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Given a 95% confidence interval why are we using 1.96 and not 1.64?

The Z-score for 90% is -1.6444 and 1.6444. , 1 sigma The z-score for 95% is -1.96 and 1.96 , 2 sigma The z-score for 99% is -2.5758 and 2.5758 , 3 sigma in your Estimation of Population Mean meaning, ...
JeeyCi's user avatar
  • 101
1 vote

What is the (fully rigorous) definition of a confidence interval?

This adds minor measure theory details to Amir's answer (which I have upvoted). For the probability space $(\Omega, \mathcal{F}, P)$, it is not $\theta$ that is random, but the set $C_n$ itself: $\...
Michael's user avatar
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1 vote

What is the (fully rigorous) definition of a confidence interval?

Just to get some notation really explicit, $\{P_\theta : \theta \in \Theta\}$ is an indexed family of probability distributions on $\mathbf{R}^n$, and $a$ and $b$ are measurable functions $\mathbf{R}^...
Damian Pavlyshyn's user avatar
2 votes

What is the (fully rigorous) definition of a confidence interval?

Here is a slightly more general notion of coinfidence set. At issue is that statements such as $P[\theta \in C(X)]$ are not really probabilistic statements about $\theta$, since in the classical (...
Mittens's user avatar
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1 vote
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Randomized Trace Estimation

Let $Y = x^{\top} M x$, where $x$ is as defined in the blog. To apply the Bernstein inequality, we need to find (an upper bound on) the following quantities: (i) A value $B$ such that $|Y - \mathbb{E}[...
sudeep5221's user avatar
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7 votes

What is the (fully rigorous) definition of a confidence interval?

Let $$T_1=g_1(X_1,\dots,X_n)$$ $$T_2=g_2(X_1,\dots,X_n)$$ be two statistics where $$X_1,\dots,X_n\sim F_\theta.$$ Then, $[T_1,T_2]$ is called a confidence interval (also called interval estimator) ...
Amir's user avatar
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0 votes

Is Standard Deviation the same as Entropy?

The standard deviation and the entropy are not the same, but a transformation of the standard deviation, the coefficient of variation ($CV_Y := \frac{\sigma_Y}{\mu_Y}$), is part of the single-...
gjmb's user avatar
  • 13
0 votes
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How to Combine Joint PMFs Using an Indicator Variable

If a function $f(k,h)$ satisfies $f(k,0)=a(k)$ for $k\in\{0, ..., n\}$ and $f(k,1)=b(k)$ for $k \in \{0, ..., n\}$, you can equally express this in the following three acceptable ways: Just say $f(k,...
Michael's user avatar
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2 votes
Accepted

Does $\sqrt{n}$-consistency imply consistency?

Let $\varepsilon,\delta >0$. By assumption, there exist $N_\delta\in \mathbb{N},M_\delta>0$ s.t. for all $n\geq N_\delta$ we have $P(\sqrt{n}|\theta_n-\theta|>M_\delta)<\delta$. So $$\...
Snoop's user avatar
  • 15.6k
1 vote

Efficient and unbiased estimation of the location ($\mu$) of truncated normal distribution with known scale ($\sigma^2$) and truncation points

One observation is not enough to infer $\mu$ with much accuracy at all. I think you are inherently going to be limited by your lack of data. I suspect the best you can do is take a Bayesian approach. ...
D.W.'s user avatar
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1 vote

When to use t-distribution?

Perhaps the ambiguous description "z-distribution or lambda as it is also know" is something associated with a Poisson random variable. There may be a cool way to solve this problem using ...
Michael's user avatar
  • 24.4k
1 vote

Probability that maximum of two iid Unif(0, 1) r.v.s is less than the minimum of two other iid Unif(0,1) r.v.s?

