Tagged Questions

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 ...
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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 ...
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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 $\operatorname{Var}[Y]$
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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 ...
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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 ...
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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 ...
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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 ...
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(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
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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 $\mathcal{E}(\lambda_i)$ ...
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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 $E\|Y_n\|^4$. ...
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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 ...
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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 ...
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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 ...
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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.
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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?
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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 $i = 1..n$. ...
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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 $a_n \left( \max_{1 \leq i \leq n} X_i - b_n \right)$ ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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 ...
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 ...
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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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$X = (X_1, X_2)$ 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 ...
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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) ...
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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 ...
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 ...
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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 ...
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 ...
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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 ...