0
votes
1answer
22 views

Find joint distribution function in region

I can't for the life of me figure this one out, I am stuck on part (c) ... I have this as my starting point ? $$ \frac{45}{304}\int_0^x\int_{2-x}^2 u^2v^2\,\mathrm{du} \mathrm{dv} $$ Here is my ...
2
votes
2answers
72 views

Log normal distribution - Where am I wrong?

Let $X$ be a R.V whose pdf is given by $$f(x)=p\frac{1}{\sqrt{2\pi\sigma_1^2}}\exp\left(-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right)+ ...
3
votes
2answers
81 views

How to give rigorous proofs of these two limit statements?

Let $X$ be a random variable with cumulative distribution function $F(x)$. Then how to rigorously prove the following two limit statements? $\lim_{x \to - \infty} F(x) = 0$. $\lim_{x \to + \infty} ...
0
votes
0answers
15 views

An example of $k$-independent distributions.

I'm trying to better understand the idea of $k$-independence in distributions. The idea is that a distribution $\mu$ over $\{0,1\}^n$ is $k$-independent if any restriction of $\mu$ to $k$ variables ...
0
votes
1answer
14 views

normality of data

Does the qqplot below suggest that the data is normally distributed? The fact that it's nearly perfectly linear is to me an indication of normality. However, the Anderson-Darling test for some reason ...
4
votes
1answer
55 views

Uniform sampling with replacement item frequency

Suppose we are sampling from $N$ distinct items uniformly with replacement $M$ times. What can be said about the distribution of frequencies of items drawn? For example, if I sort all the frequencies ...
1
vote
1answer
37 views

Question about the Bayesian Inference of a parameter

In order to understand the difference between the Frequentist and Bayesian inference, I was reading the presentation at: http://www.stat.ufl.edu/archived/casella/Talks/BayesRefresher.pdf . In order to ...
0
votes
0answers
38 views

Prove Logarithmic function is part of exponential family

The aim is to prove that the logarithmic distribution with parameter $p (0<p<1)$ is part of the exponential family and hence, give its canonical parameter. To prove a distribution is part of ...
1
vote
1answer
22 views

Show that statistic is (not) sufficient

I need to verify ifthe statistic $|X|$ is or npt sufficient for $\mu$, if $ X \sim N(\mu, 1)$ Using the definition, I've obtained the pdf of X given $ T(X)=|X|:$ $$f_{X|T}(x|t) = ...
0
votes
0answers
8 views

Given MTTF and Number of Items, how to calculate failing parts with Time?

I am wondering how to make use of MTTF Here is the situation, I am given an MTTF for an item type x and a certain demand for that item in the next 25 years, say 100 parts that will be in operation ...
0
votes
1answer
22 views

Let X be a random variable with PDF fx. Find the PDF of the random variable |X| in the following

Here's my question: X is uniformly distributed in the interval $[-1,2]$. Find pdf of $|X|$... So I did P($|X| \le x$) = P($-x \le X \le x$)... From here I'm not too sure how to proceed. I know the ...
2
votes
2answers
73 views

$X = (X_1, X_2)$ is it not a multivariate random variable?

$X=(X_1,X_2,\ldots, X_P)$ is a $p$-dimensional random variable on $(\Omega, S, P) $ iff $X_i$'s are univariate random variables on the same probability space $(\Omega, S, P)$ ." We all know ...
2
votes
1answer
66 views

Proving $E_{\theta}[T(X)] = \frac{\psi'(\theta)}{\eta'(\theta)}$

I'm trying to understand how to prove the following theorem: Let $\{P_{\theta}, \theta \in \Theta\}$ be a family of distributions in the one parameter exponential family with density (pmf) ...
0
votes
1answer
25 views

Pareto Distribution transformation

Suppose $X$ is a random variable with Pareto distribution. There pdf and cdf are: $$f_X(x) = \begin{cases} {\alpha x_m^\alpha \over x^{\alpha +1}}, & \text{if $x\ge x_m$ } \\ 0, & \text{if ...
0
votes
0answers
34 views

Determining $\sigma$ given mean and proportion of a Normal distribution?

The marks of a random sample of students with mean $\mu$ and standard deviation $\sigma$ showed that 15.87% scored higher than 70. The distribution of the marks is Normal with mean $50$ standard ...
0
votes
0answers
25 views

probability that a variable, as a function of choice variables, is among the top k out of n when ordered

Suppose $(h_1,h_2,...,h_n)'$ is an $n\times 1$ vector. Let $h_i=g_iX_i$, where $g_i$ is a choice variable which can vary across $i$ and $X_i$ is a random shock with Pareto Type I distribution. ...
0
votes
1answer
12 views

Choice of distribution given data

I have some data to be analyzed. It's histogram looks unimodal, with the support being positive reals between 0 and 100, most of the values huddled up around the mode.I want to be able to ...
1
vote
1answer
37 views

