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

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2
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0answers
8 views

Double Median of functional: does the order matter?

Suppose I have an absolutely continuous distribution $F$ and two random variables $X\sim_{\mbox{i.i.d}}F$ and $Y\sim_{\mbox{i.i.d}}F$ (this means that $X$ and $Y$ are independent realizations of ...
1
vote
0answers
5 views

Convergence of Sum of Random Variables “Independent in Limit”

Consider a sequence of random variables $X_n\sim U[-n,n]$, a random variable $Y\sim N(0,1)$, and a random variable $Z\sim U[0,1]$, all independently distributed. In addition, consider a bounded, ...
1
vote
1answer
14 views

Suppose that $N$ is an iid geometric RV and $X_i$ is an iid Bernoulli RV. Find the p.g.f. of $R=X_1+ \dots + X_n$.

Each year a tree of a particular type flowers once and produces a random number $N$ of flowers, where $\mathbb{P}(N=n)=(1-p)p^n$, $n=0,1,2,\dots $ and $0<p<1$. Each flower has probability $1/2$ ...
1
vote
1answer
19 views

Joint probability distribution (over unit circle)

A couple of two continuous random variables $(X,Y)$ is distributed uniformly over the closed unity circle (so $-1\leq x \leq 1$ , $y$ analog). $U$ is defined as the distance from $O$ to the point ...
-3
votes
3answers
31 views

Two random variables

$X$ and $Y$ are two independent identically distributed random variable with pdfs given by $0.5\big[\exp (- \vert x-1\vert) \big]$, where $x$ and $y$ range from $-\infty$. to $+\infty$. Question ...
1
vote
2answers
29 views

Sum of Binomial distribution when the success rate is different.

Is there any easy way to calculate the probability of the sum of two binomial random variable if the success rates of them are different each other? I mean that $X \sim Bin(n,p_0)%$, $Y \sim Bin(m, ...
1
vote
0answers
19 views

Mixed distribution of product of Bernoullie and Gaussian r.v

Confused with the formulation of density function of the following mixed distributed random variable $Z$. $$Z \equiv X \cdot Y,$$ where $( \cdot)$ is product operation, and $X$ and $Y$ being ...
0
votes
1answer
16 views

Sum of iid random variables with an odd distribution

I have $G_1,G_2$, iid with probability distribution function $f(y) = Ce^{-y}y^{-1/2}$ where c is a normalizing constant. I am trying to find the distribution of $G_1+G_2$. I have tried transforming ...
0
votes
0answers
28 views

A Gaussian Divided by a Gaussian Equal to A Gaussian Divided by a Constant

I have a neural-network model in which each neuron is associated with an angle $\theta$. Firing rate as a function of $\theta$ is either a Gaussian or a constant. The claim has been made using this ...
0
votes
2answers
27 views

A problem on continuous random variables

I was reading a The First course on Probability by Sheldon Ross, while I stuck at this possibly stupid doubt. The problem is : The density function of X is given by $$ f(x) = \begin{cases} 2x, ...
0
votes
1answer
18 views

Pdf of a normal variable accepted with probability dependant on the normal variable

Assume $z$ is a standard normal variable. If $z<0$, then we accept it with probability 1. if $z\ge0$, we accept it with probability $e^{-mz}$, where $m>0$. I'm trying to figure out the pdf of ...
3
votes
1answer
20 views

Density Function of Random Variable Related to Brownian Motion

Above is my question. I've done the first two parts, that's no problem. I'm stuck on finding the density of the rv $R = W_1 / M$. I have got as far as $$g(x,y) = \frac{\partial^2}{\partial x ...
0
votes
1answer
30 views

Probability of long identical substring

You have a string of $20,000$ consecutive bits. Each bit is either a $1$ or a $0$ and has a $0.5$ chance of being either. Calculate the probability that there is at least one substring of at least ...
2
votes
0answers
21 views

Probability of being between two independent Gaussian random variables

Suppose we have two independent random variables $X$ and $Y$. I am interested in calculating $P(X\leq x \leq Y)$. Is this correct? $$P(X\leq x \leq Y) = P(X\leq x)P(Y \geq x) = P(X\leq x)[1 - P(Y ...
0
votes
2answers
35 views

determing the probability distribution

I have 2 sets of elements, say $A=\{a\}$ (only 1 element) and $B = \{b_1, b_2,..., b_n\}$. The probability of picking $A$ is $0.3$ and the probability of picking $B$ is $0.7$, and all elements in $B$ ...
0
votes
0answers
8 views

How to compute the normalizing constant of the Dirichlet distribution for high value of alpha?

