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

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0
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2answers
26 views

Find the CDF of a function given its PDF

The probability density function of the random variable X is as follows $f_{X}(x) = \begin{cases} 1/4, & \text{if 0 < x < 1} \\ 1/4, & \text{if 2 < x < 4}\\ 1/4, & \text{if 6 ...
1
vote
1answer
651 views

Find the MOM estimate and the MLE of the Pareto distribution.

The Pareto distribution has been used in economics as a model for a density function with a slowly decaying tail: Assume that $X_0$ > 0 is given and that $X_1, X_2, ..., X_n$ is an i.i.d. sample. ...
3
votes
0answers
24 views

What is the limit distribution of $\frac{S_{N_n}}{\sigma \sqrt{a_n}}$ as $n\rightarrow \infty$.

Let $X_1, X_2, X_3,...$ be iid with $\mathbb E[X_i]=0$ and $\operatorname{Var}[X_i]=\sigma^2>0$, and let $S_n = \Sigma_{i=1}^{n} X_i$. Let $N_n$ be a sequence of integer valued random variables ...
0
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0answers
13 views

n is the number of Bernoulli trials with success p. Let $X_i$ be the number of attempts until success. What is the joint probability function?

n is the number of Bernoulli trials with success p. Let $X_i$ be the number of attempts until success. What is the joint probability function where $i=1,2$. Well let's figure them out separately ...
0
votes
1answer
14 views

2 Cards are picked from a deck without replacement. Let X= number of aces, and Y= number of kings. Find the joint probability function.

2 Cards are picked from a deck without replacement. Let X= number of aces, and Y= number of kings. Find the joint probability function (in a 3x3 table) X and Y are both discrete random variables ...
0
votes
2answers
15 views

Probability: Reading tables and using the data from them?

Alright probability is not as hard as I imagined yet I strugle with reading tables and applying them to the formulas. The question bellow has a table with 3 rows and 3 collumns and I am asked to see ...
1
vote
1answer
38 views

A silly question regarding a badly written exercise for probability equations.

I am doing some exercises and this silly question is bothering me even though I am familiar with probability theory and Bayes law but this question is written in a rather peculiar manner I have no ...
1
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0answers
25 views

Optimal choice for the values of money units

I just thought about how to find the optimal values for money units, given that you want your currency to come in $n$ different values (e.g. Euros come in 7 values for bills and 8 values for coins, so ...
0
votes
2answers
21 views

Central limit theorem on packs of variables

I'm trying to solve the following exercise: Let $\mu$ be a probability distribution on $\mathbb{R}$ having second moment $\sigma^2<\infty$ such that if $X$ and $Y$ are independent with law ...
1
vote
1answer
37 views

Word Problem: Probability of Y books Fitting in Book Case

Problem: You have $4600$ cm of book case. The thickness of the books are independently distributed with $X \sim N(1.8$ cm$,0.7^2)$. Approximately determine what the probability of ...
3
votes
1answer
1k views

Bus arrival poisson paradox

I have a question about the waiting time paradox for poisson processes(in this case in terms of bus arrivals). Suppose I know that buses arrive with poisson distribution(lambda). I arrive at fixed ...
2
votes
1answer
21 views

Probability - Finding the Support of a Joint Transformation

$$ f(x,y) = \left\{ \begin{array}{ll} 12xy(1-y) & \quad 0< x < 1, 0<y<1 \\ 0 & \quad \text{elsewhere} \end{array} \right. $$ ...
0
votes
2answers
20 views

Given a Poisson distribution, $2f(0) + f(2) = 2f(1)$, what is the mean of the distribution?

If for a Poisson distribution $2f(0) + f(2) = 2f(1)$, what is the mean of the distribution? I know that for X ~ POI($\lambda$), then the pdf for the random variable X is \begin{equation} ...
0
votes
2answers
10 views

Does this follow a binomial distribution?

Q. Four roads start from a junction. Only one of them leads to a mall. The remaining roads ultimately lead back to the starting point. A person not familiar with these roads wants to try the different ...
1
vote
1answer
642 views

Probability generating function for logarithmic series distribution, support $k\geq1$

I'm trying to derive the probability generating function (pgf) for the logarithmic series distribution, and not getting the expected form $\frac{\log{(1-qs)}}{\log{(1-q)}}$. It seems that pgfs are ...
-1
votes
1answer
43 views

Finding the conditional probability

enter image description here Let $(X,Y)$ be a two-dimensional stochastic vector with density $$ f_{X,Y}(x,y) = \begin{cases} \dfrac{e^{-y}} y & \text{if } 0<x<y, \\[4pt] \,\,\,\, 0 & ...
2
votes
1answer
21 views

Derivation of the Negative Hypergeometric distribution's expected value using indicator variables

