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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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A basic question about conditional expectation for sum of random variables

The question below might be very basic and just caused by my shaky foundation in probability theory, but I would at least like to get some clarification for it. Suppose we have two random variables $...
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1answer
13 views

Find minimal sufficient statistic for truncated exponential distribution

Let $X_1, ..., X_n$ be iid $f(x; \theta, \lambda) = \dfrac{\lambda e^{-\lambda x}}{1-e^{-\lambda \theta}}$ for x $\in [0, \theta]$. I want to find A minimally sufficient statistic for $\theta$ ...
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How to graph mathematically a normal distribution?

Hello and sorry for the question, it is good that in most of the books I had seen graphics but I do not know what values can be given. The fact is that if the graph on the Cartesian plane wanted to ...
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1answer
12 views

Having a problem on a paradoxical answer

Suppose that we have PN objects (disks for an example) and we have P slots (or boxes),how many ways can we distribute those PN objects on those P slots so that each slot has exactly N object? i saw a ...
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1answer
53 views

Cutting a rope randomly and taking the longer piece, cutting the longer piece and taking the shorter piece.

Cut a rope with unit length into two pieces randomly. Cut the longer piece of the first cut into two randomly again. Take the shorter piece from that second cut. What would be the PDF and the expected ...
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114 views

Sum of Random Variables and Central Limit Theorem

As we know, each random variable is responsible for associating some random events to the probability values. These random events belong to the specific population, and that random variable represent ...
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2answers
18 views

Poisson Random Variable Question

A radioactive source emits certain particles with a Poisson distribution. The probability of no particle emissions during an hour of observation is $0.4$. What is the probability that the first ...
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1answer
8 views

For game where first player to N points wins, find the distribution of win probability and total number of points between players

Two players, A and B, play a series of points in a game with player A winning each point with probability p and player B winning each point with probability q = 1 - p. The first player to win N ...
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1answer
25 views

Combined probability distribution for two hazard functions

I have two items with two different failure rates. I need to find the combined hazard function corresponding to both items failing, from which I can find the combined probability distribution. I ...
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1answer
61 views

Pdf of the product of an exponential rv and a $f_Y=Ka^{-K}y^{K-1}$ distributed rv …

Let $X$ and $Y$ are 2 independent random variables, where $X$ has an exponential distribution with parameter $1$ and $Y$ has the following Pdf: $f_Y=Ka^{-K}y^{K-1}, 0 \le y \le a $. Someone claims ...
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1answer
32 views

What is the probability to be after $n$ random jumps of unit length in space within a distance of radius $r$ from the start?

Assume a particle, at instant 0 at the origin of three dimensional euclidean space jumps at each tick of the clock exactly one unit from its current position into a random direction. By this we ...
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1answer
36 views

Joint distribution of absolute difference and sum of two independent exponential distributions

If $X\sim \rm{Exp}(1)$ and $Y\sim \rm{Exp}(1)$ are two independent random variables. What is the joint distribution of $U = |X - Y|$ and $V = X + Y$? I used the Jacobian transformation to obtain ...
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1answer
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given exponential Cumulative distribution function, finding another Cumulative distribution function with functionl connection

There is given $X$ a random variable with exponential cumulative distribution such that $X~Exp(1)$ so the exponential Cumulative distribution function is: $P(x\le t)= F_x(t)=(1-e^{-t} , 0\le t) \...
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1answer
45 views

Does MLE really care about PDF?

I am wondering, whether MLE really cares whether it operates on proper distributions. Lets take a look at the following situation: likelihood: $$L(\theta \mid x) = \prod_{n}^{N}{f(x_n \mid \theta)}$$ ...
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Random Variable expression with the distribution function

In my textbook, it is said that if we chose the probability space ( (0,1) , B(0,1) , P) where P is the Lebesgue measure, and we have the distribution function FX(x) of a random variable X we can find ...
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1answer
47 views

MLE of simultaneous exponential distributions

Given the $X_i\sim \text{exp}({\theta})$ and $Y_i\sim \text{exp}(\frac{1}{\theta})$, where $X_i$ and $Y_i$ are indpendent, with the same $\theta>0$. I have to find the MLE and its distribution. I ...
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1answer
7 views

Existence of a sequence of independent $E$-valued random variables with distribution $\mu$ given $\mu$ and $E$ Polish

I know that the following question is true for $E=\mathbb{R}$. I would like to know if it can be extended to Polish spaces. Suppose that $(E,d)$ is a Polish space. Write $\mathcal{B}(E)$ for the ...
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A random variable formed by two Normal random variables, under a conditioning process.