Making explicit the idea in Henry's comment, note that by exchangeability (which holds since the random variables are i.i.d.) we have $$P(X_{\sigma(1)} < X_{\sigma(2)} < X_{\sigma(3)} < X_{\...
Mauro's user avatar
  • 916
0 votes

the first principal component’s variance

PCA is based on the covariance matrix $S = X^T X / N$. In (3.48) the division by $N$ was omitted and in (3.49) it is again accounted for.
Markéta Makarová's user avatar
1 vote
Accepted

Calculating a Conditional expectation

But this only left me with $$E[X_i|X_{\max}]=X_{\max}\mathbb{P}(X_i=X_{\max}|X_{\max})+E[X_i\mathbf{1}_{\{X_i<X_{\max}\}}|X_{\max}],$$ which doesn't really gelp me I think. Mixing the notation ...
Graham Kemp's user avatar
0 votes

Bayesian Estimation Statistics MAP

We are given the prior density $\pi(\rho)$. Let $d$ denote the events given by the observations (data) that is 70 successes out of 110 trials. Assuming independent trials, the probability of this is ...
whpowell96's user avatar
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0 votes

Probability that maximum of two iid Unif(0, 1) r.v.s is less than the minimum of two other iid Unif(0,1) r.v.s?

Assuming that all four variables are independent, we know that $$ \mathbb{P}(\max(X_1,X_2)\le z)=\mathbb{P}(X_1\le z,X_2\le z)= \mathbb{P}(X_1\le z)\mathbb{P}(X_2\le z)=z^2 $$ for $0\le z \le 1$. ...
van der Wolf's user avatar
  • 3,089
6 votes
Accepted

Understanding summation identity

There is a typo in the RHS. The $u_{j-s}$ in the middle sum should instead be $u_{s-j}$: \begin{align} &\quad \sum_{k=0}^q \theta_k \sum_{j=1}^n u_j + \sum_{s=1}^q \sum_{j=s}^q \theta_j \color{...
RobPratt's user avatar
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6 votes
Accepted

Given the maximum likelihood function- estimate the value of the parameter

$\beta$ is the minimum of this Pareto distribution: the question says the density is $0$ below $\beta$ so there is zero probability of observing data below $\beta$. Suppose for example you knew $\...
Henry's user avatar
  • 158k
4 votes

Given the maximum likelihood function- estimate the value of the parameter

As stated in the solution, the maximum likelihood function, for a fixed $\alpha$, is increasing in $\beta$. Thus, for some fixed sample $(x_1,\ldots,x_n)$ we want to make $\beta$ as big as possible. ...
Julio Puerta's user avatar
  • 8,427
1 vote

distribution of sample mean to sample standard deviation

$\def\ed{\stackrel{\text{def}}{=}}$ Your expression $\ \frac{X_1+X_2+\dots+X_k}{\sqrt{\frac{X_1^2+X_2^2+\dots+X_k^2}{k}}}\ $ is not the ratio of the sample mean to the sample standard deviation, as ...
lonza leggiera's user avatar
1 vote
Accepted

What exactly is P-value and what is its relation with the significance level?

Here is the general form of a $p$-value computation: you decided on a summary statistic ("proportion of times $6$ was rolled") which maps the space of available data ("sequences of dice ...
Patrick Stevens's user avatar
1 vote

What exactly is P-value and what is its relation with the significance level?

It's basically giving the null hypothesis the benefit of the doubt. Essentially, for a 5% significance level, if there is a $\geq$5% chance that, if the null hypothesis is true, we would get the ...
John's user avatar
  • 175
1 vote
Accepted

Don't events have to be independent in order for you to multiply their probabilities?

I think if you see where the terms $P(A|B)P(B)$ and $P(A|B)P(B^{\star})$ come from, you'll see why it has nothing to do with independence. Event $B$ is defined as the event of high pollen levels. That ...
Paul Ash's user avatar
  • 1,350
0 votes

Aggregates and Psuedorandomness

If $0<N<2^{32}$ then there exist subsets of size $N$ with different averages, for example the subset of the least $N$ numbers has a strictly smaller average than the subset of the largest $N$ ...
Lieven's user avatar
  • 1,247
1 vote

