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

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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 ...
2
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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 ...
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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 ...
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1answer
66 views

Can you suggest a method to generate random sample from following PDF?

‎Let‎ ${‎‎\bf{\alpha}}=(\alpha_1, \alpha_2, \ldots, \alpha_m)$ ‎and ‎‎$‎‎\textbf{b}=(b_1, b_2, \ldots, b_m, b_{m+1}).$ I intend ‎to ‎generate ‎sample ‎from PDF $$ g(\alpha_1, \alpha_2, \ldots, \...
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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\...
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3answers
75 views

Why is $P(X\in[a,b])=P(X\in[a,b))=P(X\in(a,b])=P(X\in(a,b))$

I saw, for any continuous random variable $X$, $P(X\in[a,b])=P(X\in[a,b))=P(X\in(a,b])=P(X\in(a,b))$, where $a,b\in\mathbb{R}$, in my textbook. I don't quite understand why the openness/closeness of ...
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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}^{\...
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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-...
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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 ...
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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\...
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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 ...
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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. ...
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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 $$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{\mathbf{\Phi}(cz+d)}{\mathbf{\Phi}(cz+d')}e^{-z^2/2}dz$$ ...
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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,\...
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1answer
17 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 ...
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1answer
40 views

If $\mathbb{E}[f(X)]=\mathbb{E}[f(Y)]\,\,\,\forall$ continuous $f$, then $X, Y$ have same distribution

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $X, Y:\Omega \to [0,1]$ be random variables. Prove that if $$\mathbb{E}[f(X)]=\mathbb{E}[f(Y)] \text{ for all continuous }f:[0,1]\to\...
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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 ...
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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 ...
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0answers
95 views

convergence in distribution in Banach spaces

We let $\Omega$ be a compact metric space and consider $C(\Omega)$ to be the space of all continuous functions on $\Omega$. The dual space of $C(\Omega)$ can be seen as the set of all signed borel ...
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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, ...
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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....
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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. ...
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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' ...
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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. $...
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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 ...
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0answers
31 views

Probability of collision of some family of hash functions

Given $x$ and $y$ in $\mathbb{R}$, and let $\mathcal{H} = \{ h \mid \mathbb{R} \to \mathbb{N} \}$ be a family of hash functions where $ h(x) = \left\lfloor x + \sum^C_{i=1} U_i \right\rfloor$ for some ...
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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 ...
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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 ...
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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-...
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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 $c_x + \mathrm dc_x$...
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0answers
23 views

Is a particular sample a 'random sample'?

Say I have a high dimensional Bernoulli distribution $X$ (defined by $p_1,p_2,...,p_n$, independent marginals). I realized I cannot use the chi-squared test to check if a given set of $m$ samples from ...
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1answer
27 views

Prove $X$ and $Y$ are not independent

Let $X$ and $Y$ be two random variables. Their joint probability density function is $$f: (x, y) \mapsto C(y^2 - x^2)e^{-y} \mathbf{1}_A(x, y)$$ where $C \in \mathbb{R}$, $A = \{(x,y) \in \mathbb{R},...
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1answer
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5 independent traffic lights, how many is car expected to pass without getting stopped

$\newcommand{\E}{\mathbb{E}}$ I can't wrap my mind around this one. I keep thinking it is geometric probability problem, but can't get correct solution (which is $\E(X) = 0.6598)$. Problem : ...
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24 views

Shifting the mean of a composite function of deterministic and random variables

For a project I am involved in relating to communication, I have the following model: $L = f(r).X$ where $X$ is a lognormal random variable with zero mean in the logarithmic scale and standard ...
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1answer
20 views

Distribution of the minimum

I have the following problem, given a random variable $X$ with density $$f(x)=2x\text{ for }x\in(0,1)$$ and a r.s.s. $X_1, X_2, X_3$. I have to calculate the probability that $X_{(1)}=\min\{X_1,X_2,...
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1answer
118 views

Distributional equality

Let $(W_t)_{t\geq0}$ be a standard Brownian motion. I have to show that the following equality holds in distribution. Does someone has a good hint to show this? $\sup_{t \geq 0}( |W_t| -t) = \sup_{t \...
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0answers
57 views

Simple Markov property on stopping times [on hold]

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 ...
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1answer
29 views

Transformation technique to find PDF

Consider two random variables with the following joint PDF: $$ f_{X,Y}(x,y) = \begin{cases} 2, & x > 0, y > 0, x + y < 1 \\ 0, & \text{otherwise} \end{cases} $$ I need to find ...
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3answers
43 views

Generating samples from a Beta(2,2) distribution

I'm looking for a convenient way to generate $\text{Beta}(2,2)$ random variables, using independent $\text{Uniform}(0,1)$ random variables and elementary functions. I'd prefer to avoid rejection or ...
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2answers
25 views

Conditional Probability: Birth rank of children in randomly chosen families

(BH 4.7) A certain small town, whose population consists of 100 families, has 30 families with 1 child, 50 families with 2 children, and 20 families with 3 children. The birth rank of one of these ...
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1answer
58 views

Probability for a leading candidate to eventually win

Two candidates contest a close election. Each of the $n$ voters votes independently with probability $\frac12$ each way. Fix $\alpha \in (0,1)$. Show that, for large $n$, the probability that the ...
3
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1answer
31 views

Bell numbers and the Moments of expected number of fixed points

Let $X_N$ be the random variable corresponding to the number of fixed points (1-cycles) in a permutation chosen uniformly at random from $S_N$. Then, the $m^{\text{th}}$ moment, when $m < N$, is ...
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1answer
31 views

Probability distribution of order statistics

Let $X_1$, $X_2$ and $X_3$ be independent random variable with continuous distribution $$f(x;\theta)=\frac{1}{\theta}I_{(0,\theta]}(x), \ \theta \gt 0$$ I need to find distribution of $Z=\frac{X_{(...
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1answer
35 views

I am trying to find answer to this bivariate normal problem. Can anyone help. [closed]

The distribution of the heights of husband-wife pairs in a particular population is modelled by a bivariate normal distribution. The mean height of the women is 165cm and the mean height of the men is ...