1
vote
1answer
28 views

Show expectation is infinite

Let $X_1,\ldots,X_n$ be independent, identically distributed with expectation 1 and finite variance. Find the limit distribution of $\sqrt{n}(\bar{X}_n^{-1}-1)$. If the random variables are sampled ...
0
votes
0answers
14 views

Difficult Survey Sampling question

Question: A Secretary of State wants to survey the primary owners of motorcycles registered in the state to estimate the proportion who want the license plates redesigned. (Primary owner means that ...
-1
votes
1answer
51 views

Suppose $X\sim U[0,1]$ and $P(Y=1| X=x)= x = 1-P(Y=0| X=x)$. Find the expectation and variance of $Y$ [on hold]

Suppose $X\sim U[0,1]$ and $P(Y=1 \mid X=x)= x = 1-P(Y=0 \mid X=x)$. Find $E[Y]$ and $\operatorname{Var}[Y]$
-1
votes
0answers
4 views

Measure to compare quality of synthetic data generated?

What is a good measure to compare the quality of the synthetic data generated with respect to the original data? The synthetic data I have, is the scaled up version of the original data. I am confused ...
0
votes
2answers
28 views

Ratio of Gamma random variables

If $X_i$, $i=1,2$ are independent gamma$(\alpha_i,1)$ random variables, find the distribution of $\frac{X_1}{X_1+X_2}$ and $\frac{X_2}{X_1+X_2}$. Attempt: Let $Y_1 = \frac{X_1}{X_1+X_2}$ and ...
1
vote
1answer
21 views

Finding distribution of random variable if X is exponential $(1)$

Let X be an exponential (1) random variable, and define Y to be the integer part of X+1, that is $\hspace{15mm}Y=i+1$ if and only if $\hspace{5mm}i \leq X \leq i+1, i = 0,1,2,...$. Find the ...
1
vote
0answers
21 views

Correlation and First Order Stochastic Dominance

Suppose we have a random variable $X \sim [0,1]$ with a continuous distribution $F_X(x)$. Suppose $I \in \left\{0,1\right\}$ is a discrete random variable with $\text{Prob}(I=1 \ | \ X=x)$ strictly ...
0
votes
1answer
18 views

(Statistics)Probability of given sum in dice tossing [on hold]

I need some help with this problem: By tossing two dice, what is the probability of: i) Total sum of 7 ii) Difference of 5 iii) Total sum multiple of 7 Thanks everyone ~Chris
0
votes
1answer
20 views

Showing something converges, in distribution, to a normal distribution

I'm not sure how relevant the first few parts are, but I will post it just in case... $(X_i,Y_i), i=1,\dots,n$ are independent where $X_i$ has an exponential distribution $\mathcal{E}(\lambda_i)$ ...
2
votes
1answer
45 views

What is the bound on $E\|Y_n\|^4$ in terms of $n$?

Let $X_n,n\in\mathbb{N}$ be i.i.d. zero-mean random variables in some separable Hilbert space with $E\|X_n\|^8<\infty$ and $Y_n=\frac{1}{n}\sum_{i=1}^nX_n$. I need to find bounds on $E\|Y_n\|^4$. ...
2
votes
1answer
33 views

Using the inverse Gaussian integral to find percentiles

I need some help with the following: Let $$R=\mu+\sigma*\epsilon \hspace{1cm} \epsilon \sim N(0,1)$$ I want to argue that $$ \mu + \sigma*\Phi^{-1}(u)$$ are the percentiles of the model when ...
0
votes
0answers
17 views

Finding a sufficient statistic for an iid sample of the Gumbel distribution

$G(x;\alpha, \beta) = \exp\{-\beta e^{-\alpha x}\}$ for $x \in \mathbb{R}$ is a distribution (Gumbel family). Side question: is $G(x;\alpha, \beta)$ a member of the exponential family? I do not think ...
-2
votes
1answer
29 views

Probability and Induction help [closed]

Let $Y=X_1+X_2+ \cdots+X_n$ where $X_1, X_2, \ldots, X_n$ are independent Bernoulli random variables, each with probability of success equal to $q$. Use induction to prove that $Y$ has a Binomial ...
-2
votes
2answers
31 views

Proof Question- Need Help [closed]

