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

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42 views

Finding a likelihood function given binary observed data

I'm having trouble really understanding the terms used for this homework question, and what I am actually supposed to be doing, given the actual data for the problem. Below is the problem: Suppose we ...
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1answer
59 views

What is the probability that two univariate Gaussian random variables are equal?

Let $X_1$ and $X_2$ be two independent univariate Gaussian random variables, s.t. $$X_1\sim \mathcal N (m_1,\sigma_1^2)$$ $$X_2\sim \mathcal N (m_2,\sigma_2^2)$$ So now what is $P(X_1=X_2)$? I tried ...
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0answers
45 views

Distribution of sum of absolute values of 2D Gaussian

It was a while back I read probability theory and I've stumbled on a question in my work I'm not to sure about. I have a position a=(x,y)+g with a added 2D Gaussian noise g $\in ...
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1answer
153 views

Compound of Exponential and Inverse Gamma Parameter

I am trying the prove the following: Show that an exponential random variable such that the inverse of the parameter is gamma-distributed is Pareto-distributed. More precisely, show that if $$X | M = ...
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2answers
65 views

How to compute probability related to a difference of two random variables

I am studying Joint Probability Distributions and Random Samples. I have a function for a probability distribution, defined as: $ f(x, y) = K(x^2 + y^2)~~~~~~~~~ 20 \leq x \leq 30, ~~~20 \leq y ...
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2answers
63 views

Construct a random variable under given constraints

In preparation for a probability examination, I am working on the following problem. Problem A box contains three white balls and ten black balls. Balls are drawn without replacement until all the ...
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1answer
56 views

Calculating probability distribution under given constraints

I recently asked a question about the construction of a random variable under given constraints (see: Construct a random variable under given constraints). The only answer to my question suggested a ...
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1answer
26 views

How to Solve Multiple Stopping Problem with a Known Payoff Distribution

I'm interested in learning how to optimally solve a multiple stopping problem with a known payoff distribution, like the following: You are observing a sequence of forty $(40)$ opportunities, each ...
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1answer
128 views

Application of Compound Poisson Process

I am trying to solve the following application problem: The life T (hours) of the lightbulb in an overhead projector follows an Exp(10)-distribution. During a normal week it is used a Po(12)- ...
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1answer
74 views

Why is the strong law of large number stronger than weak law? [closed]

The weak law is easy to prove, but the strong law (which of course implies the weak law, by Egoroff’s theorem) is more subtle. I'd like to know for which mathematical reason is the strong law ...
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1answer
65 views

Recover the distribution of a Binomial random variable from its Characteristic Function

Hoping someone could show how to use the Characteristic Function of a binomial r.v. to recover its distribution. Using the inversion formula to recover the pdf of a r.v. with a continuous ...
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0answers
30 views

What function describes the frequency for each unique ratio for all possible expansions n over d where n<d?

I am hoping to solve the following problem for a scientific investigation, which relies on the probabilites of all possible expansions. What function $f(r)$ describes the frequency for each ratio for ...
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1answer
150 views

How to get uniform distribution with two dice rolls?

The sum of two dice rolls will not have uniform distribution. Never realized... Is there an easy way to cheat? Will this work? 1st die roll, 1-6... 2nd die roll, if 1-3, add 0 to first die, if 4-6, ...
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1answer
39 views

Distribution from moment generating function [closed]

Moment generating function for $ X ~ (\vec{\mu}, \Sigma) $ is of form $ M_x(t) = exp( t^T\vec{\mu}+\frac{1}{2}t^T\Sigma t)$ The random variable $X = [T_1, T_2]^T$ has moment generating function ...
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0answers
26 views

Improper integral over product of exponentials: $\int_{-\infty}^{\infty} e^{-\frac{(a-x)^2}{2c}} e^{-\frac{(b-f(x))^2}{2d}} dx$

I'm looking for a way to evaluate following integral $$ \int_{-\infty}^{\infty} e^{-\frac{(a-x)^2}{2c}} e^{-\frac{(b-f(x))^2}{2d}} dx $$ f(x) resembles however a complex simulation and can therefore ...
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1answer
58 views

Distribution of arcsin of a uniform random variable

Question: Find the law of $\arcsin(X)$ where $X\sim Unif[0,1]$ and where $X\sim Unif[-1,1]$ My attempt: We say $f_X(x)=Unif[0,1]$, and that $Y=\arcsin(X)$ We say $x=\phi^{-1}(y)=\sin(y)$ and have ...
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1answer
37 views

Probability of Playing Darts

We have a dartboard with radius $1$, the dart will always hit the dartboard. The hitting point of the dart is uniformly distributed, with a stochastic vector $(X,Y)$. Now I want to determine the ...
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0answers
51 views

