New answers tagged statistics
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How many ways can a limited set of natural numbers be added together $x$ times to get $y$.
You know the denominator for computing the probability, but are stuck as to how to compute the numerator.
Your original problem (with some modification of symbols to conform to the form into which I ...
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How many ways can a limited set of natural numbers be added together $x$ times to get $y$.
This answer doesn't give a method for computing the individual probabilities that's necessarily faster, but it organizes the computation into a familiar form, namely expanding a polynomial power. This ...
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How many ways can a limited set of natural numbers be added together $x$ times to get $y$.
One way to characterize this problem is as a particular statistic drawn from a multinomial distribution. Specifically, the distribution of interest here is the trinomial distribution ($k=3$) of events ...
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Can the first and second derivatives of the Exponentially Smoothed Moving Average be calculated?
Huh, this is a problem that I solved for myself last summer. It achieves the aim of having an estimated position, velocity, and acceleration of an average. At the cost of having that average be ...
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How do I calculate the quartiles for this problem?
As explained in the previously posted answers, the multiple $7\text{s}$ are easily dealt with. Just make a sorted list of all the data, each value occurring in the list the same number of times as in ...
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Is there a quicker way to solve this probability question?
You should have drawn the Venn diagram out and seen that there is tremendous symmetry in this problem. Of course, you could have also mentally realised that there is tremendous symmetry in the problem ...
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Is there a quicker way to solve this probability question?
The working is correct, but it can be much faster with a Venn diagram.
We are given
$$\frac13 = \frac{y}{y+0.12}$$
This is annoying to solve directly. But we can convert it into a ratio, much like ...
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How to use the binomial test for obtaining an accuracy threshold for a binary classifier?
The basic idea here is that you should want the test to reject $H_0 : p = 0.5$ in favor of $H_1 : p > 0.5$ if the number of correct predictions is "sufficiently high" as to suggest that ...
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The median of a frequency distribution and $g(x)=\sum_{k=1}^n f_k |x-x_k|$
All three statements of OP about median are correct and exhaustive.
The constructed function $g(x)$ is actually $n$ times the mean deviation of the frequency distribution (FD).
OP's observation about ...
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Given 100 coin tosses, the largest string of same results in a row is...?
$\def\ed{\stackrel{\text{def}}{=}}$
If $\ L_n\ $ is the length of the longest run of consecutive flips of the same outcome occurring in a sequence of $\ n\ $ tosses of a fair coin, you can obtain the ...
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Expectation under different laws
When talking about probability, you have a measure space $(\Omega, \sigma, \mu)$ (where $\sigma$ is the sigma-algebra and $\mu$ is a probability measure). Now, random variables are just measurable ...
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Question regarding independence in set theory: Does the provided statement give independence for one set of events, or all pairs of events?
You do not care whether buying the health policy affects the choice of which of the two life polices to buy.
Since you "Assume nobody purchases both life insurances" you may as well consider ...
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Understanding convergence in law to a continuous CDF
The idea of a reasoning by contradiction may work. However, the negation of "$a_n\to a$" is rather "there exists $\varepsilon_0$ and a subsequence $(a_{n_k})$ such that for each $k$, $\...
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Question regarding independence in set theory: Does the provided statement give independence for one set of events, or all pairs of events?
The event "the participant wanted to purchase exactly only one policy" can be written
$$ (H \cap (P \cup Q)') \cup (H' \cap (P \cup Q)). $$
Note that $H \cap (P \cup Q)' = H \cap P' \cap Q'$ ...
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Obtaining One of Multiple Subsets Coupon Collector Variant
Define $C$ to be the number of the first draw on which all of the ranks have been seen in some one of the suits. We want to find $E(C)$. The approach is to find the exponential generating function $g(...
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Need a review on my proof that a sequence forms a martingale
I think one should explicitly give a definition of the reversed order of the filtration to be accurate because using the same notation for the reversed order filtration make things a bit confusing. ...
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Hypothesis testing with a sample from a shifted exponential distribution (Jun Shao - Mathematical Statistics, exercise 6.20, point (v))
I think the key point is that for a test of size $\alpha$ you want $\mathbb P(\text{Reject }H_0 \mid H_0) = \alpha$.
As you say, if $a_1 \le x_{(1)} \lt a_0$, then you will reject $H_0$. Indeed $\...
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Why isn't a t test used when comparing two proportions?
I had the same question for a long time, fwiw.
The t-test assumes that the mean and the variance of the distribution are independent (which is teh case for normal distributions). But this is not true ...
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Continuous vs Discrete Variables
A variable is discrete if its set of possible values is countable (typically integers: 0, 1, 2, …). However, a variable is continuous if it can take any real value in an interval (e.g., 16.2, 16.2001, ...
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Continuous vs Discrete Variables
However, some people consider "the number of students in Harvard University" as continuous rather than discrete, because the number of students in Harvard changes per year or per term. Are ...