Calculating a Conditional expectation

Why not apply the corresponding formula, with $X=X_1$ and $Y=\max\{X_2,\ldots,X_n\}$?: \begin{gather*} E[X|max(X,Y)=u]=\frac{\displaystyle\int_{-\infty}^u\hspace{-15pt} xg(u)dP_{X/Y}(u)+\displaystyle\...
Speltzu's user avatar
  • 593
2 votes

Calculating a Conditional expectation

Let $V=\max\{X_2,\ldots,X_n\}$. For $0\leq v\leq \lambda$ we have $$ \begin{eqnarray} F_V(v) &=& \Pr(V \leqslant v) = \Pr(\max\{X_2,\ldots,X_n\} \leqslant v) \\ &=& \Pr(X_2 \leqslant ...
Julio Puerta's user avatar
  • 8,427
0 votes

Is my understanding about sufficient statistics correct?

I think you are right. That's why $T$ is said to be sufficient: It contains everything that you need to know.
Simon's user avatar
  • 349
1 vote
Accepted

Verification of a Transformed Random Variable

Integrating the density over its support is one way to check that you in fact calculated the derivative of the CDF correctly; however, you are correct that evaluating the CDF at the upper limit of its ...
heropup's user avatar
  • 139k
0 votes

Survival function for a certain population

You've calculated $\operatorname{E}[X > 41]$ but the question asked, if someone is already $41$ years old, what is the expected number of additional years of life? That is to say, it is asking you ...
heropup's user avatar
  • 139k
0 votes

To find $k$th factorial moment from Factorial Moments Generating Function (FMGF), do I evaluate at $t=0$ or $t=1$?

Considering that $X$ (with support within $\mathbb{Z}_{\ge 0}$) has fmgf/pgf given by $$ M_X(t) := \mathbb{E} \left[t^X\right] = \sum_{n=0}^\infty t^n \,\mathbb{P}(X=n) $$ and the $k$th factorial ...
PrincessEev's user avatar
  • 44.5k
0 votes

What is Expected Prediction Error (EPE) a function of?

The two are indeed different and I understood this as we take the initial expectation over the training data, $\mathbb{E}_{\tau}$, to capture the variability amongst the possible training sets and ...
InvestingScientist's user avatar
1 vote
Accepted

Estimating the conditional entropy of a discrete variable conditioning on continuous variable

This is not trivial. Calling $X$ the outcome (discrete) variable, and $Y$ the continuous one, if $X$ is Bernoulli (two values), I would write $$I(X;Y) = h(Y) - h(Y|X) = h(Y) - (1-p) \cdot h( Y | X=0) -...
leonbloy's user avatar
  • 63.9k
0 votes

Minimum squares with two different means?

This formula you know $\displaystyle\sum_{i=1}^n (x_i-\bar{x})^2$ is fine to find $\bar{x}$. But the task is to get $\theta$ which is obviously determined by both, $X$ and $Y$. Also your ansatz $\sum_{...
m-stgt's user avatar
  • 382
0 votes

Minimum squares with two different means?

In general given a set of data $X=(X_1,...,X_n)$ s.t. $X_i\sim p_i(x|\theta)$ the Least Squares esitmator of $\theta$ is the value $\hat{\theta}$ that minimizes $||X-\mathbb{E}_\theta [X]||^2_2$. In ...
cespun's user avatar
  • 49
3 votes
Accepted

Almost Sure Convergence of Rayleigh Distributed Random Variables

A simple calculation tells us that $$\mathbb{P}( X_1 > x ) = e^{ - x^2/2}.$$ It follows that $$\mathbb{P}( X_n > 1/\sqrt{n} ) = \mathbb{P}( X_1 > \sqrt{n} ) = e^{ - n/2}.$$ In particular, $$\...
Samuel Johnston's user avatar
0 votes

FOB Poker: Probability of a sequence of numbers (with the existence of "wild-card" number)

I got the following probabilities. They are counted so that if a sequence satisfies multiple hands, it is counted in all categories (for example 3-of-a-kinds are also 2-of-a-kinds). ...
ploosu2's user avatar
  • 9,366
1 vote
Accepted

Hypergeometric-like distribution but where non-successes are replaced?