The normalization constant of the Dirichlet distribution has the following form: $B(\alpha) = \frac{\prod_{i=1}^{K}\Gamma \left ( \alpha_i \right )}{\Gamma \left ( \sum_{i=1}^{K} \alpha_i \right )}$ ...
0
votes
0answers
31 views

Entropy of $\operatorname{Beta}(\alpha, \beta, a, c)$

I know that the differential entropy of the two parameter Beta distribution $X \sim \operatorname{Beta}(\alpha, \beta)$ is $$ \begin{align} h(X) = \ln \operatorname{B} (\alpha, \beta) &- ...
2
votes
1answer
49 views

Distribution of the product of a Normal and an Exponential random variable

What is the probability distribution of $M$, given $M=V*X/k$, where $X$ is Normal, $V$ is Exponential, $k$ constant? Or, in the real world, the probability distribution of (Cost/k) where ...
0
votes
0answers
26 views

Statistical distribution methods

I'm coding a program to classify a stock price movement in a given timeframe. I have around 20k data points and it ranges from 0 to around 300, with the average being 10,8. I want to use a genetic ...
1
vote
1answer
18 views

find the probability that, in the next 7 weeks, there are exactly 3 weeks in which Jan receives exactly 2 free gifts

Can you give me a breakdown of the stages you take arriving at the answer to the following question: Jan buys $5$ packets per week with a $30\%$ chance of finding a gift per packet,find the ...
0
votes
1answer
20 views

Solving a series in the proof of the expectation of the binomial distribution

I am studying the expectations and variances of the most common distributions. For the binomial distribution the mean is equal to $np$. Considering $p$ and $q$ independent variables and ...
1
vote
0answers
26 views

Find the distribution of $Z = 1/X_1 + 1/X_2$

Find the distribution of $Z = 1/X_1 + 1/X_2$, where $X_1$ and $X_2$ follow normal distribution. I have $2$ variables with normal distribution, $X_1$ and $X_2$. How can I find the distribution of: ...
0
votes
0answers
9 views

Exponential Demand Periodic Review

I have exponentially distributed demand data and I am trying to find a formula for an 'order up to level (OUL)' periodic review ordering policy. We are not using a re order point for this policy. ...
1
vote
1answer
18 views

Translate exponential distribution into normal distribution

I have a bunch of inventory management formulas that are supposed to be used with normal distributions, however my demand data fits an exponential distribution. Is there any way to translate the ...
-1
votes
0answers
20 views

Applied mathematics for Clinical Medicine [on hold]

I'm a medical graduate, looking for advice/help on a project I would like to start. I would like to use applied mathematics to deconstruct the medical SOAP note into data sets that can be reproduced ...
2
votes
0answers
21 views

Is Gaussian $(X_1, X_2)$ optimal for $h(a_1X_1+ a_2X_2+Z_1) - \mu \, h( b_1X_1+b_2X_2+ Z_2)$?

Let \begin{align} W &= h(X_1+Z_1) - \mu \, h( X_2+ Z_2) \quad (1) \end{align} where $h(\cdot)$ is the differential entropy function, $\mu\ge 1 $ is a scalar, and $Z_1$ and $Z_2$ are ...
0
votes
0answers
22 views

Bayesian statistics and Basis for continous functions

I was thinking about Bayesian statistics, and one thought bothered me: In Bayesian statistics, we assume that the pdf $p(x)$ can be described as: $p(x)=\int f(x|\theta)g(\theta)d\theta$ usually ...
0
votes
0answers
13 views

zero drift brownian motions and barriers problem [duplicate]

Given two same brownian motion with no drift and different variances: $$(dG_1/G_1)= \sigma_1dW_g $$ $$(dG_2/G_2)= \sigma_2dW_g $$ and two barriers $P_1 > P_2$ assuming that $ \sigma_1 > ...
2
votes
0answers
24 views

Laplace transform of inverse error function

I want to calculate the convolution of a function with the inverse error function. Therefore I chose to try to first find an integral transform of the inverse error function like the laplace ...
1
vote
0answers
16 views

Suggestions for dealing with these order statistics

Consider a collection of $n$ random variables $X_i \sim N(\mu, \sigma^2)$, ($i = 1,2,\ldots, n$) and a random variable $X \sim \text{Exp}(\lambda)$. All $X_i$'s and $X$ are mutually independent. Let ...
0
votes
2answers
59 views

Antiderivative of $xe^{-cx^2}$

I need to define $c$ in $$\int_0^\infty xe^{-cx^2},$$ so that it becomes a probability-mass function (so that it equals 1). Where do I even begin finding the antiderivative of this? I know the answer ...
2
votes
1answer
17 views

How to find data distribution law using MATLAB?