Probability distribution for n-th smallest value in array with random values

I have a $d$-dimensional ($d>3$) array (vector, set, ...) which is filled with random values taken from uniform distribution (interval $[0,1]$). What is the probability distribution for n-th ...
1
vote
1answer
36 views

Probability that a random variable is among the top k out of n when ordered

Suppose $X_1,X_2,\ldots,X_n $ are $n$ i.i.d. random variables with a continuous distribution $F(x)$ and density function $f(x)$. What is the probability distribution that any given $X_i$ is among the ...
1
vote
2answers
32 views

Exponential distribution - Using rate parameter $\lambda$ vs $\frac{1}{\lambda}$

Sometimes I see the exponential distribution defined as follows: $$f(x) = \lambda e^{-\lambda x}$$ when $x > 0, 0$ otherwise I have also seen it defined like so: $$f(x) = \frac{1}{\lambda} ...
0
votes
1answer
42 views

Find the distribution ,when parameter is random

Let $X$ be the number of coin tosses until heads is obtained. Suppose that the probability of heads is unknown in the sense that we consider it to be a random variable $Y \in U(0, 1)$. $(a)$ Find the ...
1
vote
1answer
30 views

Computing P-value

In a book, from a sample they derived Mantel-Haenszel chi-square statistic $$\chi_1^2=1.41$$ And it is written that : this $\chi_1^2=1.41$ is associated with a one-sided P-value between $0.10$ and ...
0
votes
0answers
22 views

Probability Distribution in Cumulative Follow-Up Study

Data layout for a cumulative type of follow-up study is : $$\text{table 01. Data layout for a cumulative follow-up study}$$ $$ \begin{array}{l|cc|l} & \text{Exposed}(E) & ...
0
votes
1answer
36 views

Variance of sum of multiplication of independent random variables

Suppose that we have $Z=\sum_{i=1}^n (a_i+b_iX_i)(c_i+d_iY_i)$. Where $a_i,b_i,c_i$ and $d_i$ are real numbers and $X_i$s and $Y_i$s are all independent random variables. How can I find the variance ...
1
vote
1answer
91 views

Relation between two distributions expressed in terms of their CDFs

Not great at stats, and having trouble wrapping my mind around this. Would love an explanation, not overly detailed, in plain english of what these transformations mean. The bias correction ...
1
vote
2answers
76 views

The sum $Y$ of independent Bernoulli variables with Poissonian upper limit $N$ is independent of $N-Y$

The random variables are $N,X_1,X_2,..$ are independent,$N \in po(\lambda)$ and $X_k \in Be(1/2)$, $k \geq 1$. Set $Y_1 = \sum\limits_{k=1}^{N}X_k $ and $Y_2 = N - Y_1$. Here $Y_1 = 0$ for ...
3
votes
1answer
40 views

Expectation of first and second order statistics in a random distribution

Let $E(f_{i}^{n})$ and $E(s_{i}^{n})$ denote the expected first and second order statistics for $n$ draws from the distribution $V_i$ .i.e set $X_{i}^{n}=\{x^1,.....,x^n | x^j \sim V_i \}$ and let ...
0
votes
0answers
15 views

Modeling Counts with Small Number of Observations

I have a large data set that contains $5$ different fields. The fields are ...
0
votes
1answer
33 views

Hypothesis Testing, P-value, T-test Statistic, Confidence Interval

I am writing a report for my class project. I am taking statistics and I am REALLY panicking with the results I have in my report. I do not think my calculations for t-test statistic or confidence ...
2
votes
1answer
40 views

Deriving a lower bound for a probability involving a random variable $X$ with the Gamma distribution.

Question Let $X$ have the $Gamma(\alpha, \beta)$ density. I.e. $$f_X(x) = \frac{1}{\gamma(\alpha)\beta^\alpha}x^{\alpha-1}e^{-\frac{x}{\beta}}$$ when $x >0$ and $0$ elsewhere. The moment ...
1
vote
1answer
19 views

Histogram with different sample probabilities

Assume we are given a list of samples $L_1,L_2,\ldots,L_n$ of some random variable $L$. By classing them into bins we can easily create a standard histogram. But now suppose that we associate a ...
1
vote
0answers
26 views

Working with the sum of two independent random variables, and estimating a parameter

A network source sends a sequence of zeros and ones, $X_1, X_2, ...$ with $X_i$(iid) Bernoulli with $p = P(X_i = 1), 0 < p < 1$. Due to disturbances the received sequence is $Y_1, Y_2, ...$ ...
1
vote
1answer
58 views

Exponential of Squared Brownian Motion

Long time lurker, first time posting! Have a problem, that looks familiar but I can't put my finger on it. Need to calculate $\mathbb{E} [\exp(aW_T^2)|F_t]$ where $W_t$ is an $F_t$ adapted standard ...
0
votes
1answer
53 views

Transformation theorem, Cauchy distribution

I have derived the density for the ratio of two independent random variables,via the transformation formula. In this way: $V = X/Y $ and $ U = X $ inversion yields: $Y = U/V$ och $X =U$ , the ...
0
votes
0answers
13 views

Sum of a truncated normal random variable and normal random variable (correlated) [duplicate]

I'm wondering if there is a closed form of pdf of sum of a "correlated" normal random variable and a truncated normal random variable. I found a paper providing the pdf for "uncorrelated case" but I ...
0
votes
0answers
34 views

How to estimate the covariance matrix if the unnormalized pdf is known but integral is intractable?

