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

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0answers
12 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 ...
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0answers
21 views
+50

Estimates for the normal approximation of the binomial distribution

I'm interested in estimates of the normal approximation for binomial distributions, i.e. in estimates of $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - ...
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1answer
15 views

When two random variables that have the same law… Can they be happily exchanges?

Imagine, $X$ and $Y$ are two random variables which have the same law, which we denote by $X\sim Y$. We have then a third random variable $Z$. Can we say that $$(X,Z)\sim (Y,Z)?$$ In what cases is ...
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0answers
23 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 ...
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1answer
9 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
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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 ...
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0answers
6 views

Maximize the net profit with probabilities — optimal purchasing

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 ...
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0answers
22 views

Pareto distribution,fisher information, confidence interval

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, dont know how to do that on a keyboard. The rest is ...
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3answers
279 views

Finding unknown values from discrete probabilities.

(I am confused here with the limits. It says x = 0,1,2,3... So what is my end limit her? Thanks.)
1
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0answers
16 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 ...
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1answer
22 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 $y_1$, ...
1
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2answers
105 views

Proving that a function is monotone

Here is the setting: We have a middleman that buys a product from the producers, and sells the product to the customers. The middleman charges a price $R$ to the customers, and pays a price $p(R)$ to ...
0
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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
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1answer
19 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 ...
11
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2answers
641 views

Is There a Continuous Analogue of the Hypergeometric Distribution?

As the title states, is there a continuous analogue of a Hypergeometric distribution? If $ X \sim H(m,n,N)$ is a common Hypergeometric distribution, where $N$ is the population size, $n$ is the ...
2
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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
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0answers
21 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 ...
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1answer
48 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 ...
1
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1answer
28 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
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0answers
26 views

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, ...
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2answers
834 views

How to find $E(X|X+Y=k)$ for geometrical distribution?

$X$ and $Y$ are independent and geometrical distributed with same parameter $p$. How to find $E(X|X+Y=k)$ for all $k =$ $2,3,4$.... I thought $$E(X|X+Y=k) = \sum_{x=1}^{k-1} xp(x,x+y=k)/p(x+y=k)$$ ...
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2answers
19 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 ...
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0answers
5 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?
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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? ...
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
1answer
12 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 ...
3
votes
1answer
461 views

Maximum likelihood estimators, hypergeometric and binomial

I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red ...
2
votes
1answer
447 views

Sum of Wishart matrices

Considering two matrices, $H_1$ and $H_2$, that are independent of each other and follows complex wishart distributions as $\mathcal{CW} _m(n_1,\Sigma_1)$ and $\mathcal{CW} _m(n_2,\Sigma_2)$ ...
0
votes
3answers
52 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 ...
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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 ...
0
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1answer
342 views

Product of two exponentially distributed random variables

I am trying to find the close form expression of probability distribution of $Z$ such as $Z=X_1X_2$ where $X_1$ and $X_2$ are two independent exponentially distributed variables with PDF ...
5
votes
1answer
20 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 ...
2
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2answers
36 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
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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 ...
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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 ...
0
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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 ...
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0answers
18 views

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

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 ...
3
votes
1answer
53 views

Stationary distribution of an increasing stochastic process with a cut-off

I have a discrete time stochastic process $\{X_t : t \in T\}$ with continuous state space. Assume $X_0=0$ and increments $\delta_t$ are exponential with mean $\alpha$ (so its parameter is ...
0
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0answers
20 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 ...
2
votes
1answer
216 views

Conditions for positive dependence

Consider two random variables $X$ and $Y$ with joint distribution $F_{X,Y}$ and strictly positive density function $f_{X,Y}$. Additionally, let $x^*$ be the value of $x$ that solves: $$ \Pr[Y\leq ...
0
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1answer
38 views

chi square distribution probability

I am having a problem with this. Suppose a stock's returns are normally distributed with mean $m$ and variance $\alpha^2$ and we compute the sample variance from a sample of $41$ periods and find ...
0
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1answer
25 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 ...
2
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1answer
28 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 ...
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1answer
19 views

Expected Value: how to understand this expression?

So I have come across a question asked by my peers. Define: $$g:=\sqrt{E[|y_r(t)|^2]}$$ Given that $$y_r(t)=\sqrt{t}\cdot h+b+k+c,$$ where $h$, $b$, $k$, and $c$ are independent random variables. ...
3
votes
1answer
21 views

Independence of random variables and covariance in the limit.

Consider two sequences of random variables $(X_n)$ and $(Y_n)$ which converge in distribution to $X$ and $Y$ respectively, where $X$ and $Y$ are independent, but each pair $(X_n, Y_n)$ is not ...
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0answers
16 views

Copula theory on discrete random variables [on hold]

How can I find the joint pmf on two discrete random variables using the copula theory
4
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0answers
137 views

About the total number of twin primes in the vicinity of twin primes

Just for curiosity's sake, I did a test regarding twin primes, and I have doubts about the meaning of the results. Test: calculation of ${\pi_2}$(n) and the twin primes density in the vicinity of ...
1
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1answer
536 views

Solving Probability Density Function for continuous random variable

The probability density of a random variable $x$ is $$f(x)=a\ \cdotp x^2\ \cdotp \mathrm{e}^{−kx}\ (k>0,\ 0\leq x\leq \infty)$$ Then, the coefficient $a$ equals $$(i)\frac{k^3}{2}\ \ \ \ (ii)\ k^3 ...
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0answers
13 views

Posterior probability estimation in MAP model

I have a question about probability. I am using Bayes rule to determine which class the $x$ belong to. According to Bayesian formula, the MAP estimation is equivalently found by $$p(x \in \Omega_i|x)= ...
2
votes
2answers
534 views

How to prove that the binomial distribution is approximately close to the normal distribution when $np(1-p) \geq 10$

I would like a formal proof for this "rule of thumb." Can you assist me in getting to this solution? I require the insights and creativities of mathematicians. We know that if $np(1-p) \geq 10$ the ...