Having a random variable $T \geq 0$ and a set of discrete data represented by $t=t_i$ and $P(T \leq t-i)$. My aim is to find the distribution law of $T$. Is there any fast method in Matlab that can ...
2
votes
1answer
25 views

Derive probability mass function from probability-generating function

Given the probability generating function $$G(z) = \frac{1}{2} \frac{3+z}{3-z}$$, how can one derive the pmf? I know that I have the manipulate the function into a series: $$G(z) = ...
2
votes
0answers
23 views

Permutation and combinations (Sheldon M.Ross)

We need to divide 8 new teachers among 4 schools , how many such divisions are possible ? I tried to solve this by the Distribution Method , that is : $x_1$ + $x_2$ + $x_3$ + $x_4$ = 8 , which ...
0
votes
1answer
23 views

determining distribution composed of uniform distributions

Let $X,Y,Z$ be i.i.d. $U(0,1)$ distributed. How can I determine the distribution of $$ \frac{X}{X+Y+Z}?$$ I have no idea how to go about this problem. Obviously this expression also has values ...
1
vote
0answers
13 views

Product of CDF and CCDF (or survival function)

Suppose we have two independent random Gaussian-distributed variables X and Y. X and Y represent thresholds for activation and deactivation, respectively. I'm interested in ensemble averaging over ...
1
vote
1answer
17 views

Multidimensional convergence in probability

If I have a vector $X^n=(X^n_1,...,X^n_m)$ is it true that $ \mathbb{P}(X^n\geq\epsilon)\rightarrow 0$ if $ \mathbb{P}(X^n_i\geq\epsilon_i)\rightarrow 0\ \forall i =1,...,m$ As $n\rightarrow \infty$?
0
votes
1answer
7 views

Terminology for probability matrix.

I have two related questions about terminology. If a matrix contains probabilities such that each column (or row or both) sums to $1$ , is this matrix always called a stochastic matrix i.e. even if ...
0
votes
1answer
19 views

How do I find the marginal probability density function when the interval is dependent of one of the variables?

I'm trying to find $f_x$ and $f_y$ given a joint probability distribution $$f(x,y) = \frac18 (y^2 -x^2)e^{-y}$$ defined on the interval $0 \leq y \leq \infty$, $-y \leq x \leq y$ Naturally I've tried ...
-1
votes
1answer
27 views

Cumulative distribution function of the ratio of the maximum and minimum of two random variables [on hold]

Let $X_{1}$ and $X_{2}$ be independent, absolutely continuous random variables, each uniformly distributed between 0 and 1. I want to find the cumulative distribution function of the random variable ...
0
votes
0answers
11 views

Cardinality of maximum independent set for a given degree distribution

Consider undirected graph $G(V,E)$. Assume that $f_n(k)$ be the probability mass function of degree of a vertex in $G$. Further, assume that $f_n(k)$ is an strictly decreasing function of $k$ with ...
2
votes
3answers
43 views

If two different linear combinations of two random variables are Gaussian, can we deduct both of them are Gaussian.

If two different linear combinations of two random variables are Gaussian, can we deduct both of them are Gaussian. Mathematically, if we know that $a_1X+b_1Y$ and $a_2X+b_2Y$ have Gaussian ...
1
vote
0answers
22 views

Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions?

I am wondering whether Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions? Any comments or references would be greatly appreciated.
0
votes
1answer
12 views

Reconstructing a restricted distribution from its mean and standard deviation

For simplicity lets assume we have a continuous distribution from 0 to 100. If the mean is 60 and std is 10, then it would make sense to simply model it as a gaussian with those parameters. However ...
2
votes
1answer
30 views

Intuition/proof that $E(X)= \int X(w) dP = \int x d\alpha$, where $\alpha$ is the cumulative distribution function of X

Looking for more intuition/help explaining the equivalence of the following two integrals. I know that the push-forward measure, or the CDF, of a random variable $X$ on a prob. space $(\Omega, \cal ...
0
votes
2answers
37 views

Integration limits of a Marginal Probability Density Function with a Triangle-Shaped Boundary

I have given a triangle shaped boundary $M$ of my probability density function in $\mathrm{R}^{2}$, with the limitations beeing: $$y = 0$$ $$y = x$$ $$y = 2-x$$ and a probability density function $$ ...
-1
votes
0answers
47 views

Find the density of a ratio of random variables

$X$ has density $2x, 0 < x < 1,$ and $Y$ has density $1/10$ over $0 < y < 10$. $X$ and $Y$ are independent. I have to find (a) density of $Y/X$ (b) $E[Y/X]$ (c) $E[Y^2/X]$ I let $Z=Y/X,$ ...
0
votes
0answers
27 views

What is the name of a “Bernoulli” distribution with values in $\{1,2\}$ instead of $\{0,1\}$?

I have a "Bernoulli" distribution, but instead of $X$ taking values in $\{0,1\}$, it takes values in $\{1,2\}$. So $Pr(x=1)=p$ and $Pr(x=2)=1-p$. Is there a specific name for this distribution? ...
1
vote
1answer
103 views

Confidence Interval for Pareto Distribution

A random variable is said to have probability density function $$f_X(x)=\frac{\alpha k^\alpha}{x^{\alpha +1}},\quad \alpha , k>0 \; \text{ and }\; x>k.$$ 1. Compute the MLE estimators ...
0
votes
1answer
11 views

Multivariate Quantiles

I am interested whether a concept for the multivariate equivalent to quantiles exists. In the univariate case, a $p$-quantile can be computed via the inverse of the cumulative density function, ...