Imagine two independent random variables, $X$ $\sim$ $N$ $(\mu_1$,$\sigma_1^2)$ and $Y$ $\sim$ $N$ $(\mu_2$,$\sigma_2^2)$. Now imagine a process whereby one observation of $X$ and one observation of $...
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Change of variable to calculate expected value

$X$ and $W$ are independent random variables. $$ Z=X+W $$ $$ W \sim \mathcal{N}(0,\sigma) $$ $$ E[X]=\bar{x} $$ I want to calculate $E[Z]$ with respect to the joint pdf $p(z,x)$ $$ E[Z]=\int\int (x+w)...
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1answer
14 views

Expectation of sum is less than the second moment

Given $E[f^2(X)] < \infty$ and $X_i \sim_{iid} X$, need to show $$ E\left[ \frac{1}{n} \left( \sum_{i=1}^{n} (f(X_i) - E[f(X)] \right)^2 \right] \leq E[f^2(X)]. $$ My try: $$E\left[ \frac{1}{n} \...
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1answer
28 views

Why does $ \mathbb{P}\left(X < -z\right) = \alpha \Rightarrow -z = \chi^2_{1 - \alpha}(2n) $ hold?

Assume $X_i$ are generated by $\Gamma(\theta_0,n)$ distribution, and $S_n = \sum X_i$. Further, it is known that $2 \theta_0 S_n$ follows a $\chi^2(2n)$ distribution, $\theta_0$ is known, $\theta_1 &...
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Independence of sample mean and sample variance

It is well known that under normality assumption, the sample mean and sample variance are independent, by Basu's Theorem. My question is that, is the normal distribution the only distribution whose ...
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1answer
33 views

Sampling subsets of positive numbers with sum close to a given number

Given a set of $N$ positive numbers $a_1 \ldots a_N$, I am trying to generate random subsets of this set such that the sum of numbers in subset is close to a given number $M$. Suppose I can only ...
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2answers
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Expectation of sum of geometric random variables vs. expectation of Pascal r.v.

Let $\{X_i\}$ be a Bernoulli process, i.e. $X_1, X_2, X_3, \dots$ are i.i.d. Bernoulli variables with parameter $p$. Let $T_k$ be the time at which the $k$th success occurs. I can reason about the ...
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Convergence of Random Variable (convergence in probability)

Let $(x_{n} )$be a sequence of real random variable defined on probability space ,converge in Probability to $x$ . Let $y$ be a random real variable on probability space. For $\varepsilon>0$ ...
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Convergence of random variable ! (Probability) [on hold]

Let $(x_{n} )$be a sequence of real random variable defined on probability space ,converge in Probability to $x$ . Let $y$ be a random real variable on probability space. For $\varepsilon>0$ ...
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Inference regarding the mean lifetime of a bulb using a new technique

The lifetime in hours of each bulb manufactured by a particular company follows an independent exponential distribution with mean $\lambda$. We need to test the null hypothesis $H_0: \lambda=1000$ ...
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3answers
25 views

Probability generating function of exponential distribution

The exponential distribution is given by: $$PDF: \lambda e^{\lambda x}$$ And the formula for probability generating function is given by: $$G(z) = \sum_{x=0}^\infty p(x)z^x$$ where $p(x)$ is a ...
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Wasserstein distance between centered Gaussian mixtures

We use $\mathcal{W}_2(\cdot, \cdot)$ to denote the quadratic Wasserstien distance as defined here. Now, let $X,Y = \mathcal{N}(0,1)$ be two standard normal random variables and for $ a \in[0,1]$ let $...
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1answer
52 views

ML estimation for Weibull

What are the maximum likelihood estimators of $\eta$ and $\beta$ ($\eta>0$ and $\beta>0$) for an i.i.d. sample of size $n$ from the following density: $f(x_i)=\frac{\beta x_i^{\beta-1} }{\eta ^ {...
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Lexicographical order in context of identifiability of mixture of two Normal distributions

I want to understand a method used in a paper on identifiability of mixture of two Normal distributions. This is Teicher 1963 "Identifiability of finite mixtures", fragment of the proof The author ...
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2answers
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How to determine Y(n)

A random variable $x$ from the set $\{1, 2, ... ,n\}. $ Let $x$ has distribution function $f(k) = Y(n) · g^k$ where $g$ is a fixed number within $0$ and $1$. Find $Y(n)$ which is a constant term in ...
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1answer
34 views

Problem to calculate a marginal function in probability

I have a problem in probability. I have $f(x, y) = \frac 14 \cos(y) $, if $x$ is between $0$ and $\pi$, and if$ y$ is between $-\frac x2$ and $\frac x2$. I have to calculate $f(y)$. I calculated ...
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2answers
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Linear transform of bivariate normal distribution

Suppose that $Y_1$ and $Y_2$ follow a bivariate normal distribution with parameters $\mu(Y_1)= \mu(Y_2)= 0, {\sigma^2}(Y_1)= 1, {\sigma^2}(Y_2)= 2$, and $\rho = 1/\sqrt 2$. Find a linear ...
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How do I find if the probability of the sample proportion is greater than something?

