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

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2answers
49 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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0answers
23 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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0answers
20 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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1answer
24 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 ...
2
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0answers
18 views

Law of a supremum of random variables

Let $(B_t)_{t\geq 0}$ the standard brownian motion (with $B_0=0$), $p$ be a real number greater than $1$ and $q$ its conjugate number. Prove that $X_p=\sup _{t\geq 0}(|B_t|-t^{p/2})$ is a.s. strictly ...
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0answers
13 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?
1
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1answer
32 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 ...
0
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0answers
47 views

Mixture of Discrete Binomial Distributions

Let $B\left(p,N\right)$ be a Binomial distribution with parameters $p$ and $N$. We define a Mixture of Discrete Binomial Distributions by $\left\{ \left(B\left(p_{i},N\right),\alpha_{i}\right)\right\} ...
2
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1answer
23 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
20 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 ...
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0answers
32 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 ...
1
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1answer
31 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 $\mu$ ...
2
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0answers
22 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 ...
1
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0answers
24 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 ...
1
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2answers
30 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 ...
0
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0answers
17 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 ...
1
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0answers
31 views

Product of Wishart and inverse Wishart distributions

Let $$ X \sim \mathcal{W}_{q} (n, \Sigma) \; \; n > q$$ and $$ Y \sim \mathcal{W}^{-1}_{q} (n, \Sigma^{-1}) \; \; n>q$$ Where $\mathcal{W}$ denotes the Wishart distribution and ...
0
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1answer
51 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 ...
2
votes
3answers
56 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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1answer
45 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 ...
2
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2answers
35 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 ...
1
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1answer
40 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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1answer
28 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 ...
0
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0answers
23 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 ...
1
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2answers
31 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 ...
3
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1answer
43 views

Joint distribution from two gamma distributed random variables

Let $X$ and $Y$ be two independent random variables with distributions $\Gamma(a,c)$ and $\Gamma(b,c)$ respectively. $a,b,c>0$. Set $S=X+Y$ and $T=\frac{X}{X+Y}$. What is the law of the couple ...
0
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0answers
14 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 ...
3
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1answer
18 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 ...
1
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1answer
30 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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0answers
8 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 ...
1
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1answer
32 views

Likelihood of a roulette player, using a proposed system, winning big before going bankrupt

I recently came across this deceptively simple question: how likely is a roulette player, who always bets $10\%$ of his current bankroll, to increase his holdings from $\$1,000$ to $\$10,000$ before ...
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0answers
26 views

Why use rejection sampling in Monte Carlo simulations?

I've noticed that a lot of physics Monte Carlo simulations make extensive use of rejection sampling, rather than inverse transform sampling. In my research, I'm sampling random energy transfers from ...
2
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0answers
63 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 ...
1
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2answers
21 views

Proving that the maximum values of these different, Normal distribution, curves are different.

In a question, one variable X is Normally distributed with mean=100, variance=25 and Y is Normally distributed with mean=110, variance=36. The question asks to sketch the p.d.f of each on the same ...
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0answers
20 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 $U_2$ ...
1
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1answer
34 views

Joint Distribution, find $P(X-Y>z)$

Suppose $X$ and $Y$ have joint density $f(x,y)=2$ for $0<y<x<1$. I want to find this expression: $P(X-Y>z)$. So: \begin{align} P(X-Y>z) = P(X \geq z, Y \leq X-z) &= ...
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2answers
32 views

$X$ and $Y$ have Joint density, what is $c$?

Suppose $X$ and $Y$ have joint density $f(x,y)=c(x+y)$ for $0<x$, $y<1$. Now my question is, what is $c$? I tried to solve, which is reasonable in my opinion. But it don't seem to work: ...
2
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0answers
34 views

In the space of probability distributions, the set of discrete distributions is dense?

Is the following true: In the space of probability distributions, the set of discrete distributions is dense wrt the Levy metric. Can some one point me to any ...
3
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0answers
22 views

Joint distribution of degrees of Erdös Renyi random graph

The marginal degree distribution of any particular vertex is $$Bin(n-1,p)$$ in an Erdös Renyi random graph G(n,p). Denoting the degrees of the n vertices as d1,d2,...,dn, can you please let me know ...
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1answer
35 views

Solve for $K$, numerical method

I have been given the following function: $f_Y(y; \lambda, \mu) = K\exp[λ\cos(y−μ)]$ if $0≤y<2\pi$; $0$ otherwise $λ = 1$ and $μ = \frac \pi 2$ and I must solve for $K$. I have been told ...
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0answers
23 views

Joint transformation of a gamma distribution

I have a question regarding the transformation of a gamma distribution. I think I solved the problem, but I am not sure whether it is correct. Let $X$ and $Y$ be independent and Gamma distributed ...
3
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1answer
39 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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1answer
32 views

Density Function $Y=X(2-X)$

Suppose $X$ has density function $\frac{x}{2}$ for $0<x<2$ and $0$ otherwise. Now I am wondering what the density function of $Y=X(2-X)$ will be. I tried to compute $P(Y \geq ...
1
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0answers
29 views

Showing $e^{2B_t - 2t}$ is a martingale

A process $M_t$ is a martingale, if $1(a): \space \space \mathbb{E}[M_t | \mathcal{F}_s] = M_s$ for all $s \leq t$ Or equivalently, $1(b): \space \space \mathbb{E}[M_{t+s}| \mathcal{F}_t] = M_t$ ...
0
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1answer
18 views

Coefficient of variation of an hyperexponential

This question is on my mind for days and I haven't find the answer. Can someone help me? Suppose there exists a super awesome hyperexponential random variable $X$ with $k$ exponential variables with ...
0
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1answer
37 views

Using the symmetry assumption in this familiar probability problem

I'm revising some probability and have run into this old problem (context: Monte Carlo tests): Suppose there are random variables $t_0,t_1,\ldots,t_B$ that are independently and identically ...
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2answers
50 views

Double exponential function

Suppose $X$ has an exponential distribution with parameter $1$ and $Y=ln(X)$. The distribution of $X$ will be $f(x)=e^{-x}$. Now I want to find the distribution of $Y$. So say: ...
1
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1answer
38 views

Distribution function?

Let $F(x) = e^{-1/x}$ for $x>0$ and $F(x)=0$ for $x\leq0$. Now I am investigating if $F$ is a distribution function. Say: \begin{align} \int\limits_0^\infty e^{-1/x} \, dx = \left[ ...
3
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1answer
40 views

Distance between a Poisson and Normal distribution.

Let $X_a$ be a random variable Poisson distributed with intensity $a$. That is $$\mathbb{P}(X_a=k)= e^{-a} a^k / (k!)$$ for any $k\in \mathbb{N}$. Let $$Y_a=(X-a)/\sqrt{a}$$ the normalization of ...
1
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0answers
35 views

Compute (a) P(X = 2|Y = 3) and (b) P(Y = 3|X = 3) for the following joint distribution

I have some trouble understanding a question from my testbook. The question is as follows: Compute (a) P(X = 2|Y = 3) and (b) P(Y = 3|X = 3) for the following joint distribution: Y X=1 2 3 1 ...