I'm trying to understand how to derive the Negative Hypergeometric's expected value using indicator variables. Note, in the problem below, we are only interested in the expected value before the first ...
17
votes
4answers
911 views

very elementary proof of Maxwell's theorem

Maxwell's theorem (after James Clerk Maxwell) says that if a function $f(x_1,\ldots,x_n)$ of $n$ real variables is a product $f_1(x_1)\cdots f_n(x_n)$ and is rotation-invariant in the sense that the ...
-5
votes
0answers
19 views

Multivariate Gaussian distribution [on hold]

I done parts (a) and (b) but I am stuck on part (c). I think the joint distribiution of R^2 is a chi squared r.v but I am not sure
-1
votes
0answers
15 views

Finding joint pdf from marginal pdf's

I have $N$ samples $(X_1,\cdots X_N)$ of exponential random variables with parameter 1. The samples are ordered such that $X_N \geq X_{N-1} \geq \cdots X_1$. I know the individual pdf's of $X_N$ and ...
0
votes
0answers
7 views

How to prove $2d_H(\{XY\},\{X\}\{Y\})^2 \le I(X,Y)$?

Let $X$ and $Y$ be discrete random variables. Denote the joint distribution of $X$ and $Y$ by $\{XY\}$ and their marginal distributions by $\{X\}$ and $\{Y\}$. Let $\{X\}\{Y\}$ denote the product of ...
0
votes
1answer
18 views

Let $X$ have pdf $f(x) = e^{-x}$. Find the pdf of the integer part of $X$.

A continuous random variable has a pdf defined by $$f(x) = e^{-x} , x > 0.$$ The discrete random variable $Y$ is defined as the integer part of $X$, that is the largest integer less than or ...
-1
votes
0answers
29 views

Generating a Uniform R.V with specified correlation [on hold]

I understand that it involves copulas, but I'm looking for a specific methodology for a specific correlation. I want to generate $U$ and $V$, random variables that are $~Uniform (0,1)$ with ...
0
votes
1answer
18 views

Calculating the magnitude of random numbers from normal distribution

Statement: Given an array of 80 random numbers, normally distributed between 0 and 1, we can expect that the numbers are all of similar magnitude, on the order of $80^{-1/2} \approx 0.1$. Question: ...
-1
votes
2answers
53 views

Conditional probability of a Joint distribution

Let $(X,Y)$ have joint density $f(x,y)=e^{-y}$ , for $0<x<y$, and $f(x,y)=0$ elsewhere. What is $f_{X\mid Y} (x,y)$ for $0<x<y$? I think that the answer is $1/y$, however, I am having ...
0
votes
2answers
21 views

Show that this MC is ergodic?

Suppose I have a Markov Chain with States, $S = {1,2,3,4}$ and a PTM given by $P =$ $\begin{pmatrix} .25 & .25 & .25 & .25 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\\ 0 ...
0
votes
1answer
31 views

Probability function and random variables

Given a Bernoulli r.v., $W$, which is derived from r.v. $T$ (Poisson) (a) if $T=0$ then $W=1$ and (b) if $T>0$ then $W=0$. One has to show that the sample mean (the proportion of $0$s in the ...
0
votes
1answer
32 views

Heavy tailed discrete distribution infinite mean

I'm looking for an example of a discrete distribution with infinite mean $f_n = P(X = n)$ for $n=1,2..$ such that the sequence $r_n = \sum\limits_{k=n+1}^{\infty}f_k$ satisfies the relation $$r_n = ...
1
vote
0answers
45 views

Cumulative distribution function of two independent and uniform on $[0,1]$ random variables is a surjective map for $t\in [0,1]^2$?

I am trying to argue that the cumulative distribution function of two independent and uniform on $[0,1]$ random variables is a surjective map for $t\in [0,1]^2$. Below the argument I have developed. ...
0
votes
0answers
13 views

Similarity measure for uncertain point sets

Imagine that we have two sets of points $M=(x_{1}, x_{2},...x_m)$ and $N=(x_{1}, x_{2},...x_n)$. These are actually lists of $x$, $y$ (and $z$) in 2D (or 3D) space so $x_i\in\ \mathbb{R}^2$ (or ...
0
votes
0answers
24 views

Difference of dependent central Chi-Square random variables with 2 degrees of freedom

Suppose we have $X$ and $Y$, both are dependent and complex Gaussian random variables with zero means and the same variance $\sigma^2$. The real and imaginary parts of $X$ and $Y$ are independent, ...
-3
votes
0answers
92 views

Mathematics Homework 2 Question 8d :What is the probability you and your partner are now able to meet the new deadline? [on hold]

You are working on a programming project with your partner for a computer science course. The project is due in 48 hours. Together, you are to produce a computer program and each of you are assigned ...
0
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0answers
21 views

Distribution of Double Stochastic Integral

Assume that $f(s)$ is a $C^\infty$ univariate function and that $\{ (W_{1,t}, W_{2,t})\}_{t \geq 0}$ is a two-dimensional, correlated Wiener process. Then, does the random variable $X_T \equiv ...
0
votes
1answer
23 views

Convergence a.s. and convergence in $L^1$ don't imply each other [on hold]

I'm trying to get two examples that convergence a.s. and convergence in $L^1$ don't imply each other. Now, I only know the examples that convergence a.s can't implied by convergence in probability, ...
3
votes
1answer
523 views

Fast generation of Pareto-distributed randoms.

