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

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2
votes
1answer
39 views

What is the probability that 5 randomly chosen cards in a deck add up to 40 or greater than 40?

I have made a probability game, where you have to pull out any 5 cards without looking (from a deck of 52 cards), and if all five cards add up to 40 or more, they player pulling the 5 cards from the ...
0
votes
1answer
20 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
9 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 ...
0
votes
1answer
14 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
24 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
9 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
26 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
20 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
11 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
29 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
35 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 $$ ...
0
votes
0answers
25 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
10 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, ...
1
vote
0answers
38 views

Skellam CDF Increasing vs Decreasing in a parameter

I'm working with the following Poisson difference distribution: $$\text{Prob}\{X_1-X_2 \geq 0\} $$ where $X_1 \sim$ Poisson $(\mu_1)$ is independent from $X_2 \sim$ Poisson $(\mu_2)$. I need to ...
0
votes
1answer
12 views

Calculate Probability of a Range of Dice Rolls given their Distribution

I'll prefix this with - I'm not particularly great at Maths, so I might ask for an explanation of some of the answers. What I'm trying to do is convert this into something I can code. I've got a ...
0
votes
1answer
24 views

What will be the pdf from Mixture of Gaussians

In Euclidean $R^M$ space, I want to compute the pdf of the Euclidean distances between $d^M(\mathbf{z_i}) $= $||\mathbf{z_i -z_j}||^M = r_i^M , i \neq j$. What will be the pdf $f(r)$ ? Let there be ...
2
votes
1answer
27 views

Convergence of probability for $t$-distribution

Assume that $Z_0,Z_1,Z_2,\dots$ are i.i.d. RVs, $Z_j\sim N(0,1)$, and set $$T_n:=\frac{Z_0}{\sqrt{\frac1n(Z_1^2+\cdots+Z_n^2)}}$$ (a) Compute the limit $$\lim_{n\to\infty}\text{P}(T_n^2+2T_n\leq ...
0
votes
0answers
9 views

Poisson Distribution Optimization Problem

A retailer buys $n$ cookies and has to pay $\zeta_1$ for each. He wants to sell them for a price of $\zeta_2$ (with $0$ < $\zeta_1$ < $\zeta_2$). Let X be a random variable which states, how ...
-2
votes
0answers
204 views

Pareto distribution,fisher information, confidence interval [on hold]

Having a bit of problem at these questions, greatly appreciated if anyone can solve them. For the notation, k^ is k with a hat on top of it, don't know how to do that on a keyboard. The rest is ...
1
vote
1answer
71 views

Density functions and estimators

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
26 views

upper bound and a lower bound on the number of points that are uniformly distributed on a surface

Can I calculate an upper bound and a lower bound (or max or min) on the number of points that are uniformly distributed on a surface, knowing the area of the surface ? More precisely, I have a sector ...
0
votes
0answers
15 views

Can we calculate the derivative of a distribution function with respect to its parameters?

I am asking a very basic question. Can we calculate the derivative of a density function with respect to its parameters, mean and variance? Can we calculate the derivative of a distribution function ...
1
vote
1answer
79 views

Fisher information matrix of MLE's

I know what it means to compute the fisher information matrix of a vector of parameters. However, how does one compute the fisher information matrix of a vector of MLE's? Specifically, I am working ...
1
vote
0answers
27 views

References for the following functional

In many of the types of problems Ive looked at the following quantity keeps arising and I was wondering if anyone knew any references I could look at to learn some its properties. Take any function ...
2
votes
0answers
1k views

Sum of F Ratio distributed random variables

Where $X$ follows an F Ratio distribution F$(1,\alpha)$ with pdf: $$ f(x)= \frac{\alpha ^{\alpha /2} (\alpha +x)^{\frac{1}{2} (-\alpha -1)}}{\sqrt{x} B\left(\frac{1}{2},\frac{\alpha }{2}\right)},\; ...
1
vote
1answer
31 views

Probability distribution for a geometric distribution don't add up to 1

Say I'm rolling 2 dies,numbered 1 to 10. A successful outcome is considered rolling a multiple of 4. Therefore,probability of success=0.25 and prob of failure=0.75. This is an example of a geometric ...
1
vote
1answer
49 views

Questions about integration

I'm still a bit confused about definite integration although got the basic idea of how to do integration. The problem is to integrate functions on a uniform distribution over [50, 150]. Firstly ...
0
votes
0answers
6 views

Inverse Gaussian versus inverse Normal distribution [on hold]

I am wondering what is the difference between the inverse Normal and the inverse Gaussian distribution?
0
votes
2answers
20 views

CDF of a Uniform probability density function

I want to find Cumulative distribution function (CDF) of the following density function: $ f(x)= \begin{cases} 3/20 & \text{for } 2 \leq x \leq 4 \\[8pt] 4/20 & \text{for }4 < x \leq ...
0
votes
0answers
16 views

Distribution formed by taking two random points on an open disc and graphing their midpoint

I am wondering about the following distribution: Take an open disc and choose two points at random. and then take its midpoint in the new distribution. What does the resulting distribution look like? ...
1
vote
0answers
58 views
+50

Probability of absorption of a biased random walk when the starting point has binomial distribution

Consider a random walk $\{0,1, ... , N\}$ with up probability $p$ and down probability of $p-1$ where $p \neq 1/2$. Suppose there are absorbing barriers at $0$ and $N$ and that the starting point, ...
0
votes
0answers
16 views

Joint cumulative density function of two independent Gaussian random variables

Assume we have two independent random variables $\theta_1$ and $\theta_2$ which each have separate Gaussian distribution functions $D_{\theta_1}$ and $D_{\theta_2}$. $\theta_1$ describes a threshold ...
4
votes
2answers
49 views

Can some probability triple give rise to any probability distribution?