I have this problem and I have no clue how to solve it. In 2012, 31% of the adult population in the US had earned a bachelor’s degree or higher. One hundred people are randomly sampled from the ...
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1answer
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Tail difference of quantiles of (symmetric) distribution functions

Assume, for example, $z_\alpha$ are $\Phi^{-1}(\alpha)$ quantiles from standard normal distribution, $\alpha > 0$. If we are interested in the sum$$z_\alpha + z_{1 - \alpha}$$ for standard normal ...
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1answer
769 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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Correct formula to calculate Chi square statistic [on hold]

I came across two formulas for calculating Chi square statistic. Method-1: Chi square statistic, X²= [(n -1)*s² ]/σ², where n is the sample size, s denotes standard deviation of the sample and σ is ...
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Probabilities of choosing a red ball:Case 1) 10 distinct balls,8 red and 2 black Case 2) 10balls ,8 identical red and 2 identical black

I know the probability in both cases evaluate out to be 8/10. I want to know the intuition behind solving the probability of choosing a red ball when there are 8 identical red balls and 2 identical ...
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78 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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How to calculate Poisson probability with float $k$?

Ok, so I see the equation is $\frac{λ^k * e^{-λ}}{k!}$ but what if my $k$ is not an integer but a float possibility? for example, $k = 2.5$ and $λ = 5$. I have tried to multiply $k$ and $l$ with the ...
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Probability of having picked every item from the set at least once after n turns, while picking 3 per turn

Let's say I have a set of 100 items. Each turn, I pick three items at random, note which ones I've picked, and put them back. What is the probability I've picked every item at least once after $n$ ...
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1answer
23 views

Does this discrete probability distribution have a name?

I was wondering whether the probability distribution $$P(X = k) = \frac{\lambda}{(1+\lambda)^{k+1}}, \quad k= 0, 1, 2, \dotsc,$$ where $\lambda$ is a fixed positive number, has a name.
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Convergence of KL-divergence along a convergent sequence of measures

My question is about Lemma 12 and 13 (page 6) of of this paper https://arxiv.org/abs/1802.09583. The Lemma 13 in particular proves, ``Let $\log(g)$ be bounded. If $P_n \rightarrow P$, then $KL((P_n)_g ...
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When does $E[f(X_i)]=E[f(X_j)], i\neq j$?

Suppose we have random variables $X_1, \dots, X_N$, with joint probability distribution $F_{X_1,\dots,X_N}$. Under what conditions does the following equality holds? $$E[f(X_i)]=E[f(X_j)],\ \ i\neq ...
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Reverse engineering distributions

Suppose I am given a measurable function $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and a probability distribution $\mathbb{P}$ on the Borel or Lebesgue sigma algebra of $\mathbb{R}^n$. Assume that the ...
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1answer
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How can we approximate a function by sampling a distribution proportial to it and making a histogram of samples?

I've read the following (here on page 2): Suppose that you want to approximate a function $f$. One way to do this is to produce a sampling distribution proportional to $f$ and then make a histogram ...
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1answer
25 views

Probability distribution of a moving particle

I am having a issue with the wording of this question. Find the probability of the following. The velocity $v$ of a randomly selected particle, whose distribution obeys the probability density ...
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Distribution of life time of a serial circuit with bulbs

Assume that we have a serial circuit with three bulbs. Each bulb's life time is exponentially distributed: $$f_{bulb}(t) =\left\{ \begin{aligned} &\lambda e^{-\lambda t} & t \ge 0\\ &0 &...
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1answer
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Transforming sum of exponential variables to chi-squared distribution

Assume $X_i$ are generated with the following distribution: $$ f(x; \theta, c) = \theta^{-c}cx^{c-1}e^{-(x/\theta)^c}$$ $\theta>0$ and $c>0$ is known. Further, assume $T(X)=\sum^{n}_{i=1} X_i^...