You can compute the exact distribution by tracking the state distribution of a Markov chain. The states could consist of the total number of cards picked so far and the number of specials picked so ...
HighDiceRoller's user avatar
0 votes

Distribution of sample standard deviation

A "noncentral chi-squared distribution" comes from the sum of squares of $k$ iid $N(\mu,1)$ distributed random variables. If $\mu=0$ then you get an ordinary chi-squared distribution, with $...
Henry's user avatar
  • 158k
1 vote
Accepted

How to find the power function given exponential distribution?

You are right. There is an error. The upper limit of integration must be $0.05$. The power is defined by the formula $\inf_{\theta \in \Theta_1} \mathbb{E}_{X \sim \theta}[T(X)]$ for any test $T$. In ...
温泽海's user avatar
  • 2,400
0 votes

Questions on Box and whiskers diagram

Heropup wrote a pretty good answer, but I wanted to add an answer under the assumption that all the data points would be evenly distributed within their respective quartiles. Since each box and each ...
VV_721's user avatar
  • 182
0 votes
Accepted

What is the value of $\operatorname{Var}(S^2)$, when $S^2$ is given as sample variance with denominator $n$ instead of $(n-1)$?

I will assume that $X_i$'s are IID, having finite 4th moment. Note that \begin{align*} S^2 = \frac{1}{n} \sum_{i=1}^{n} (X_i - \overline{X})^2 = \frac{1}{n} \sum_{i=1}^{n} (X_i - \mu)^2 - (\overline{X}...
Sangchul Lee's user avatar
0 votes

Is Stochastic dominance of the second degree order preserving.

Given $X\preceq^2 Y$ by definition, we have that: $$\int_{-\infty}^x F_X(u)d u \leq \int_{-\infty}^x F_Y(u)du \quad \forall x$$ Since $F_X(u) = 1-F_{-X}(-u)$: $$\Rightarrow \int_{-\infty}^x 1-F_{-X}(-...
Paolo Conorto's user avatar
1 vote
Accepted

Why do I get two different answers (probability vs statistics approach)?

The maximum likelihood principle results in a point estimate that seeks to maximize the likelihood function for the unknown parameter, given the observed data. But this is certainly not the only ...
heropup's user avatar
  • 139k
2 votes

Questions on Box and whiskers diagram

Skewness is difficult to discern from a box-and-whisker plot, so I would not assume that in general one can assert extent of skew from such diagrams. In cases where we can make assumptions about the ...
heropup's user avatar
  • 139k
0 votes

Modelling a tennis match winner using set winning probabilities

These are odds set by a company that makes money offering the chance to bet. They set those odds by looking at what people are betting along with other public or proprietary information they may have. ...
Ethan Bolker's user avatar
  • 96.3k
1 vote

concentration of maximum of gaussians

For the sake of completeness, I will prove here the inequality \begin{equation} P(\|X\|_\infty \geq \sqrt{2 \log (2n)}+t)\leq \frac{1}{2}\exp(-t^2 /2) \quad (t>0), \end{equation} which is slightly ...
Maurizio Barbato's user avatar
2 votes
Accepted

Approximating the poisson distribution using normal distribution

I think the point of the question is to appeal to the central limit theorem (CLT), which states that for i.i.d. observations $X_{i}$ with $E[X_{i}^{2}] < \infty$ (so that the variance exists), we ...
minginator's user avatar
1 vote

How to derive likelihood function

Here is a somewhat intuitive answer to hopefully help you understand the concept: the likelihood of an event is simply the probability of that event being observed: the likelihood of "Heads"...
plywood98's user avatar
1 vote
Accepted

How to find the percentage of a group that experienced a change in color.

It may be instructive to consider a numerical example. Suppose you initially have $r = 3$, $g = 2$, and $b = 5$ red, green, and blue balls, respectively. Then: $2$ red balls change to green $1$ ...
heropup's user avatar
  • 139k
0 votes

How to find the percentage of a group that experienced a change in color.

You can't know the answer without more information. To see this, assume you have two colors, red and blue, each of which have been applied to half of the balls. After the change, you still have the ...
Robert Shore's user avatar
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