Show that if $P(A|E) \geq P(B|E)$ and $P(A|E^c) \geq P(B|E^c)$, then $P(A) \geq P(B)$. I am reviewing for test, and I came across this problem in the textbook. I need help with this question.
0
votes
1answer
13 views

Related to chi-squared functions

I'm finding difficulty in finding what type of function it is in continuous distributions in probability.Mainly how can i identify whether a function is chi-squared or not?
0
votes
0answers
15 views

MAP for exponential function (Maximum a posteriori)

I am trying to find the MAP for an exponential function of the form $p(y) = \theta.e^{{-\theta}y}$ Given that $\theta$ is constant, I want to estimate maximum $y$ = $p(y).p(X=x_i|y)$ for $i = 1..n$. ...
1
vote
0answers
18 views

Finding $a_n, b_n$ so that a sequence converges in distribution to a nondegenerate random variable.

Now, $X_1, X_2,\dots$ are iid with the same distribution as the chi-squared distribution with one degree of freedom. Find $a_n$ and $b_n$ so that $a_n \left( \max_{1 \leq i \leq n} X_i - b_n \right)$ ...
0
votes
1answer
26 views

What are the odds that every team that lost the prior week would be facing a team that won the prior week?

I apologize in advance for a potentially elementary question but I cannot figure out how to even begin with this. We have 10 teams Half lost the first week, the other half won Second week, we all ...
0
votes
1answer
19 views

Expected Residual lifetime

I have a 2 part question. I was able to figure out part 1. I need some help with part 2. I will write out part 1 (and my solution) for completion. Let $T$ be a continuous survival time with survival ...
0
votes
1answer
17 views

Average Waiting Time for a General Process

The time between the arrival of two consecutively buses are independent and averages out to be $T$. A passenger arrives at a uniformly distributed random time independent of the bus arrival time. Can ...
0
votes
0answers
13 views

Finding test of critical region for sum/variance of normal distributions

Let $Y_1,....,Y_n$ denote independent, identically distributed random variables such that $Y_1$ has a normal distribution with mean $\theta$ and standard deviations $\theta$, where $\theta$ > 0. ...
2
votes
0answers
20 views

Showing distribution has a $\chi^2$ distribution with df = n

Let $X_1,X_2,....,X_n$ denote independent identically distributed random variables such that $X_1$ has density $p_1(x;\theta)$ where $\hspace{15mm}p(x;\theta) ...
5
votes
0answers
70 views

Finding an upper bound for $\frac{d}{d\theta}\beta^*(\theta)|_{\theta=\theta_0}$

Suppose that a random variable X has a distribution depending on a parameter $\theta$, $\theta \in \Theta$, and consider a test of hypothesis $H_0: \theta = \theta_0$ versus the alternative $H_1: ...
1
vote
1answer
28 views

Survival function in terms of the hazards

Let $T$ be a discrete random variable assuming values $x_1<x_2<\ldots$ with probability $f(x_i) = \text{Pr}\left\{T=x_i\right\}$. Show that the survival function $S(t)=\prod_{i:x_i\leq ...
1
vote
1answer
14 views

Consistency of kernel density estimator with constant bandwidth

Let ($x_1, ..., x_n$) be i.i.d. samples drawn from some distribution $P$ with an unknown probability density function $f$. Its kernel density estimator is \begin{align} \hat{f}_h(x) = ...
0
votes
2answers
51 views

A random number of random variables, (expectation help)

everyone First a definition let $S_N = X_1 + X_2 \cdots + X_N$ where $X_i$'s are random variables and $N$ is also a random variable. Also assume that the $X_i$'s(integer valued),independent ...
0
votes
1answer
14 views

How to calculate the median and the quantiles of this distribution?