Assume a $d$-dimensional random vector $x$, whose unnormalized pdf is known as the product of N multivariate t-distribution: $$Pr(x)\propto\prod_{i=1}^nt_{\nu_i,\mu_i,\Sigma_i}(x)$$ Is there any ...
0
votes
1answer
36 views

Understand step in computing marginal distribution of restricted Boltzmann Distribution

Proof taken from http://image.diku.dk/igel/paper/AItRBM-proof.pdf (page 24) I understand everything up to and including: (1) $$p(\textbf{v}) = \frac{1}{Z}e^{\sum_{j=1}^mb_jv_j} \prod_{i=1}^n\sum ...
0
votes
1answer
26 views

A question about $\chi^2$ distribution

Ok, i have a question but i start with a definition first so that one can get the context. (All variables in question have the same variance and under $H_0$ which we are considering - they have the ...
0
votes
1answer
38 views

Method of moments for Beta $(\alpha_1,\alpha_2)$ distribution

I am trying to solve for the first two moments of a Beta$(\alpha_1,\alpha_2)$ distribution. We know that the first moment is equal to: $\mu_1 = \frac{\alpha_1}{\alpha_1+\alpha_2}$ and the second ...
1
vote
1answer
26 views

generating random samples with a PDF

I have the PDF of a distribution from which it is not possible to get a closed from for the CDF or inverse CDF. Is there a technique that would allow me to generate samples from this distribution ...
0
votes
1answer
19 views

Infimum of Gamma distribution

Let $X$ be a Gamma random variable with the CDF $F_X(x)=\frac{1}{\Gamma(\alpha)}\gamma(\alpha,\beta x)$ where $\Gamma(x)$ represent the gamma function and $\gamma(a,b)$ denotes the lower-incomplete ...
3
votes
0answers
31 views

Gamma distribution Norming constant for extreme minima

the norming constants for extreme maxima of Gamma distribution is known and is give in link.springer.com/article/10.1007/s10687-010-0125-3. I would like to know is there reference or paper that states ...
3
votes
1answer
64 views

What is the joint probability distribution of number of balls after $n$ draws?

The following problem came into my mind when I am studying the Polya Urn Model. At the beginning, from a bin containing $c_1$ balls labeled $1$, $c_2$ balls labeled $2$, … , $c_m$ balls labeled $m$, ...
0
votes
1answer
37 views

Normalizing constants for Extreme value distributions

I have a question regarding the normalizing constants $\mu$ and $\sigma$ that appear in the following problem. Let the random variable $Y_n$ be $Y_n=max(a_1,a_{2},\cdots, a_n)$ and $X_{n}$ be ...
0
votes
0answers
26 views

Notation related to Markov kernels

We wish to jointly construct two copies $(X_n)_{n \in \mathbb{N}}$ and $(Y_n)_{n \in \mathbb{N}}$ of a Markov chain on general state space, s.t. for $n=1,2,...$ $\mathcal{L}(X_{n+1}|X_n) = ...
0
votes
1answer
30 views

Limiting distribution of $n(T_n-4p^3(1-p))$

I want to find the limiting distribution of a $n(T_n-4p^3(1-p))$, where $T_n=\displaystyle\frac{4(n-t)t(t-1)(t-2)}{n(n-1)(n-2)(n-3)}$ with $t=\sum X_i$ is the UMVUE of $4p^3(1-p)$ that I found, where ...
5
votes
2answers
50 views

Deriving Moment Generating Function of the Negative Binomial?

My textbook did the derivation for the binomial distribution, but omitted the derivations for the Negative Binomial Distribution. I know it is supposed to be similar to the Geometric, but it is not ...
3
votes
1answer
49 views

Interesting Problem - Computing CDF

A rv X is an exponential distribution with parameter 1 and Y is a uniform distribution between 0 and 1. X and Y are independent. Define Z = min {X, Y}. Compute the CDF of Z ? I really have no idea ...
0
votes
0answers
36 views

One double integral elated problem

The bit I am stuck is the limits in the double integral. I tried X from 0 to uy and Y from 0 to infinity, this is obviously incorrect. I just want to know the complete double integral in the order ...
8
votes
2answers
136 views

Lies, damned lies, and statistics

A story currently in the U.S. news is that an organization has (in)conveniently had several specific hard disk drives fail within the same short period of time. The question is what is the likelihood ...