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

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3
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1answer
445 views

example of convergence in distribution but not in probability

While I was looking for an example of a sequence of random variables which converges in distribution, but doesn't converge in probability, I have read that it should be enough to consider a sequence ...
0
votes
1answer
24 views

Finding the Integral´s interval for a Probability Function

The probability density function of a given random variable is given by the graph below. How can I set up the integral in order to find P(X>0, P(X>3/4). I tried to set up the integral and this is my ...
0
votes
1answer
35 views

Use of language on wikipedia - what kind of distribution?

I have an interesting problem and was wondering whether anyone would be able to point me in the right direction. I am wondering whether the use of a word in the english language on Wikipedia is ...
0
votes
2answers
690 views

Bivariate Normal Distributions

Let X and Y have a bivariate normal distribution with parameters μ1 =3, μ2 = 1, σ1^2 = 16, σ2^2 = 25, and ρ = 3/5 . Determine the following probabilities: (a) P(3 < Y < 8). (b) P(3 < Y < ...
0
votes
0answers
30 views

Algebraic manipulation of probability distributions

Let $X$ and $Y$ be random vectors that have the same continuous distribution. If $A$ and $B$ are constant matrices and $AX$ and $BY$ have the same distribution does this imply that $A=B$? Are there ...
2
votes
1answer
37 views

Joint distribution of dependent Bernoulli Random variables

I have $N$ Bernoulli random variables $X_1, ..., X_{N}$ with known parameters $p_1, ..., p_{N}$. I want generate a joint distribution in which these random variables are not independent as I know that ...
0
votes
2answers
413 views

Determine the probability distribution of a ratio of two random variables?

Setting You are given two independent random variables $X_0,X_1$ with common exponential density $f(x) = \alpha e^{-\alpha x}$. Let $R = \frac{X_o}{X_1}$. Determine $\Pr[R > t]$ for $t > 0$. I ...
1
vote
1answer
38 views

Correspondence between AB-divergence and Kullback-Leibler divergence

I'm reading up on AB-divergence (alpha-beta-divergence) based mainly on the exposition given in Chichoki et al. (2011), "Generalized Alpha-Beta Divergences and Their Application to Robust Nonnegative ...
1
vote
0answers
24 views

How to generate multivariate random variables given probability distribution?

Suppose you can generate uniformly distributed random numbers $x_i\in[a,b]$. To shape probability distribution of these numbers as you like using inverse transform sampling. But what if you need to ...
1
vote
1answer
4k views

Understanding the difference between normal distribution and lognormal distribution

I'm having trouble understanding the difference between a normal distribution and lognormal distribution. Here's what I've done so far. Definitions of lognormal curves: "A continuous distribution in ...
1
vote
1answer
28 views

Why $P\left(Y>X\right)=\sum\limits_n P\left(X=n\right) \cdot P\left(Y\geq n+1\right)$

Joint Distribution Chapter of P exam book—Discrete case. Problem 41.7 (p exam book by M. Finan) Part of the question's solution was already posted here. Michal's answer was: \begin{align} P\left(X=n\...
0
votes
1answer
21 views

Percentage of Sum of 2 Continuous distributions

In a factory, there are Independents 2 pipe-cutter machines. The length of the pipes from the first machine is $X_{1}$. The length of the pipes from the second machine is $X_{2}$. I know that $X1\...
0
votes
1answer
19 views

Relationship of $L_1$ distance between CDFs and PDFs?

Let $F:(-\infty,\infty)\rightarrow[0,1]$ and $G:(-\infty,\infty)\rightarrow[0,1]$ two CDFs with PDFs $f$ and $g$, respectively. Is there a connection/inequality between: $$d_1 = \int_{-\infty}^{\...
1
vote
2answers
43 views

Rolling a k-sided fair die n times and not see all k numbers fall

If we throw a k-sided fair die n times, what is the probability that we will never get one of its k numbers? Further, what is the probability that two, three, or more of its numbers will never occur ...
0
votes
2answers
33 views

How to generate a random variable $r_i$ such that $\sum_{i=1}^n |\frac{r_i}{\sigma_i}|^2\leq\chi^2_{n,\alpha}$

How can I generate $r_i$ for $1 \leq i \leq n$, such that $\sum_{i=1}^n |\frac{r_i}{\sigma_i}|^2\leq\chi^2_{n,\alpha}$, where $\sigma_i^2$ is the variance of $r_i$ and, $\chi^2_{n,\alpha}$ is a chi-...
0
votes
0answers
17 views

Derive a probability distribution to transform scalars for weighted random sampling

Let's say I have a case with only two choices: $N_a$ = 20 $N_b$ = 10 I want to find a probability distribution that I can map these two values too, such that the probability of one variable being ...
0
votes
1answer
507 views

