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

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2
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2answers
49 views

Are there two different notions of “conditional probability”?

This question comes from reading the discussion here. (1) If one is given a "probability measure" $P : F \rightarrow [0,1]$ mapping a Borel $\sigma$-algebra $F$ to $[0,1]$ then for two ``random ...
4
votes
1answer
107 views

Show $\int_{-\infty}^{\infty}\,f(u,t)dG(u)$ is a ch.f. where $G$ is a d.f. ; $f(u,\cdot)$ is a ch.f. and $f(\cdot,t)$ is continuous.

Show $$\int_{-\infty}^{\infty}\,f(u,t)dG(u)$$ is a ch.f. where $G$ is a d.f. ; and $f(u,\cdot)$ is a ch.f. for each $u$ and $f(\cdot,t)$is continuous for each $t$. Note that ch.f. means ...
2
votes
2answers
85 views

Does the distribution of a process on $\mathbb{R}^{[0,\infty)}$ uniquely define it?

Question: Can I have two different stochastic processes $(A_t)_{t \in [0, \infty)}$, $(B_t)_{t \in [0, \infty)}$ having the same distribution on $\mathbb{R}^{[0, \infty)}$ differ in some ways? ...
2
votes
0answers
25 views

The uniqueness of solution of an equation that involves CDFs

I have two monotone CDFs $F(x)$ and $G(x)$. The functions are symmetric in a sense that $F(x)=1-G(1-x)$, $f(x)=g(1-x)$. I am trying to show that equation $xF(2x)+(1-x)G(2x)=1/n$, $n\geq2$ has a unique ...
0
votes
1answer
17 views

Cumulant-Legendre

I have a short question: So suppose $b=\text{ess sup} X<\infty$, where $X$ is a random variable on $\mathbb{R}$. Now take $\Lambda (u)=\ln \mathbb{E}[e^{uX}]$, the cumulant, and ...
1
vote
1answer
33 views

Comparing Percentiles of 2 Samples Drawn from the Same Distribution

Suppose I have two sets of numbers: $A=\{a_1,a_2,...a_{N_1}\}$ and $B=\{b_1,b_2,...b_{N_2}\}$ with $N_1<N_2$. WLOG assume that $a_i<a_j$ for all $i<j$ and similarly for $b_i$ and $b_j$. ...
0
votes
1answer
25 views

Let X and Z form a random sample from a poisson dist.If Y=min( X,Z), what is P(Y=1)??

Let X and Z form a random sample of poisson distribution and define Y=min( X and Z) What is P(Y=1)?? I think Y is minimum of two. If X=1, then Z can be any number except 0 If Z=1, then X can be ...
0
votes
1answer
135 views

Lifetime of light bulbs is modeled as a Poisson Process - using excel

I have a homework question that I can't seem to figure out. Any help is appreciated! The lifetime of light bulbs (in days) is modeled as a Poisson Process with expected lifetime of beta = 200 days. A ...
1
vote
1answer
37 views

Using PMF and CDF to calculate probability

Given the following CDF what is $$P(T > 3)$$ and according to my answer key it's 1-1/2 = 1/2. Can someone explain to me why it is 1-F(3), and would subtracting F(3) be subtracting 4 as well? ...
1
vote
1answer
29 views

Given Nd6, what is the probability that the two highest are minimum 4?

So, my statistics knowledge is rather poor, so I would welcome a formula explanation to the question: given Nd6 (6-sided dice) what is the probability that the two highest numbers are at least a 4? ...
2
votes
0answers
96 views

Application of Slutsky's Theorem to the Convergence of Sum of R.V.

Let $X_1, X_2,…, X_n$ be i.i.d. $U(−\theta,\theta)$. Show that $Z_n \to N(0,\sqrt{5/9})$ in distribution, where $Z_n ...
0
votes
1answer
36 views

How to show that $E(X^k)=npE((Y + 1)^{k-1})$ where $X\sim\mathrm{Bin}(n,p)$ and $Y \sim \mathrm{Bin}(n-1,p)$.

Show that $$E(X^k)=npE((Y + 1)^{k-1})$$ where $X\sim\mathrm{Bin}(n,p)$ and $Y \sim\mathrm{Bin}(n-1,p)$. I am looking for suggestions on where to start? Or any resources someone may have. I am not ...
0
votes
2answers
22 views

Probability of X given the sum

I am given that $X \sim P(\lambda)$, $Y \sim P(\gamma)$, and told to calculate the distribution of $P(X | X+Y = n)$ I proceed as follows $$ \begin{equation} \begin{split} P(X=i|X+Y=n) &= ...
1
vote
1answer
84 views

Exponential distribution question!

