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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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Understanding the connection between the chi-square and the gamma distribution

If $Z_1,\ldots, Z_n$ are independent standard normal random variables, then the random variable $X = \sum_i Z_i^2$ is said to have a chi-square distribution with $n$ degrees of freedom. If you ...
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1answer
30 views

Variance equations that yield inconsistent results

Consider a game of chance were a player places a bet, and coin is flipped if the coin lands on heads the player wins $2b$ where $b$ is the bet value. If the coin lands on tails the player looses the ...
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3answers
62 views

Where did I go wrong in proving $\mathbb E[X^{2n}] = \prod_{1 \leq k \leq 2n, k \operatorname{odd}}k$

Let $X$ ~ $\mathcal{N}(0,1)$ Show that: $\mathbb E[X^{2n}] = \prod_{1 \leq k \leq 2n, k \operatorname{odd}}k$ Idea: $\mathbb E[X^{2n}]=\frac{1}{\sqrt{2\pi}}\int_{\mathbb R}x^{2n}e^{-\frac{x^2}{2}}...
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Showing monotonicity for ratio of binomial pmf and tail cdf

I'm interested in showing for $X\sim\text{Bin}(n,p)$ that when $x\geq np$, $$ \frac{P(X=x)}{P(X\geq x)}\leq \frac{P(X=x+1)}{P(X\geq x+1)} $$ I've verified using numerical simulations, but can't seem ...
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1answer
53 views

What is the expected wait times of Poisson arrivals?

Suppose customers arrive at a system as a Poisson process with rate $\lambda$, given a specific time interval $[0,t]$, what is the expected wait times for those customers who arrive in this interval? ...
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7 views

Joint PDF & CDF

LET X & Y be continuous random variables with $\int_0^\infty\int_0^\infty cxe^{-x(y+1)} dydx$ PART ONE We are asked to solve for c, $1=\int_0^\infty\int_0^\infty cxe^{-x(y+1)} dydx$ $1= c\int_0^...
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14 views

Random perturbation of uniform distribution on the unit sphere

Assume that $\theta$ is uniformly distributed on the unit sphere in $\mathbb R^d$ and let $w \sim N(0, I_d)$, that is. a canonical Gaussian vector. Let $\alpha \in \mathbb R$ be a random variable. ...
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16 views

Show that the family of beta distributions where parameters $α$ and $β$ are unknown is an exponential family.

Show that the family of beta distributions where parameters $α$ and $β$ are unknown is an exponential family. I know that the beta distribution is$f(x; \alpha, \beta)={1\over B(\alpha, \beta)}x^{\...
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22 views

Show that $e^{-\mu+\frac{\sigma^2}{2}} $ is estimable if $\frac{\mu}{\sigma}$ is estimable.

Suppose $X_1,X_2,...,X_n \sim^{i.i.d} N(\mu,\sigma^2)$.Show that $e^{-\mu+\frac{\sigma^2}{2}} $ is estimable if $\frac{\mu}{\sigma}$ is estimable. I am utterly confused, in fact I can think of this ...
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2answers
31 views

Finding $E(Y\mid X)$ if the PDF of $(X,Y)$ is $f_{X,Y}(x,y)=e^{-y}$ on $y>x>0$

The random variables $X$ and $Y$ have the joint density $$f_{X,Y}(x,y)= \left\{\begin{matrix}e^{-y}, \mbox{ } 0\leq x \leq y \le \infty \\ 0, \mbox{ otherwise}\end{matrix}\right.$$ Evaluate the ...
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29 views

Discrete PDF for N number of Coin Toss where the probability of Head is p [on hold]

This is a question from my Estimation Theory class and it would a great help if someone can solve the math for me. Consider a coin where the probability of heads is p. Determine the discrete pdf for ...
2
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1answer
31 views

X and Y random variables. How can I interpret $\sqrt{E(Y-X)^{2}}$ as the distance between X and Y?

X and Y random variables. How can I interpret $\sqrt{E(Y-X)^{2}}$ as the distance between X and Y by showing (prooving) that i) and ii) below works: if $\sqrt{E(Y-X)^{2}}$ = 0 then $P(Y=X)=1$ The ...
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23 views

How is the probability-generating function computed when we change the area in this problem?

I have a great difficulty understanding the following problem. Suppose we have an area of a forest $A$ units large. Let $N$ be the random variable of the number of animal nests in the area. We know ...
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13 views

Finding joint pdf of two functions, one min of two geometric functions and other defined based on difference of variables

I'm really stuck on this problem and I have no clue how to proceed right now. Let $X,Y$ be random geometric variables. Let $Z = min(X,Y)$, and $W = { \left\{ \begin{array}{lll} 0 & \mbox{if ...
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1answer
117 views

Ten balls are thrown randomly into three buckets.

how would you solve this problem? Ten balls are thrown randomly into three buckets. Compute the probability that each bucket contains at least one ball.
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0answers
15 views

Statistical sampling in audit

I am working on statistical sampling in audit. I would like to summarize some criteria and some theory about sampling in order to properly estimate sample sizes from which then is it possible to draw ...
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0answers
7 views

upper bound for KL divergence of two sub-gaussians with different mean and variance