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Probability question on drawing balls
Chancing upon the question, I am posting a palpably simpler way to arrive at the answer, avoiding casework.
Drawing out balls from the right, the leftmost ball must be red for the condition to be met,...
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What is probability distribution function of the sum of two independent random variables when one variable is correlated with itself?
If $X_i \sim U[-d,d]$ and $Y_{ij}\sim N(0,1)$ are independent of each other, then the distribution of $Z_{ij} = X_i + Y_{ij}$ is indeed the convolution to the two individual distributions, with ...
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Statistics: Using Stirling's Approximation with $3 N$
So, as Donald has noted above, when it comes to Multinomial coefficients, there's more than one degree of freedom when it comes to the relationship between each $N_i$, and thus we need to be explicit ...
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Deconvolution of distribution of diffraction reflexes
What helps is to re-express everything in variables that add linearly. Set
$$
X=\frac{1}{a^{2}},\qquad Y=\frac{1}{b^{2}},\qquad Z=\frac{1}{c^{2}},
$$
and for a given reflection $(h,k,l)$ write
$$
W=\...
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Definition of a Statistical model
Not all $\mu_\theta$, $\theta\in \Theta$ but only one$^1$ special $\mu_{\theta^*}$ (the unknown real one) is related to your $(\Omega, \Sigma, p)$ and $X: \Omega \to \mathbb{R}$. This $\mu_{\theta^*}$ ...
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Does Bayes’ Theorem apply in determining the probability of selecting a biased coin after flipping 4 heads?
To add to Lulu's answer, imagine we were conducting regular statistics and we observed the height of blades of grass. Assume I have ten blades and measure them, getting $a$ metres. I then gather five ...
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Optimal strategy for answering multiple choice test with positive expected value.
With your professed objective clarified in comments as "to score maximum points,"
and a net expected gain per question of $4\times\frac14 - 1\times\frac34 = \frac14$ even with total ...
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How to estimate variances for Kalman filter from real sensor measurements without underestimating process noise.
Precision Evaluation in Discrete Kalman Filtering: A Posteriori Perspective, Journal of Global Positioning systems, www.cpgps.org/vol20i1-2.html, Joint No. 1 & No. 2: 69-87
How to estimate the ...
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Multinomial Distribution to Binomial Distribution and Joint Probability Function
$E(z_1^{X_1}\ldots)z_k^{X_k})=(p_1z_1+\cdots p_kz_k)^n$ implies by doing $z_2=\dots=z_k=1$ that $E(z_1^{X^1})=(1-p_1+p_1z_1)^n$ and $X_1\sim B(n,p_1).$
For the joint distribution of $(X_1,X_2),$ ...
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How much less is the arithmetic mean than the max given the average deviation?
Fix wlog $\mathrm{AM}=\mu$ and $\max=b$. The maximal AAD is obtained for $n-1$ values equal to $b$ and only one value equal to $a=n\mu-(n-1)b$, as you already noticed, and in this case $\mathrm{AAD}=\...
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Covariance of Unbiased Sample Variance Estimators with Overlapping Samples
We have the following formula:
\begin{equation}
\begin{aligned}
\left(N+d-1\right)v(N+d) = &
\left(N-1\right)v(N)
+\sum\limits_{i=N+1}^{N+d} \left(x_i - \frac{1}{d}\sum\...
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Double negatives in hypothesis test conclusions
Aside from the concern that you have the responsibility of assigning grades for examinations without sufficient statistical background (in the sense that you need to post a question online), the idea ...
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What is the name of the distribution of the maximum number of distinguishable balls in a single distinguishable box?
With the scale of numbers you are talking about, two approaches seems reasonable: simulation or approximation.
For example for simulation using $10^4$ cases of $10^5$ balls going into $10^2$ bins, ...
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Predicting $Y$ from a correlated variable $X$
The problem is easier when both $X$ and $Y$ are jointly lognormal. Denoting the logarithms of $X$ and $Y$ by $x$ and $y$ respectively, both $x$ and $y$ are normally distributed. Let the means of $x$ ...
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Mixture Model Expectation
It is not clear what you mean by "we are computing the expectation of the sample mean with respect to the model $p_\theta$".
My understanding is that $X_i$ is a random variable which assumes ...
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Proof for the distribution of a two-sample t-test with unequal population variances.
This is called Welch's $t$-test and there are numerous mentions in statistical textbooks as well as the links for the Wikipedia article.
The numerator is normally distributed under your assumptions ...
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“Central limit theorem” for symmetric random variables with no finite mean
I commented that with stable distributions with heavier tails than a Cauchy distribution, the distribution of $\frac{X_1+\cdots+X_n}{n}$ is not stable as $n$ increases and you have to divide by ...
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Dominated statistical models
Yes, this is true.
For 1. $\Longrightarrow$ 2., simply take $\nu = \mu^{\otimes n}$ to get the dominating measure for the product. This is a $\sigma$-finite measure which dominates the product of the ...
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