# Tagged Questions

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

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 < ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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\...
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### 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 ...
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### Integration of ratio of cumulative normal distribution

I am trying to see whether there is a closed form solution to the following integral $$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{\mathbf{\Phi}(cz+d)}{\mathbf{\Phi}(cz+d')}e^{-z^2/2}dz$$ ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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....
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### 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 ...
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### 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 ...
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### 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-...
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 ...
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 $c_x + \mathrm dc_x$...