Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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16 views

Using CLT to determine the sample size to achieve a given power.

Consider a distribution having a pmf of the form $f(x;\theta)=\theta^x(1-\theta)^{1-x}$ $x=0,1$, zero elsewehre. Let $H_0: \theta=\frac{1}{20}$ and $H_1: \theta>\frac{1}{20}$. Use the Central Limit ...
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16 views

The quotient of two chi distributions

The quotient distribution of two chi-squared distributions is F-distribution. What would be the quotient distribution of two chi distributions? Is there a general distribution for this?
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29 views

Models for Probability Density Functions with unknown parameters and given mean and variance

The PDF $f(x)$ of a non-negative random variable $x$ has the structure $$f(x)=\exp (a-bx-cx^{2})$$ where $a$, $b$ and $c$ are any model parameters. It is assumed that $c\ge 0$ so that $f(x)$ does not ...
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1answer
18 views

Bernoulli distribution with non integer number of trials

Can we generalise the Binomial distribution for a non-integer number of Bernoulli trials?
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1answer
38 views

finding the marginal density of Y

Question . The joint probability density function of X and Y is given by $f(x, y) = (1/8)(y^2 − x^2)e^{-y} , -y\leq x\leq y, 0\leq y \leq \infty $ Find the marginal density of x. So i know that we ...
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1answer
75 views

Probability that at least two of three uniform random variables~[0 1.5] add up to >2 [closed]

There's a problem I've been stuck on for a while regarding the sum of two uniformly distributed, independent random variables. The problem goes like this: You find some old batteries in a drawer. ...
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1answer
19 views

Distribution of linear combination of discrete variables

Assume $X,Y$ are discrete independent random variables with known distribution $P_X(x), P_Y(y)$ and $c_1, c_2$ constants. Can we determine the shape of the distribution of: $Z = c_1~X+c_2~Y$
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1answer
18 views

Sum of two random variables converging with different modes [closed]

Is it true that if X_n converges in distribution to X; Y_n converges in probability to Y; X_n, Y_n, X and Y are real-valued random variables defined on the same probability space, then X_n + Y_n ...
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11 views

Is there a tool/program out there that allows you to draw probability function and it spits the equation, runs simulations, etc…?

Is there a tool that allows you to draw a probability function (with some parameters pre-set such as the # of polynomials or curves, discrete/continuous, etc...) and the program: Spits out the ...
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25 views

Upper bound for most likely category in multinomial random vector NOT being max count realized

Let $(X_1,\dotsc, X_k)$ be distributed multinomial with parameters $n, (p_1\dotsc,p_k)$ and suppose $p_1>p_j$ for $j\neq 1$ so that category 1 is the most likely outcome from any given realization. ...
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25 views

how can I find the PDF of g(x) when the Characteristic function is known?

Suppose that the characteristic function of X is given ($sin(\alpha\omega/2 \pi)$ ) how can I find the PDF of the $y=x^2$ ?I think we should find the PDF of the function X (using the related ...
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1answer
29 views

Unknown distribution of a random variable

$X_1, X_2, \ldots, X_{400}$ is a random sample from given distribution with median of m ($P(X_i \le m)=0.5$). Calculate $P(X_{220:400} \le m)$. How to calculate that? I am lost with this question. ...
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1answer
23 views

What is the average number of random selections it would take to have picked every element of a set and the size of that set, n?

I've been discussing this question with my AP statistics teacher and we're both racking our brains as to how this probability distribution would look. The problem came up when looking at the scenario ...
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0answers
16 views

Convergence in distribution of the following sequence of random variables

$X_n\sim Beta\left(\frac{\alpha}{n},\frac{\beta}{n}\right)$ with $\alpha>0$ and $\beta>0$. Does $X_n$ converge to a distribution?
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22 views

Max of independent Poisson random variables with widely different means

Suppose $0<\lambda_1\leq \lambda_2\leq \ldots\leq \lambda_k.$ For each integer $n>0,$ and $1\leq i\leq k,$ let random variable $X_{n,i}$ be distributed as a Poisson random variable with mean ...
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29 views

Probability density function with two peaks and skewness

I have plotted a probability density function on a graph. With one line from $(0,1)$ to $(1,0)$ and the second line from $(1,0)$ to $(2,1)$. The area under the lines sum up to $1$ and all values of ...
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1answer
28 views

memoryless property of exponential distributions with random variables

It is true that $P(X>t+s|X>t)=P(X>s)$ for certain values $t$ and $s$. However, how can I show that this still holds if: $T$ is a continuous random variable. That is ...
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26 views