Exlain the significance of the uniform random variable for the simulation of random variables

I can think of the "Universality of the Uniform": Given an Unif(0,1) r.v., we can construct an r.v. with any cts distribution we want. Conversely, given an r.v. with an arbitrary cts ...
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0answers
21 views

# of Crossing of pairs continuous distribution functions and # of crossing of their inverse

Suppose $F_X$ and $F_Y$ are two continuous probability distributions that cross only twice. Does that imply that $F_X^{-1}$ and $F_Y^{-1}$ also only cross twice?
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1answer
31 views

Joint density of normal random variables

Let $Z=X+Y$ where $X$~$N(\mu,\sigma^2)$ and $Y$~$N(0,1)$ are independents. Find the joint density of Z and X. This is the first time I see something like that, look what I did below: I know ...
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1answer
31 views

Numerical approximation to the Wasserstein metric?

Are there numerical methods for approximating/calculating the Wasserstein metric in particular cases? Suppose that $f$ and $g$ are two density functions with the same support. How can I calculate the ...
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1answer
30 views

Question about uncorrelatedness of random variables and distributions

I was wondering, if two random variables are dependent, does that mean that they must be correlated? does one imply on the other or vice versa? Also, if I know that a joint distribution of two ...
2
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1answer
68 views

Laplace transform of stopping times

I am nearly done with a question: Let $(B_t)$ be a Brownian motion on $\mathbb{R}$. For a fixed $x >0$, let $\tau$ be a stopping time defined by $$ \tau = \inf \{t \geq 0 : B_t \not \in (-x,x) ...
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0answers
40 views

Find the distribution of $Z=\frac{X_1+X_2}{X_1X_2}$, where $X_1$, $X_2$ follow normal distribution

Lets assume $X_1$, $X_2$ follow normal distribution. I am looking for the distribution of: $$Z = \frac{(X_1+X_2)}{X_1*X_2} $$ This is what I have thought so far: The distribution of the ...
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2answers
91 views

Bayesian statistics and Basis for continous functions

I was thinking about Bayesian statistics, and one thought bothered me: In Bayesian statistics, we assume that the pdf $p(x)$ can be described as: \begin{equation} p(x)=\int ...
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1answer
43 views

Expectation of the time difference between starting times in queueing theory

Consider 2 independent, parallel $M/M/1$ queues $Q_1, Q_2$ with identical arrival rate $\lambda$ (corresponding to an exponential random variable $A \sim \text{Exp}(\lambda)$) and service rate $\mu$ ...
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0answers
37 views

The probability that two or more successive tasks with Weibull distributed lengths have completed?

I have a set of independent tasks whose lifespan/time it takes to complete seems to fit nicely into a Weibull distribution. The tasks are to be handled one by one, sequentially. As far as I ...
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0answers
31 views

Suppose that $U$ is uniformly distributed on $[0,1]$. Given its p.d.f. and c.d.f, find $P(U<a|U<b)$ for $0<a<b<1$.

Suppose that $U$ is uniformly distributed on $[0,1]$. Find $P(U<a|U<b)$ for $0<a<b<1$. We know that the p.d.f. of $U[a;b]$ is $f_X(x)=\begin{cases}\frac{1}{b-a} & :\text{for }a ...
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2answers
45 views

Sufficient parameters for a probability distribution

We know that a Gaussian distribution can be constructed if its first two moments i.e. its mean and covariance are known. Is there any other standard distribution whose construction requires the ...
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0answers
45 views

Geometric Mean of Random Variables

I measure a series of $n$ objects [O_1, O_2, O_3, ..., O_n]. Because those measurements are quite hard to perform, I have quite a lot of measurement error and ...
2
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3answers
71 views

Prove that $f(x)=exp(-x-e^{-x})$ for $x\in \mathbb{R}$ is a p.d.f and find the c.d.f.

Prove that $f(x)=exp(-x-e^{-x})$ for $x\in \mathbb{R}$ is a probability density function and find the cumulative density function. I think that by proving that $f(x)$ is a pdf, it should be fairly ...
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2answers
1k views

Question on uniform distribution

Two people agree to meet each other on a particular day, between 5 and 6 PM, They arrive independently on a uniform time between 5 and 6 and wait for 15 mintues. What is the probability that they meet ...
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1answer
54 views

Coutinuous distribution in Probability

If suppose there is an interval $[a,b]$ then choosing a number from it is equal probable and a number can be any real number within the interval. Is it a case of continuous distribution ? How to ...
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1answer
29 views

Distribution combinations [closed]

How many ways can $25$ identical pencils be distributed between two people? Each pencil must be given out. a) Each person must have at least $5$ pencils. b) Each person must have at least $7$ ...
2
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1answer
60 views