Suppose that the time between calls from your best friend has an exponential distribution with a mean time of $3$ days. (a) If you just received a call from her, what is the probability that you will ...
2
votes
2answers
36 views

Essential supremum via cumulant

Let $p(t)=\log \mathbb{E}[\exp (tX)]$ for $X$ real valued random variable. Now it holds (assuming that $p$ is smooth and finite on $\mathbb{R}$) that $p'(\infty)=\text{ess}\sup X$. How can I prove ...
2
votes
0answers
34 views

convolution of two probability density functions

Please no one call me dumb - I am not a mathematician and haven't done proper math for the last ten years. But I have a problem at work where I need to perform a convolution of two probability density ...
2
votes
1answer
29 views

Finding this Probability Density Function

I would much appreciate if you help me out with this problem Let $X \sim Unif(0,1)$ Find the density of $Y = -\lambda^{-1} \log(1-X)$ with $\lambda > 0$ Then calculate $P(Y>t+s|Y>t)$ for ...
1
vote
0answers
39 views

Number of same degree vertex pairs between two random graphs

I am considering the random graphs generated by the Erdős-Rényi model for this question. Random Graphs as Models of Networks by Newman is a reference on this topic. A random graph $\Gamma_{n,p}$ has ...
1
vote
1answer
22 views

If $E(|X|)<\infty$, how do we show that it can be expressed as below

$F(x)$ is the distribution function of $\mathbb X$, and $f(x)$ is the derivation of $F(x)$, Prove that $\int_{0}^{\infty}(1-F(x))dx-\int_{-\infty}^{0}F(x)=E(X)$. Note that ...
0
votes
2answers
309 views

Why birthday distribution is not uniform. [closed]

I was reading about birthday problem and I found a statement that real-life birthday distributions are not uniform since not all dates are equally likely (last line ...
1
vote
1answer
52 views

Statistics - Exponential distribution

There are $n$ machines. Each has durability given by exponential distribution with $EX = 10$. If a dead machine is replaced with new one immediately, find minimal $n$ so we can say with $P = 0.99$ ...
1
vote
1answer
66 views

Is this a misuse of the term “probability space”?

Let me first state the definitions as I am using them. Do correct me if I am wrong here! A "probability space" is a triple $(\Omega, F \subseteq 2^{\Omega}, \mu : F \rightarrow [0,1])$. The ...
0
votes
0answers
40 views

Family of parameterized distribution functions with separable inverse

I am looking for parameterized families of distribution functions that have a separable inverse. For example, the inverse of the exponential distribution is $H^{-1}(p;\gamma)=\frac{1}{\gamma}ln(1-p)$ ...
0
votes
2answers
33 views

Find the density of their average

If $f_{X,Y,Z}(x,y,z)=e^{-(x+y+z)}I_{[0,\infty]}(x)I_{[0,\infty]}(y)I_{[0,\infty]}(z)$ find the density of their average $\frac{X+Y+Z}{3}$ I'm a little lost on how to solve this exercise, ...
2
votes
1answer
88 views

the probability density function (PDF) of concatenation of two Gaussian variables

Gaussian variable $x$ follows from $N(u_x,\sigma_x^2)$ and $y$ follows from $N(u_y,\sigma_y^2)$. Assume we have the vector $\bf{z}=[x,y]^T\in R^2$, then it seems that no matter whether $x$ and $y$ are ...
-1
votes
1answer
46 views

Statistics - normal distribution problem

Two random variables $X$ and $Y$ are i.i.d. normal$(\mu, \sigma^2)$. If $P(X > 3) = 0.8413$ find $P((X+Y)/2 > 3)$. The result must be exact number, so normal distribution parameters are ...
2
votes
1answer
34 views

Inequality for the derivative of a density of a random variable convolved with a normal r.v.

I have a question about the following proof. The statement is: Let $X$ be a random variable and $Z_\tau \sim N(0,\tau)$ be an independent random variable. Then $Y_\tau := X + Z_\tau$ has a ...
2
votes
1answer
56 views

Non-Linear System. Find the conditional expectation.

I've had my test for this course and I think I failed it again. The hardest part for me is findig the correct distributions. This is a test exercise I couldn't figure out or at least, I probably ...
0
votes
0answers
24 views

(Almost) Gaussian distribution

I need to find a distribution for random variable $\boldsymbol \epsilon$ goverened by parameter $\alpha >0$, such that: for any given $\boldsymbol \pi \in \mathbb [0, 1]^M: (\boldsymbol ...
3
votes
0answers
81 views

Random matrices, eigenvalue distribution.

I just investigated randn(1024) + 1i*randn(1024), a 1024x1024 complex valued matrix with elements both real part and imaginary part drawn from $\mathcal{N}(\mu = 0, \sigma = 1)$. I was a bit surprised ...
0
votes
3answers
252 views

Cumulative distribution function of Cauchy distribution

Let X be a Cauchy distribution with X~Cauchy(1) (so a=1). Prove that Y=1/X has the same cumulative distrubtion as X. Now I've tried taking F_X(x) for a=1 combined with the identity ...
1
vote
1answer
93 views

How to interpret a p-value that's significant from Fisher's Exact test

Given a binomial distribution with p=.03, n=902, the $.025$ and $.975$ quantiles are $17$ and $38$ respectively. I interpret ...
1
vote
2answers
34 views

Is my probability reasoning here correct?