Gap probability for i.i.d. random variables

Given a set $\{X_1,\ldots,X_N\}$ of real i.i.d. random variables, drawn from a common parent pdf $p_X(x)$, what is the probability that, given one random variable taking value in $(t-dt,t)$, there are ...
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1answer
22 views

Product of discrete random variable with constant

Say that we have a random variable $X$ and the distribution $P_X$. I know that is $X$ in continuous then, from the principle of conservation of probability, we get $P_{cX} = \frac 1 {|c|} P_X(\frac ...
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1answer
22 views

Why is chi-square distribution with 2 degrees of freedom an exponential distribution?

Is there any explanation on why these two distributions are equivalent? How can the sum of two square of Gaussians represents the limit of a geometric distribution? I found an answer here, which ...
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1answer
47 views

Probability of two IID random variables

Let $X$ and $Y$ be independent and identically distributed. Show that if $X$ and $Y$ are continous, then $P(X<Y) = 1/2$ Give an example of two IID RVs $X$ and $Y$ such that $P(X<Y)\neq 1/2$ ...
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17 views

Is there a notation to designate a random variable given it's distribution and vice versa?

Most of the times, I find it unnecessary and tedious to name the corresponding random variables of a given bunch of probability density functions. For example, one must write things like this even ...
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17 views

Pdf of multivariate Gaussian random variable raised to a power?

Let $X$ be a vector that has a multivariate normal distribution. Is there a formula for the density of $|X|^p$?
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13 views

Variance of Distributions from the Exponential Family

I want to understand how the variance of an exponential family behaves. To take a very concrete example. Let consider the unit ball $B$ in d dimensions. Consider the following distribution over unit ...
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11 views

Convexity of Copulas in n-dimensions

I am trying to understand the convexity and concavity of Copulas in n-dimensions and I would like to understand the approach to use to prove whether a copula is convex or concave. I am looking for a ...
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1answer
19 views

Beta distribution density function - integration problem

I want to calculate $EX$ of beta distribution given by certain formula (if necessary i will post it) i am stuck with an integral of this sort: $$\int_0^1\frac{\Gamma ...
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0answers
29 views

What notation can I use to indicate probabilities of probabilities?

What notation should I use to describe a uniform probability conditional upon a binomial distribution? In addition, I would like to describe the range of the uniform distribution and that each sample ...
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1answer
64 views

Assuming the two people wait for each other, what is the expected waiting time?

Two people agree to meet at a restaurant. Assume their arrival times are independent and uniformly distributed on the one hour interval from 1:00–2:00 p.m. Assuming the two people wait for each other, ...
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0answers
37 views

Probability distribution of maximum of absolute value of a random walk

Suppose that we have a random walk $\{B_t\}_{t\ge0}$. The maximum of $B_t$ is well known: $M_t=\sup_{0\le s\le t} B_s$ has probability $Pr(M_t>x)=2Pr(B_t>x)$. Is there a known result for the ...
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1answer
40 views

Prove the consistency of Gamma distribution estimators

Given $X$ a random variable in a Gamma distribution, $f(x ; \alpha,\beta)$, and: $E(X) = \alpha \beta$ $Var(X) = \alpha \beta^2$ $\hat \alpha = $$\bar X \over \beta$ $\hat \beta = $$\frac {n \bar ...
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13 views

Tight upper bound for expectation of function of a positive and bounded random variable

This problem popped up in my research. Let $X$ be a positive and bounded ($X\in (0,B) \ a.s.$)random variable with degrees of freedom $d$ and noncentrality parameter $\lambda$. My goal is to find ...
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19 views

Find pmf for binomial distribution with prior

Let $X$~$Bin(n,P)$ where $P$~$Beta(\alpha,\beta)$. How do I find the pmf for $X$? I have a vague idea that I have to condition on $P\leq \tilde{p}$ to find $Pr(X=x|P\leq\tilde{p})$ but I'm not ...
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2answers
31 views

Finding a PDF for X

So let's say I pick 4 balls out of a hat with no replacements. Let's say there's 3 black balls and 7 red balls. Let X denote the number of red balls I pick then what's P(X = 3 | X >= 2). I know what ...
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1answer
37 views

Expected number of rolls for fair die to get same number appear twice in a row?