Joint probability of sum of two random variables and one of its terms

Let $X$ and $Y$ be two independent random variables (weibull distributed) and $Z=X+Y$. I am trying to find $\mathbb P\big(Z\geq z ~\cap~ X\leq x\big)$. I know that $$ \mathbb P(Z\geq z \cap X\leq x) = ...
1
vote
2answers
40 views

Derive the value of this probability analytically

Forgive me if this question is very basic but I genuinely tried to search around including this site and could not find anything that I could adapt to my understanding. ...
0
votes
1answer
33 views

Prove that: $\frac{\Sigma_{i=1}^{n}X_i}{\sqrt{n \log n}}\rightarrow N(0,\sigma^2)$ in distribution. [closed]

Let $X_1,X_2,X_3,...$be i.i.d with density $$f(x)=\begin{cases}|x|^{-3} \text{ if |x|>1}\\0\text{ otherwise}\end{cases}$$ Prove that: $\frac{\Sigma_{i=1}^{n}X_i}{\sqrt{n \log n}}\rightarrow N(0,\...
2
votes
1answer
18 views

Find the PMF for number of heads following the first tail on a four consecutive coin toss expriment

Suppose a fair coin is toss four times consecutively. Find the PMF for random variable of number of heads following the first tail. My take: Let random variable $X$ be the number of heads in this ...
0
votes
0answers
24 views

Integration of ratio of cumulative normal distribution

I am trying to see whether there is a closed form solution to the following integral $$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{\mathbf{\Phi}(cz+d)}{\mathbf{\Phi}(cz+d')}e^{-z^2/2}dz$$ ...
0
votes
1answer
38 views

What is the inverse of the integrated $\chi^2$ function?

I am implementing some preprocessing of variables in the context of a paper called A Neural Bayesian Estimator for Conditional Probability Densities. It states: 1.) Given a non-linear, a monotonous ...
0
votes
0answers
23 views

Average distance between remaining un-hit targets as targets are progressively hit

Consider targets arranged in a regularly spaced array across a near-infinite X-Y plane (area of plane is large relative to area of a target). Each target is initially a unit distance from adjacent ...
1
vote
2answers
61 views

Why does the normal distribution describe data collected in real life so well? [closed]

$$ P(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp \left( - \frac{(x-\mu)^2}{2\sigma^2} \right) $$ Is there any intuition behind choosing $e^{-x^2}$ instead of some other function?
1
vote
1answer
27 views

Compute expected received balls from boxes

I have 6 boxes: $A,B,A',B',C \text{ and } D$. The box $A$ has $n_1$ red balls that are numbered from $1, \cdots, n_1$. The box $B$ has $n_2$ green balls that are numbered from $1, \cdots, n_2$. Make a ...
0
votes
1answer
40 views

When are conditional expectations equal?

As a sort of a follow-up and a generalization from a previous question, suppose that we have two independent, identically distributed random variables $X, Y$ and a third random variable $W$. Is it ...
-1
votes
0answers
57 views

Simple Markov property on stopping times [closed]

Suppose $(S_n)_{n\geq1}$ is a Markov chain on the two dimensional lattice of the integers. Then define the stopping time $\tau_A'=\inf\{n\geq1:S_n\notin A'\}$ and consider the following for $A\subset ...
4
votes
2answers
130 views

What is the expected value of $\min\{|X|,|Y|\}/\max\{|X|,|Y|\}$ assuming $X$ and $Y$ are independent?

So I need to compute $$E\left[\frac{\min\{|X|,|Y|\}}{\max\{|X|,|Y|\}}\right]$$ given $X,Y \sim$ Normal$(0,1)$ and independent. What I am having trouble seeing is whether $\min\{|X|,|Y|\}$ and $\...
0
votes
1answer
176 views

Is Lottery probability really the same for all combos?

http://justwebware.com/uklotto/uklotto.html Test run quickpick Test run 1,2,3,4,5,6 Test run (single digit,teens,twenties,twenties,thirties,forties) 1000 times or more each cycle for as many ...
0
votes
2answers
42 views

Find $a$ so that $a(e^{-2x}-e^{-3x})$ is a probability density function. [closed]

Let $f(x) = a(e^{-2x}-e^{-3x}),$ for $x\geq 0$, and $f(x) = 0$ elsewhere. (a) Find $a$ so that $f(x)$ is a probability density function. (b) What is $P(X\leq 1)$? Image. If it is possible, ...
0
votes
2answers
38 views

given a graph of density function, what can we conclude about expected value

given the following graph (the density function), what can we conclude about the expected value? I got stuck a little bit with that question and I would appreciate your help! I found out that C must ...
0
votes
3answers
39 views

given the following CDF, find the expected value

I got stuck at the middle of the question. would appreciate your help. first of all, given the CDF as follows, I had to find parameters $a$ and $b$ such that the CDF is a function of a continuous ...
1
vote
1answer
33 views

