitters, with each emitter encoding a unique identification signal to ...
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2answers
49 views

Sampling Distribution Disturbing Answer

I am currently studying about Sampling distribution of Sample means, and came across below example here. Question: The average male drinks 2L of water when active outdoors with a standard deviation ...
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2answers
19 views

Given $f(y) = \lambda e^{-\lambda y}$ , for $y>0$, find $P(Y>s|Y>t)$

Random variable y with pdf as follows. Given $f(y) = \lambda e^{-\lambda y}$ , for $y>0$, find $P(Y>s|Y>t)$. I am having difficulty computing this. I have used bayes and integrated to get ...
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2answers
30 views

What is the distribution created by this?

So I have a group of people. Each of them is 60% likely to vote on A and 40% likely to vote on B. What type of distribution does this create if I'm looking for amount of people that vote on A - the ...
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0answers
22 views

Produce an perpendicular vectors that follows a specific distribution

I am interested to produce a random Gaussian distributed vector perpendicular to a single or a collection of orthogonal predefined vectors. In the simple case of a single predefined vector, my first ...
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2answers
50 views

Practical physical processes for various random distributions?

You can draw from the Cauchy distribution by attaching a stick to a spindle somewhere on the y-axis, spinning it, and reading off the x-intercept as your drawn value. Where you place the spindle on ...
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2answers
63 views

Expected value for 2 dice roll

Imagine that we roll two fair six-sided dice (i.e., all six sides have equal probability). Let X1 and X2 be the random variables representing these outcomes. Now, imagine we take one of the dice rolls,...
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1answer
78 views

Probable grade of a student guessing a test with $100$ multiple-choice questions? [on hold]

Assuming there is a test a student has to take with multiple choice questions: Each question has 4 choices (a,b,c,d) The student chooses the answer per question randomly There are 100 questions Each ...
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0answers
27 views

Getting the best fitting distribution using the p value

I have a set of data points, and I want to get the best theoretical distribution that fits the data. For that I'm using Kolmogorov-Smirnov test for goodness of fit. This test reveals the p value of ...
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2answers
173 views

How to find the median of a PDF with a continuous random variable given the mode of it?

So the question is to find the median of $X$ if the mode of the distribution is at $x = \sqrt{2}/4$. And the random variable $X$ has the density function $$f(x) = \left\{ \begin{array}{ll} ...
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1answer
21 views

Stochastic orders of summands when sum has fixed distribution

Suppose $X,X',Y,Y'$ are independent random variables. We know that $X+Y \overset{d}{=} X'+Y'\overset{d}=Uniform[0,1]$ and $X\prec X'$ in the sense that $P(X> x)\le P(X'>x)$ for any $x\in \mathbb{...
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1answer
102 views

probability of prime factor

Question, what are the chances for obtaining the same, prime factor 55049? The SUM of List A gives a factor of 17 x 55049 and the sum of list B gives a factor of 19 x 55049 I want to understand how ...
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0answers
36 views

Dividing triple integral to evaluate the integrals in closed-form

I have the following triple integral $$\int_{y_1=0}^{\infty}\int_{y_2=0}^{\infty}\int_{x_2=0}^{zy_2}\exp\left(-\min\left[x_2,\,y_1(z-\frac{x_2}{y_2})\right]\right)\,dx_2dy_1dy_2$$ I want to divide ...
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60 views

Is there a generalization of the concept of variance for a collection of probability distributions?

If I have a collection of numbers, I can obtain a measure of how much they're "spread" by computing the sample variance of them, i.e. $$\frac{1}{n}\sum_{i=1}^n(x_i-\mu)^2~,$$ where $\mu$ is the sample ...
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1answer
39 views

Solution to $x = \frac{(1-F\left(x\right))}{f(x)}$ with gaussian random var

I want to find the solution to: $x = (1-F(x))/f(x)$ where $F()$ is the distribution and $f()$ the density of a gaussian random variable. Notice that it is also equivalent to solving $x = 1/h(x)$ ...
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3answers
65 views

Illustrating Normal Approximation to Binomial (CLT)

For a sampling distribution of sample proportion problem (Bernoulli distribution - $\mathbb P(\mathrm{yellow ball}) = 0.6$ out of $10000$ balls, say), I get below discrete distribution (LHS) with $\mu=...