# Tag Info

### 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, ...
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### 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 (...
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### 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.
1 vote
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### 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 ...
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### 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 ...
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### 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 $0\le z \le 1$. ...
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### 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{...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### concentration of maximum of gaussians

For the sake of completeness, I will prove here the inequality $$P(\|X\|_\infty \geq \sqrt{2 \log (2n)}+t)\leq \frac{1}{2}\exp(-t^2 /2) \quad (t>0),$$ which is slightly ...
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### 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 ...
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### 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"...
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### 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 $1$ ...
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