A median of a distribution of one random variable $X$ of the discrete or continuous type is a value of $x$ such that $P(X < x) \leq 1/2$ and $P(X \leq x) \geq 1/2$. If there is only one such $x$, ...
-1
votes
0answers
26 views

Hypercontractivity of Markov Operator

I have been reading a paper by Ahlswede and Gacs on hypercontractivity of Markov operator (see here 1) and its application to information theory. To be honest, I could not fully understand the ...
1
vote
0answers
38 views

Determining the Cramer-Rao lower bound

Let $X = (X_1,\dots,X_n)$ be a vector of iid variables from the smooth density $f(x,\theta_0), \theta_0 \in \Theta \subset \mathbb{R}$. Let $L(\theta)$ be the likelihood and $I(\theta)$ the ...
3
votes
2answers
82 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} ...
4
votes
1answer
59 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 ...
3
votes
0answers
59 views

Jacobian for a matrix transformation: Example of Cholesky decomposition

I would like to generally understand how the Jacobian of a matrix transformation can be computed. As a concrete example, consider the Transformation from a (correlation) matrix to its Cholesky ...
0
votes
0answers
50 views

Writing probability as log

I have a question regarding the log probability and I am confused on this. The question is: $$\hat P^{(t)}(x)=\sum_{i=1}^N v_i^{(t)}P_i^{(t)}(x)$$ which is some function of size $N$. The question ...
0
votes
1answer
31 views

How to check hypothesis in statistical data?

I have a statistical problem. In a city there are some hostels which differ by the number of rooms. The input data are the following. In a table there is information about hostels and corresponding ...
4
votes
1answer
145 views

Unbiased asymptotic variance

Problem: Let $X_1,...,X_n$ be indep. r.v.'s that satisfy, for $i = 1,...,n$, $E(X_i) = \mu_i(\theta)$ & $\mathrm{Var}(X_i)= \sigma_i^2(\theta)$. $\theta$ is the parameter of interest and the ...
1
vote
2answers
62 views

rationalwiki on “Extraordinary claims require extraordinary evidence”

I don't have a strong background in probability/statistics and I'm trying to understand the example at ...
0
votes
1answer
32 views

Probability for incomplete information

Let's say there are 10 teams: A-J. Only 1 team wins, others lose. Probability of any team to win is unknown (different for each team) and to be calculated. Not all teams participate in each game. ...
0
votes
1answer
38 views

How to calculate probability that a team will win

Let's say there are 10 teams: A-J. Each team always participate in each of the game. Only 1 team wins, others lose. Probability of any team to win is unknown (different for each team) and to be ...
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
71 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) ...
1
vote
1answer
23 views

How to make this bet fair?

A person bets $1$ dollar to $b$ dollars that he can draw two cards from an ordinary deck of cards without replacement and that they will be of the same suit. How to find the value of $b$ so that the ...
1
vote
1answer
25 views

How to compute of each player winning this sequence of games?

Players A and B play a sequence of independent games. Player A throws a die first and wins on a "six." If A fails, then player B throws and wins on a "five" or "six." If B fails, then A throws and ...
1
vote
3answers
48 views

How to compute these probabilities?

A pair of dice is cast until either the sum of seven or eight appears. How to compute the probability of a seven before an eight? Now, if this pair of dice is cast until a seven appears twice or ...
0
votes
2answers
50 views

How to compute this probability?

A drawer contains eight different pairs of socks. If six socks are drawn at random and without replacement, how to compute the probability that there is at least one matching pair among these six ...
0
votes
0answers
19 views

How to establish the independence or otherwise of these compound events?

Suppose that $C_1$, $C_2$, $\ldots$, $C_n$ are mutually independent events in a sample space $S$. Then how to establish the independence or otherwise of these combinations of events? $C_1^c$ and ...
0
votes
4answers
117 views

What is the probability that at least one letter is in the correct envelope?

A secretary types three letters and the three corresponding envelopes. In a hurry, he places at random one letter in each envelope. What is the probability that at least one letter is in the correct ...
1
vote
2answers
33 views

How many bulbs should be inspected for probability to exceed $1/2$?

In a lot of $50$ lightbulbs, there are $2$ bad bulbs. How many bulbs should be examined so that the probability of finding at least $1$ bad bulb is at least $1/2$? My effort: Suppose $n$, where $0 ...
1
vote
1answer
42 views

Confidence interval for estimating probability of a biased coin

Suppose we have a coin with a probability $p$ of coming up heads and $q = 1-p$ of coming up tails on any given toss. (A coin is biased unless $p=0.5$). But we are not given what $p$ or $q$ are. We ...
1
vote
0answers
7 views

Sampling with an “oversampling” factor, in K-Means||

I'm trying to understand K-Means||, a scalable version of K-Means++, which itself is an "improved" version of the clustering algorithm K-Means. Please find here the link to K-Means|| paper ...
0
votes
1answer
41 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 ...