Suppose we have a probability triple $(\Omega,\mathcal{F},P)$ and random variable $X:\Omega\to(\mathbb{R},\mathcal{B})$ with $\mathcal{B}$ denoting the Borel $\sigma$-algebra. Then, the distribution ...
0
votes
0answers
25 views

Find limiting distribution

The question is like this: $X_i$ are i.i.d with $P(X_i\leq x)=1-e^{-x}$. $S_n=X_1+\cdots+X_n$. Find the limiting distribution of $\sum_{i=1}^nI(X_iS_n>1)$. It seems that the problem is related ...
0
votes
3answers
53 views

If $X$ has a Poisson distribution with $E[X]=\lambda$, does $Var[X^2]=4\lambda^3+6\lambda^2+\lambda$?

Suppose $X$ has a Poisson distribution with mean (and therefore variance) $\lambda$. Using Excel to explore properties of the distribution of $X^2$ with some small integer values of $\lambda$ I ...
1
vote
0answers
17 views

What is the pdf of $X$, where $dX_t = -aX_t + d N_t, N_t$ is a compound Poisson process?

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where (I'm not sure this notation makes sense, I'm not very familiar with the stochastic calculus ...
5
votes
1answer
21 views

Density of stochastic integral

I am working on finding the PDF of $X_t^2$, where $X_t = \int_0^t A(u) \,dW_u$, a Wiener integral, i.e., $W_t$ is Brownian motion and $A(t)$ is a deterministic function. Here, would like to ask that ...
0
votes
0answers
30 views

Monotonocity of ratios of normal CDFs

I am solving a problem in decision theory under uncertainty and need to establish whether $\frac{\Phi(x)-\Phi(x-\varepsilon)}{\Phi(x+\varepsilon)-\Phi(x-\varepsilon)}$ $(\ast)$ is monotonically ...
2
votes
2answers
37 views

Binomial distribution central moment calculation

If for a binomial distribution the mean is $4$ and variance is $3$, find th $3^{\text{rd}}$ central moment. I understand that the first and second central moments are mean and variance ...
0
votes
0answers
23 views

Support lemma - Game theory

Let α be $a$ mixed strategy profile, $a_i ∈ supp(\alpha _i), a_i \notin B_i(\alpha _{−i}), a_i' ∈ B_i(\alpha _{−i})$ and $a_i'$ defined by $\alpha_i'(a_i)=0$, $\alpha_i'(a_i')=\alpha _i ...
1
vote
1answer
24 views

Computing expectation of a function of two random variables

I have two arrays $X$ and $Y$ of length $N$ each. In array $X$, I have random numbers $x_1$, $x_2,\ldots,x_N$, whose sum is $S_x$. Similarly in array $Y$, I have random numbers $y_1$, ...
0
votes
1answer
17 views

uniform angular distribution-change of origin

Given a variable which is uniformly distributed for $0<\theta<\pi$ on, let's say, a circle around the origin $O$ with radius $R$($\theta$ starting on the positive x-axis and turning ...
-2
votes
0answers
18 views

What is the p-value of this problem? [closed]

Over a 7 year period, an event happens 126 times during 154 opportunities for this kind of event to happen. Over the next 8 years, the same event happens 142 times during 169 opportunities for this ...
0
votes
0answers
21 views

Sum of Gaussian and Binomial distribution

I need to calculate the probability of sum of two probability variable, each of which is distributed as binomial distribution and Gaussian respectively. I mean how to calculate the probability of ...
0
votes
1answer
26 views

Probability of winning a simple game

Consider two players, A and B start with 8 and 6 stones respectively. A rolls a six-sided die to determine how many stones to take from B. B performs the same task to determine how many stones to ...
2
votes
1answer
29 views

How to represent $Prob(X_1+X_2 \leq a, X_2+X_3 \leq b, X_3 +X_4 > c)$ with mutually independent random variables?

There are four mutually independent random variables: $$X_i : \Omega \to \mathbb R$$ for $i= 1,2,3,4$ The cumulative distribution function of them is given as $F_i(x_i)$. How to represent ...
-1
votes
1answer
32 views

Finding distribution function of the ratio of two continuous uniform random variables where the denominator random variable is squared.

Let $X_{1}$ and $X_{2}$ be independent and uniformly distributed between 0 and 1. I want to find the distribution function of $X_{3}=\dfrac{X_{2}}{X_{1}^{2}}$. Denote this distibution function by ...