I was wondering if there is any upper bound for KL divergence of two subgaussian with means $\mu_1$ and $\mu_2$ and variances $\sigma_1$ and $\sigma_2$? The definition of KL divergence is $$D_{KL}(...
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3answers
27 views

Alternative approaches to obtain the expected value of the geometric distribution

Given that $X$ has geometric distribution with $p_{X}(x) = p(1-p)^{x-1}$, determine $\textbf{E}(X)$. MY ATTEMPT \begin{align*} \textbf{E}(X) = \sum_{x=1}^{\infty}xp(1-p)^{x-1} = p\sum_{x=1}^{\infty}...
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X and Y are i.i.d with Uniform distribution in $[0,1]$. Let $M=XY$. What is the distribution of $X|M=1$

X and Y are i.i.d with Uniform distribution in $[0,1]$. Let $M=XY$. What is the distribution of $X|M=1$ and $X|M=0$? so, $M=XY$ and $U=Y$ $g_{1}(x,y)=xy$ and $g_{2}(x,y)=y$ $h_{1}=m/u$ and $h_{2}=u$...
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1answer
36 views

Additivity of Value at Risk

I'm struggling with the following exercise: Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $X,Y$ be two real-valued random varaibles such that their corresponding cumulative ...
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1answer
32 views

Expected value of X*Y

I'm stuck on the following probability problem and would welcome any help: Consider three random uniform variables on [0,1]. Let X be the minimum of these three variables and Y be the maximum. What ...
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0answers
18 views

bernoulli distribution has a density with respect to dirac measure

We consider E={0,1} and the Bernoulli distribution $P_{\theta}$. We want to show that $P_{\theta}$ admit a density with respect to the measure µ= $\delta_0+ \delta_1$ I didn't understand the answer ...
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0answers
9 views

Translated uniform unit ball pdf

Let $X$ be a random variable distributed uniformly in the unit ball, then we know that the pdf w.r.t. spherical coordinates is a constant multiplied by the Jacobian inside the sphere (and 0 outside). ...
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0answers
15 views

Questions about the interpretation of system-dynamic simulation

I try understand simulation modeling in R using DeSolved package. As source i used this article https://api.rpubs.com/rsmard05/sysDynR. It is detailed, but some things are incomprehensible to me from ...
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11 views

What is the efficiency of the machine (probability theory)

A machine is made of 5 parts. Its efficiency deppends on the efficiency of each of the 5 parts. What is the efficiency of the machine, if the probabiliy that each of the parts is broken equals $P=10\%$...
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1answer
16 views

Skewness and Kurtosis of the Degenerate Distribution

If a random variable $X$ has a degenerate distribution, that is it takes a given value $k$ with probability $1$ and every other value with probability $0$, what is the skewness and kurtosis of $X$? ...
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1answer
18 views

Probability calculations verification request: a distribution function related problem

Let $X$ have distribution function $F(x)$ expressed by \begin{cases} 0 & \mbox{if } x < 0 \\ x/2 & \mbox{if } 0\leq x \leq 2 \\ 1 & \mbox{if } x \geq 2 \end{cases} and let $Y = X^{2}$...
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1answer
32 views

Expectation of a function of two independent random variables.

Suppose X and Y are independent and uniformally distributed around [0, 1]. Define $Z = (X-Y)^2$. I'm interested in the variance $\sigma_Z^2 = \mathbb{E}[Z^2] - \mathbb{E}[Z]^2$, and suppose I already ...
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8 views

Effect of a convolution with a Bernoulli distribution on Rényi divergence

Let $P$ and $Q$ be two probability distributions on $\mathbb{Z}$. Let $D_\alpha(P\|Q)$ be the Rényi divergence of order $\alpha$ of $P$ and $Q$: $$ D_\alpha(P\|Q)=\frac{1}{\alpha-1}\sum_i\frac{P(i)^\...
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1answer
20 views

A Problem related to the Cauchy Distribution

The following problem is from the book, "Introduction to Probability" by Hoel, Port and Stone. My answer does not match the back of the book. What did I do wrong? Thanks, Bob Problem: Let $X$ have a ...
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33 views

CDF after Transformation.

A random variable $Y$ has lognormal ($\mu,\sigma$) distribution if its probability density function is $$ f(y)=\frac{1}{y\sigma\sqrt{2\pi}}exp-\frac{(\ln y-\mu)^2}{2\sigma^2}$$ its CDF will be $$\hat ...
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1answer
36 views

If $\ln(x)$ is gamma distributed, what is the distribution of $x$?

Additionally, if someone could help calculate the mean and variance of $X$, that would be greatly appreciated.
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1answer
34 views

Integral of a standard gaussian distribution

Please see the image. I know that the integral of P(x) tends to 1. But the quadratic equation next to P(x) seems confusing.
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27 views

E[(X+Y)^3], where X follows exponential distribution with a, and Y follows exponential distribution with b.