Sheldon Ross theoretical exercice A jar contains $n$ chips. Suppose that a boy successively draws chips from the jar, each time replacing the one drawn before drawing another. The process continues ...
0
votes
0answers
72 views

Find the Lipschitz constant of a multi-variate Gaussian density function

I would like to find the Lipschitz constant of a multi-variate Gaussian density function: $$f_{\mathbf x}(x_1,\ldots,x_k) = \frac{1}{\sqrt{(2\pi)^{k}|\boldsymbol\Sigma|}} ...
1
vote
0answers
37 views

A Question about the Kurtosis

Problem: Show that if a binomial distribution with $n = 100$ is symmetric, its coefficient of kurtosis is 2.9. Answer: First, I am interpreting the term symmetric to mean that $p = q = \frac{1}{2}$. ...
0
votes
2answers
108 views

The distribution of the product of Gaussian variable and Rademacher variable.

I have two independent variables: $X$ follows from standard Gaussian distribution $N(0,\sigma^2)$; $Y$ follows from Rademacher distribution, i.e., $Y$ can be either $-1$ or $1$ with the same ...
0
votes
1answer
88 views

Assume a die is rolled repeatedly. Find the markov matrix $P$ for the random variable of the time until the next $6$.

Assume a die is rolled repeatedly. Find the markov/transition matrix $P$ for the random variable $X_r$ = the time until the next six at time $r$. My solution was: For $i,j \geq 0$, $P$ is given ...
2
votes
1answer
376 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
0answers
9 views

generate binomial binary data

How does one generate correlated binomial data when one is given marginal probabilities of each and also the correlation coefficient. The following code in SAS for example works best when we want ...
-1
votes
1answer
147 views

Prove that the increments of the Brownian motion are normally distributed

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
0
votes
0answers
18 views

Optimal decision for sampling a distribution.

I was wondering which probability distribution is best sampled with $\pm\alpha^n, n\in\{1,2,\cdots\}$ for various values of alpha. Sampling means to pick the one which is closest, store the sign and ...
11
votes
1answer
213 views

About the “Cantor volume” of the $n$-dimensional unit ball

A simple derivation for the Lebesgue measure of the euclidean unit ball in $\mathbb{R}^n$ follows from computing $$ \int_{\mathbb{R}^n}e^{-\|x\|^2}\,dx $$ in two different ways. See, for instance, ...
1
vote
0answers
43 views

Closeness in distribution implies closeness in statistics?

I am aware that convergence in distribution does not necessarily imply convergence in the mean. I browsed through some examples of statistical distances here ...
0
votes
0answers
34 views

Copula vs Exprimental Copula

I have read some texts about finding/approximating copulas for a given sample based on known (famous) copulas. My question is: when we have the experimental CDF of (X, Y), why we should try to find a ...
1
vote
1answer
58 views

Determining Probability Generating Function from Probability Mass Function and Convergence

I am trying to solve the following: Suppose $X_{nk}, k=1,2,\ldots,n, n≥ 2$ are i.i.d. random variables $$P(X_{nk}=0)=1-\frac{1}{n}-\frac{1}{n^2}\\P(X_{nk}=1)=\frac{1}{n}\\P(X_{nk}=2)=\frac{1}{n^2}$$ ...
2
votes
1answer
88 views

Application Problem: Conditioning Poisson Process

I am trying to solve the following application problem: There are $n$ components with independent lifetimes which are such that component $i$ functions for an exponential time with rate $\lambda_i$. ...
0
votes
2answers
63 views

probability density functions and cumulative distribution function

Suppose $X$ is an absolutely continuous real random variable, (that is, there exist a non-negative integrable function $f$, such that $\int_\mathbb{R} f=1$ and for every interval $I\subseteq ...
0
votes
1answer
55 views

When do I use Law of total variance?

For example, at the beginning of doing this problem (http://math.illinoisstate.edu/krzysio/3-6-10-KO-Exercise.pdf), I was thinking of using $\text{Var}(\text{Total loss}) = \text{Var}(N \cdot L)$, ...
2
votes
0answers
115 views

Is the product of two sub-Gaussian random variables a sub-Gaussian random variable?

If not, is there any way to make it hold? Note: the random variable $x$ is called $σ^2$-sub-Gaussian if $E[e^{tx}]≤e^{t^2σ^2/2}$.
2
votes
1answer
137 views

What distribution has $X^n$ if $X$ is normal distributed?

Let $X$ be a random variable with mean $0$ and variance $\sigma ^2$, i.e. $X \sim \mathcal{N}(0, \sigma ^2)$.What is the distribution of $Y= X^n$, $n \in \mathbb {N}.$ ? I know what distributribution ...