We repeatedly roll a fair die until any number appear twice in a row. I want to find the expected number of rolls until we stop. I am thinking this is a geometric distribution, but how would I apply ...
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16 views

poisson distribution - mode and/or maximum…?

"often we get observations $x_1 , ..., x_n$ out of a family of distributions and we want the best parameter. Therefore we assume that the probability function of those observations is given by $p(x_1 ...
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28 views

Placing 2N bricks on two piles of N

$2N$ bricks are to be placed on two piles of $N$. The bricks are placed one by one. If one of the piles is already complete(has $N$ bricks) all the successive bricks will be placed on the other pile. ...
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9 views

Proof and precise formulation of Welch-Satterthwaite equation

In my statistics course notes the Welch-Satterthwaite equation, as used in the derivation of the Welch test, is formulated as follows: Suppose $S_1^2, \ldots, S_n^2$ are sample variances of $n$ ...
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1answer
36 views

Determining the values a random variable takes

Let $(X_n)$ be IID bernoulli random variables and set $$Y_n = \sum_{i=1}^n \frac{X_i}{2^i}$$ I am trying to show this converges weakly to the uniform distribution on $[0,1]$. I am given a hint that I ...
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1answer
32 views

T-Distribution, Normal Distribution, and Confidence Intervals

In my probability class we were given the following problem: Suppose you take a sample of your friends and measure their heights. You calculate the sample mean to be 5 feet tall and the sample ...
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1answer
113 views

Sum of numbers on 36 dice

If 36 dice are rolled, find approximately the probability that the sum of the numbers appeared on them is between 110 and 130. How do you approach such a problem? Any help would be appreciated.
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1answer
21 views

Joint distribution of largest and last sample

Suppose 12 i.i.d. integer samples are taken from 1 to 10 with uniform probability. What is the joint distribution of the 3 last numbers ($x_{10},x_{11},x_{12}$) and the 3 largest numbers ...
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8 views

Assume we have whole bunch of i.i.d samples (points) in 2d where x and y are natural numbers. Can we define median s.t in belong to the set of points?

Assume we have n samples in 2d space where x and y coordinates are natural numbers. Would it be reasonable to define 2d median in this case as a point such that its distance to all other points is the ...
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Sampling distribution : mechanics of chi-squared variables

I try to understand the mechanics of the variables that are obeying to chi-squared distributions. To what distribution obey the square root of a chi-squared variable. For example, $\sqrt{X}\sim \,?$ ...
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28 views

What is the limit distribution of this sequence of random variables? [closed]

Find the distribution of the limit when $n\to\infty$ of$${S_n\over n^2}$$ where $S_n=X_1+X_2+...+X_n$, and $ X_1,X_2,...$ are random variables i.i.d. with characteristic function ...
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1answer
20 views

Incrementally adding to a uniform distribution of samples

I want to simulate the generation of objects over a defined area $A$, where the objects have a uniform spatial distribution, $D$ objects per unit area within $A$. I initially thought that, for $A$ ...
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28 views

What is the joint distribution of order statistics and samples?

If samples $X_1, X_2, ... X_t$ are picked independently and identically from the uniform distribution $[1,2, ..., P]$, what is the joint distribution ...
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0answers
30 views

Tukey's symmetrical lambda distribution

U ~ Uniform(0,1) $$Z_\lambda = \frac{U^\lambda-{(1-U)}^\lambda}{\lambda} $$ I have to find the first four moments and two ($\lambda_1,\lambda_2$) such that they have the same four moments.
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2answers
106 views

What is probability that at least $2$ people have same birthday from group of $N$ people?

Question is not that simple. There are also leap year included.Leap year will be $366$ days and normal year will be $365$ days. There is a statement in question that : there are exactly ...
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41 views

Compute a joint probability

Let $Y$ be a random variable defined as the sum of $5$ independent Bernoulli trials in which the probability of each Bernoulli taking the value $1$ is given by $r$. Suppose that prior to the $5$ ...
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1answer
45 views

M. Ross problem 12 chapter 5 - Exponential distribution

I have a question regarding problem 12(b) and (c) of chapter 5 of M.Ross "Introduction to probability models". The question is as follows: If $X_1, X_2, X_3$ are independent exponential random ...