I'm developing a library of routines for generating random numbers for simulations (it's on GitHub). I've included fast routines for normally distributed and exponentially distributed randoms, using ...
0
votes
1answer
24 views

Uniform distribution on $\{\frac{i}{n}: 1 \leq i \leq n \}$

I am trying to do a problem in which there is a type of distribution I am not familiar with, the exercise says: Find the limit in distribution of the sequence $(Z_n)_{n \in \mathbb N}$, where for each ...
0
votes
1answer
15 views

GRE Quantitative problem on distributions

I was doing some problems on this .Can some one please help me with the following: Here the given answer is that quantity B is grater than Quantity A. How is this obtained? Do we know anything ...
0
votes
0answers
20 views

Expectation of the trace of An Inverse Wishart random matrix

Assume that Σ~IW(Α,Τ,Ν) with T>N. Σ,Α are positive definite symmetric matrices and IW stands for Inverse Wishart. What is the following Expectation? $E(tr((Σ^{-1}B)$=? let B=bb', where b is a ...
3
votes
3answers
2k views

How is logistic loss and cross-entropy related?

I found that Kullback-Leibler loss, log-loss or cross-entropy is the same loss function. Is the logistic-loss function used in logistic regression equivalent to the cross-entropy function? If yes, can ...
1
vote
1answer
28 views

Probability that X is greater than the mode of X?

How do I solve this problem: Let X be a continuous random variable with density function \begin{equation} f(x) = \begin{cases} \hfill ax^2e^{-10x} \hfill & \text{for x $\geq$ 0} \\ ...
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votes
0answers
24 views

The conditional probability density function with a specific condition [on hold]

Assume the following discrete time model: $x(t+1)=Ax(t)+w(t)$ where $w(t)$ is zero mean, iid white noise with bounded covariance matrix $Q$. Let $s=x(t)+x(t-1)+x(t-2)$. How I can find ...
0
votes
1answer
1k views

PDF of the ratio of two independent Gamma random variables

Let $X \sim \operatorname{Gamma}(a,\lambda)$ and $Y \sim \operatorname{Gamma}(b,\lambda)$ being independent. Find the PDF of the ratio $W=X/Y$. I found $$ f_W(w) = \frac{\Gamma(a+b)}{\Gamma(a) + ...
1
vote
2answers
53 views

How do I find the cdf of $X_1 + X_2$?

$X_1$ uniform $(0,1)$ and $X_2$ uniform $(0,2)$ $$ \begin{cases} f(x_1,x_2) = \frac{1}{2}, &\quad \mbox{for} \ 0<x_1<1, 0<x_2<2 \\ 0, & \quad \mbox{otherwise} \end{cases} $$ ...
3
votes
2answers
39 views

Finding the median given the PDF $f(x) = cx^2$.

I'm new to stats and I facing problems in finding the median of a PDF. I have to find the median of this PDF $$ f(x) = \begin{cases} cx^2, & \text{if 0 $\le$ $x$ $\le$ 3} \\ 0, & ...
0
votes
2answers
33 views

Computing the probability of waiting someone - Uniform distribution

I have the following problem and I having trouble in finding it solution. I need a hint. The problem: Two people arranged to meet between 12:00 and 13:00. The arriving time of each one is i.i.d. and ...
0
votes
0answers
47 views

How to calculate the probability that $X_n$ is not the largest observation in the sample?

I am trying to solve the following problem: Let $X_1,\dots, X_n$, where $n > 4$, be independent random variables such that $X_i ∼ N(i, i)$ for $i = 1, \dots, n$. Let $\bar{X} = ...
3
votes
1answer
706 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 ...
1
vote
1answer
38 views

Flip $n$ coins, discard tails, and continue until $k$ heads remain

Consider the following game: $n$ participants have a fair coin each, on a given round, the not already discarded participants flip their coins, those who flip a tail are discarded from the game, the ...
3
votes
1answer
47 views

Showing 2 Distributions are the Same

Let $X_1, X_2, \dots$ be i.i.d. exponentially distributed RVs. For $n = 1,2,\dots$ consider: $Y_n = \max(X_1, \dots X_2)$ $U_n = \sum_{i=1}^{n}\frac{X_i}{i}$ Show that $Y_n$ and $U_n$ have the same ...
1
vote
1answer
779 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 ...