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
44 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? ...
0
votes
2answers
18 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
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1answer
32 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 ...
0
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1answer
25 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 ...
2
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0answers
26 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 ...
0
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0answers
26 views

distance distribution in Poisson point process

Consider a homogeneous Poisson point process in 2D space with density $\lambda$ per unit area. Let $\mathcal{B}(o,R)$ denote a disk centered at origin with radius $R$. Let $n$ be the number of points ...
1
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1answer
22 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
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1answer
57 views

Why does symmetry happen in reset-based random walks?

Studying the basic concepts about random walks / brownian motion, and based on the idea of a Möbius-based walk in Wolfram's website, I wanted to try my own version of it in Python to compare it with ...
1
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0answers
24 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
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2answers
24 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 ...
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1answer
21 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 ...
-1
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2answers
60 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
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1answer
42 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 $P = 0.99$ ...
0
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0answers
22 views

Find marginal distribution (Integral Solution)

I have derived bivariate exponential distribution in term of polar coordinate system. Now I need to derive marginal distribution of $f(\theta)$ from joint $f(r,\theta)$ for this we have to eliminate ...
0
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2answers
32 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, ...
0
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0answers
15 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 $H^{-1}(p;\gamma)=\frac{1}{\gamma}ln(1-p)$ ...
-1
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1answer
40 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
31 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
36 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
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1answer
52 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 ...
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0answers
17 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
26 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 ...
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2answers
31 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
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0answers
35 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
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0answers
25 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
29 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 ...
1
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1answer
27 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
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0answers
6 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
32 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
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0answers
16 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 ...
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0answers
29 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
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0answers
18 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
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1answer
30 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
1answer
37 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
43 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
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2answers
32 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
18 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
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0answers
17 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 $E[e^{tx}]≤e^{t^2σ^2/2}$.
0
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1answer
47 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 ...
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0answers
48 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
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0answers
27 views

Find/estimate variance

Let $w_{11},\ldots , w_{nm}\in [0, 1]$ be a set of constants and $H_1(t), \ldots , H_m(t)$ be some cumulative distribution functions (CDFs). Consider a sample of independent random variables $\xi _1, ...
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1answer
14 views

Find a distribution for this plot

Please help me find a formula that fits the distribution. It does not need to be exact, a simple approximation would suffice. Bonus points if you can tell me which predefined distribution in the ...
0
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1answer
21 views

Cumulative Distribution Funciton to pmf

I am still quite new to cdf and pmf. When we only have pmf for x = 1, 2 and 4 , how should I understand the corresponding cdf as in the pmf for x = 3 doesn't exist. Also I tried to draw the piecewise ...
1
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1answer
27 views

Let $X_1$ and $X_2$ be two independent random variables each with probability density function $fX_i(x_i) = 1$, for $0 < xi < 1$ for $i = 1, 2$.

Find: (a) $E(X_1 X_2)$, and (b) $Var(X_1 X_2)$. Isn't (a) = zero, since this are independent? How do I go about (b)
1
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1answer
37 views

Let X and Y be a random variables with $E(X) = 5$, $Var(X) = 30$, $E(Y ) = -􀀀5$, $Var(Y ) = 10$ and $Cov(X, Y ) = 7$

(a) Find $E(2X-3Y+1)$. (b) Find $E((X-2Y)^2)$. (c) Find $Var(3X-Y+pi)$ First I found $E(X^2)$ and $E(Y^2)$ using the given values for (a) I have $2E(X)-3E(Y)+1$ for (b) I come up with: ...
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1answer
28 views

CDF and Convergence of Maximum of Sequence of i.i.d. R.V. of Random Length

Let $X_1,X_2,...$ be i.i.d random variable $U(0,1)$ distributed. Let $N_m$ be $Poisson(m)$ and independent of each $X_i$. i)Find the cumulative density function of ...
0
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2answers
22 views

Operations with probability distributions

I had an idea that passes by declaring a new type of computer variable (like Integer, Double, etc.) that represents a statistical probability distribution (PDF), for that I would need to define the ...
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0answers
21 views

Direct computation of $\operatorname{log}(\operatorname{cdf})$ for a normal distribution

This question is linked to the normal distribution for a random variable. The probability density function (pdf) is expressed as: \begin{equation} \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - ...
1
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2answers
21 views

Given E(X) and Var(X) find the Expectation of $E[x-2(X-1)^2]$

Let X be a r.v. with $E(X) = 5$ and $Var(X) = 30$. Find $E[X-2(X-1)^2]$. I'm not sure as to how to approach this problem, any tips on how to approach it would be appreciated!
2
votes
0answers
26 views

limiting distribution of a function of joint normals

Let $Z_n=(X_{1,n},X_{2,n})\sim N(\mu,\Sigma_n)$ where $\mu=(0,0)'$ and $$\Sigma_n=\begin{bmatrix}a^2+\frac1n & ab \\ab & b^2+\frac1n\end{bmatrix}$$ Then where does ...