First, of all, forgive my English terminology. If you need any clarification let me know. I do understand how E[X+Y] is calculated into E[X]+E[Y], but I'm having trouble when it is ^3. Also, how can ...
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2answers
23 views

Probability of an event for a continuous random vector of three coordinates

Let $X=(X_1,X_2,X_3)$ be a continuous random vector of the joint pdf $$f(x_1,x_2,x_3)= 12x_2 \;\mathrm f \mathrm o \mathrm r \; 0<x_3<x_2<x_1<1$$ and $0$ elsewhere. I need to find the ...
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0answers
28 views

What are the odds to return a product based on its probability to be defective?

We know that diskettes made by a factory have defects with probability $0.3$, independently of each other. The factory sells packages containing $10$ diskettes and offers a guarantee that at most one ...
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0answers
22 views

Uniform PDF of offset sphere

Note that throughout this I use the spherical mapping convention: $$(x,y,z) = (r\cos\phi\sin\theta,r\cos\theta,r\sin\phi\sin\theta)$$ I have derived that the uniform pdf for a sphere $S_1$ with ...
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2answers
26 views

Probability exercise about allocating bottles of wine

The exercise states the following: We have 15 bottles of wine that we will randomly distribute among three customers: A will get 2, B will get 8 and C will get 5. We've learned later that 4 of this ...
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2answers
33 views

Notation and major confusion with the definition of the probability simplex

So it starts like this: Given a discrete set $N$, the probability simplex over $N$, denoted $∆(N)$ is defined to be: $$Δ(N) = \left\{ x \in \mathbb{R}^{|N|} \;:\; x_{i} \geq 0 \text{ for all } i, \...
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1answer
18 views

What is the probability distribution of this AR(1) function?

I'm preparing the exam for "stochastic models" and I encountered this exercise which is giving me a lot of problems: Let $X_t \sim AR(1)$, with $$X_t=-0.8X_{t-1}+ \epsilon_t, ~~~~~~~~~~\epsilon_t \...
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1answer
13 views

Discrete distribution where mgf exists only at zero but all moments are finite

Does such a distribution exist and if it does, what does it look like? For the continuous case, there is the log-normal distribution, so my gut says there must be an analogous discrete distribution ...
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0answers
24 views

show $F(a)=\sideset{_{-\infty}}{^a}\int f(x)dx$ is a distribution function

If $f$ is a function satisfying- $_{1)}$ $f(a)\ge 0$, $\forall a\in\mathbb{R}$ $_{2)}$ $\sideset{_{-\infty}}{^\infty}\int f(x)dx =1$ then, $f$ is a density function of the distribution function $...
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45 views

Proof of a lower bound on probability

How to prove that the probability of simultaneous occurrence of more than $\frac n 2$ events from $n$ independent Bernoulli trials is greater than or equal to:$$1-e^{-2n\left(p-\frac{1}{2}\right)^2}$$ ...
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1answer
29 views

Reasoning check on geometric distribution question

Consider a biased coin with probability $p$ to show head. Then you toss it twice and consider the result $(H,T)$ as a "head" and the outcome $(T,H)$ as a "tail". If it does not occur neither "head" or ...
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0answers
17 views

Truncated conditional expectation of multivariate normal distribution

Assume that $y_1 = \alpha_1 s_1 + u_1$ and $y_2 = \alpha_2 s_1 + \alpha_3 s_2 + \alpha_4 u_1 + \alpha_5 u_2$ where $s_1 \sim N(0,\Sigma_1)$, $s_2 \sim N(0,\Sigma_2)$, $u_1 \sim N(0,\sigma^2)$, $u_2 \...
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1answer
19 views

Bounded function with finitely many discontinuities is integrable $\overset{?}{\Rightarrow}$ density of continuous distribution function is not unique

The density function of the distribution function of a continuous random variable is not uniquely defined. A new density function can be obtained by changing the value of the function at finite ...
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0answers
18 views

How can I compute a distance function for two sets of vectors?

Let's say I have two sets of vectors, $A$ and $B$. The cardinalities of $A$ and $B$ are not the same, i.e. $|A| \ne |B|$ however, each of the vectors from either set are of the same size, i.e. have ...
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1answer
45 views
+50

Expectation and variance of the number of elements of a random non-empty set selected from a finite power set

Let $S$ denote a finite set of cardinality $|S| = N$. Select randomly a non-empty subset of $S$. Let $X$ indicate the number of items belonging to this subset. (a) Describe the probability mass ...
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2answers
21 views

Ordering of maximums from exponential random variables

Let $E_1,\ldots,E_n$ be exponential random variables with parameters $(\alpha_1,\ldots,\alpha_n)$. Further for $1\leq z\leq n$, let $M_1=\max\{E_1,\ldots,E_z\}$ and $M_2=\max\{E_{z+1},\ldots, E_{n}\}$....
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1answer
58 views

Probability distribution for an infinite number of fair coin flips?

Given an infinite number of unbiased coin flips, what is the probability that some portion $x\in(0,1)$ of all coin flips will be heads? Edit: Alright, I've added what I've done so for. Please keep in ...