Ratio of two beta random variables

I'm working on a problem for an hour and I wanted to get some hints. Suppose: $y_1, y_2, y_3, y_4 \sim Dir(\alpha_1, \alpha_2, \alpha_3, \alpha_4)$ what is the distribution of $\frac{y_1}{y_1 + ...
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0answers
83 views

Probability and sums of prime factors

Of the first $N$ natural numbers, we select two different numbers at random. We'll call the greater one $A$ and the lesser one $B$. What is the probability $P$ that the sum of $A$'s prime factors is ...
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1answer
115 views

Jar and Ball Probability Distribution

If I have 8 jars, each jar contains 5 unique ball types. However, I know that I have 20 unique ball types out there. So, I have balls labelled from B1, B2, B3, ...B20 to put into 5 jars. Let's say ...
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1answer
47 views

Inequality in differential entropy

In the book on "Network Information Theory" by El Gamal, there is a question to choose the correct relation ($\geq,\leq,=$) for the following: Let $X$ be a continuous random variable. Let $Y\sim ...
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2answers
43 views

Can linear combinations of any Gaussian random variables be independent?

Suppose that $X=[X_1\; X_2]^t$ is Gaussian vector. My question is whether $U=a_1X_1+a_2X_2$ and $V=b_1X_1+b_2X_2$, where $a_1b_2-a_2b_1\ne 0$, can be independent Gaussian random variables, if ...
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1answer
38 views

How can I find the PDF of this function of normal variables? Or what is the distribution of distances between two random points on a unit sphere?

How can I find the probability density function of the random variable $D = \frac{\sqrt{\left(x-\sqrt{x^2+y^2+z^2}\right)^2+y^2+z^2}}{\sqrt{x^2+y^2+z^2}}$ If x, y, and z are all independently ...
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1answer
71 views

Cebîsev Inequality or Central limit Theorem

I have the following problem. Let $(X)_{n>0}$ be a sequence of $4000$ independent random variables (discrete), all of them have a Bernoulli distribution with $p = 0.8$. Let $X$ be the sum of these ...
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0answers
50 views

What is the distribution of the dot product of two vectors of unit length with nonnegative elements

Let $X = A \cdot B$, where $A$ and $B$ are unit length vectors with $m$ elements, and no element of $A$ or $B$ is negative. What is the distribution of $X$? If it helps, we can assume that the ...
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2answers
53 views

Poisson distribution where each event can lead to different outcomes

I'm trying to tackle the following problem. Suppose that customers arriving at a bank follow a Poisson distribution with rate λ=5 every 15 minutes. For every customer, the probability that he ...
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1answer
96 views

Sampling distribution of sample trimmed (truncated) mean

It is elementary probability theory that the sample mean of an i.i.d. sample follows normal distribution, if the background distribution is normal. But what about the trimmed mean? Is there any result ...
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0answers
98 views

Request for a comparison between these 3 (advanced?) functional analysis books?

It would be helpful if I can get some comparison between these three books, T. Tao, An epsilon of room, I, Graduate Series in Mathematics 117, American Mathematical Society (2010). T. Tao Analysis ...
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0answers
22 views

Sufficiency of First Order Stochastic Dominance if MLRP not satisfied.

Consider a density $f(x;a)$ on support $x>0$ for which the CDF is not known in closed form. Also suppose the ratio of two such densities $g(x) = \frac{f(x;a)}{f(x;b)}$ does not satisfy the ...
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0answers
29 views

Transformation Theorem and Showing Independence of N(0,1) Random Variables

I am trying to solve the following problem: Show that the following procedure generates $N(0, 1)$-distributed random numbers: Pick two independent $U(0, 1)$-distributed numbers $U_1$ and $U_2$ ...
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52 views

Support lemma - Game theory

Let α be $a$ mixed strategy profile, $a_i ∈ supp(\alpha _i), a_i \notin B_i(\alpha _{−i}), a_i' ∈ B_i(\alpha _{−i})$ and $a_i'$ defined by $\alpha_i'(a_i)=0$, $\alpha_i'(a_i')=\alpha _i ...
3
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1answer
30 views

Transference of properties from marginals to joint density functions

Let $(X,Y)$ be an absolutely continuous random vector and denote by $f_{(X,Y)}(x,y)$ its joint density function and $f_X(x)$, resp. $f_Y(y)$ the marginal density functions. If $f_X$ and $f_Y$ are ...
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0answers
19 views

distance between two PMFs with zero elements

Consider two discrete random variables $N$ and $M$ with probability mass functions (PMFs) $f_N(n)$ and $f_M(m)$. Let $p_n = \Pr[N=n] $ and $q_m = \Pr[M=m]$. $p_n$ has non-zero values for all integer ...