Conditional Probability for a Poisson Distribution: X = 1 | X $\geq$ 1

Suppose X has a Poisson distribution with a standard deviation of 4. What is the conditional probability that X is exactly 1 given that X $\geq$ 1? I know that for this problem $\lambda$ is 16 ...
1
vote
1answer
1k views

Uniformly Most Powerful Test and Rejection Region of Poisson Distribution

Let $X_1, \dots,X_n$ be a random sample from a Poisson$(\lambda)$ distribution where $\lambda > 0$. (1) Find the Uniformly Most Powerful (UMP) level $\alpha$ test for the following set of ...
3
votes
2answers
57 views

Determine whether a random binary sequence was generated by human or natural process

Given a binary sequence, how can I calculate the quality of the randomness? Following the discovery that Humans cannot consciously generate random numbers sequences, I came across an interesting ...
0
votes
0answers
26 views

Transformation of Laplace distribution that preserves conditional distribution

Suppose, we have a $X\sim {\rm Lap}(0,a)$ with Laplace distribution with parameter a. That is \begin{align} f_X(x)= \frac{1}{2 a}e^{-|x|/a} \end{align} Now suppose we have two independent Laplace r....
2
votes
1answer
122 views

Calculation of $\ln\left( \frac{S_{1}(t)}{S_{2}(t)}\right)$ where $S$ are stocks

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq \tau}$ is a Filtration of an incomplete finance market with stocks $S_j(t)$ for $...
5
votes
5answers
19k views

Combining two probability distributions

I have a variable $X$. In a measurement $A$, $X$ follows the normal distribution $N_1$ with mean $m_1$ and standard deviation $\sigma_1$. In a similar measurement $B$, $X$ follows another normal ...
3
votes
4answers
75 views

Conditional expectation of independent variables

Claim. Let $Z_1, Z_2$ be two independent and identically distributed random variables. Then we have: $$ \mathbb E[Z_1|Z_1+Z_2] =\frac{Z_1+Z_2}{2}. $$ Proof. To see this, I have proceeded as follows. ...
0
votes
0answers
14 views

What is the spatial distribution of waiting time?

Suppose rimu trees are spread in the territory of some area according to a time homogeneous Poisson process. Suppose a rimu tree is at point $x$, what is the distribution function of the distance to ...
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vote
0answers
10 views

Finding $P(S<0)$ with standard Normal Cumulative Distribution function

I know I'm supposed to use the the Standard Normal Cumulative distribution function. But I can't seem to get everything I need. Let $X$ be a random variable with $P(X=-1)=P(X=0)=0.25$ and $P(X=1)=0.5$...
3
votes
1answer
1k views

Bus arrival poisson paradox

I have a question about the waiting time paradox for poisson processes(in this case in terms of bus arrivals). Suppose I know that buses arrive with poisson distribution(lambda). I arrive at fixed ...
2
votes
2answers
38 views

Are a uniformly random polynomial's roots are distributed uniformly in the field?

Assume we have a $\mathbb{F}_p$, where $p$ is a large prime (e.g. 128-bit value). We define all polynomials over the field, and pick a polynomial,$P(x)$, of degree $d$, where the polynomials' ...
1
vote
0answers
58 views

Making sense out of the method for finding posterior distributions.

I have been recently studying Bayesian statistics and more precisely the problem of finding posterior distributions. I am able to understand the my textbook's problems, but I realize that I understand ...
0
votes
1answer
22 views

How to make the probability that two random sets have any intersection close to zero (negligible)?

This question is related to one of my question: Probability that two random sets have at least one element in common Assume we have a field $\mathbb{F}_p$, where $p$ is a large prime number i.e. $...
0
votes
1answer
28 views

Expected number of same numbered balls in a box

I have two boxes: A,B. The boxes A contains $n_1$ red balls which numbered from $(1, \cdots, n_1)$. The box B includes $n_2$ green balls which also numbered from $(1, \cdots, n_2)$. Throw balls from ...
0
votes
0answers
28 views

The discrete Laplacian

I am working on the $d$-dimensional integer lattice. Let $S$ be a random walk with increment distribution $p$. Given the distribution $p$ we can define the discrete Laplacian just as in Wikipedia is ...
1
vote
0answers
41 views

Transformations of two Laplace distributions resulting in a Laplace distribution

Suppose we have two independent identical random variables $X_1$ and $X_2$ with Laplace distribution \begin{align} f_X(x)=\frac{1}{2b}e^{-\frac{|x|}{b}} \end{align} I am looking for a non-...
3
votes
2answers
2k views

Assumption of a Random error term in a regression

In one of my recent statistics courses, our teacher introduced the linear regression model. The typical $y=\alpha + \beta X + \epsilon$, where $\epsilon$ is a "random" error term. The teacher then ...
2
votes
1answer
19 views

An issue with the distribution function

I am reading a book about Boltzmann equation, here is a quotation: For a gas of $N$ particles, the number of particles having velocities in the $x$ direction between $c_x$ and $c_x + \mathrm dc_x$...