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$33$ students in clubs

You did a silly and thought $\displaystyle\sum_{k=1}^{11} k=33$ but actually $\displaystyle\sum_{k=1}^{11} k=66.$ So now try the problem for yourself again and see if you get it before reading my ...
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$33$ students in clubs

The number of total memberships is: $1+2+3+\cdots + 11 = 66$, and each student is in exactly two clubs, so there are exactly $11$ clubs for $33$ students. If $M$ is the number of maths clubs, then $11-...
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1 vote
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UMVUE of $\theta$ for Negative Binomial family

With your $(\theta, k)$-parameterisation, it seems that $X_i$ counts the number of successes $\in \{0, 1, 2, \dots \}$ until $k$ failures, where each success has probability $p = \frac{\theta}{k + \...
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  • 1,254
1 vote

Optimal stopping in red vs black card game deck of 52 cards, quick way to reasonably estimate the answer

For large $r, b$ with $\frac rb$ close to $1$, we should expect the game to be approximately unchanged after replacing each $+1$ card with four cards $+\frac12, +\frac12, +\frac12, -\frac12$, and each ...
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1 vote
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Estimating variance of population with variance of sample mean

Let $X_i$ be i.i.d. Then the estimator is $s^2=n\cdot Var(\overline X)=n\cdot Var\left(\frac1n\cdot\sum\limits_{i=1}^n X_i\right) =\frac1n\cdot Var\left(\sum\limits_{i=1}^n X_i\right)$ Now you can ...
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Estimating variance of population with variance of sample mean

This question does not make sense because $nV(\bar{X})$ is a deterministic quantity equalling $\sigma^2,$ which you have assumed is unknown.
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4 votes

Evaluating the limit of $\lim_{n \to \infty} \frac{4n^2 +9}{2n^2+2n+3}$ with definition

In your approach you need to find an upper bound for the numerator and a lower bound for the denominator. Your calculation was ok but I would present it as below to make it a little bit more explicit: ...
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5 votes
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Evaluating the limit of $\lim_{n \to \infty} \frac{4n^2 +9}{2n^2+2n+3}$ with definition

Your working has no errors as far as I can tell. Sometimes proofs like these have multiple ways to approach it; this is one such instance. To get to the line about $\newcommand{\ve}{\varepsilon} \ve/...
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1 vote

Optimal stopping in red vs black card game deck of 52 cards, quick way to reasonably estimate the answer

Look for the 'pivot points' and make continuous approximations Here's part of an answer... If we do some handwaving and approximations we can get something more compact without actually doing the ...
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How to get a good estimate of a calculation full of prime numbers and decimals?

You could try "zeroth and first order" approximation: $$\dfrac{1927.5 \times 273}{311 \times 760} = \dfrac{1900\big(1+\tfrac{27.5}{1900}\big) \times 300\big(1-\tfrac{27}{300}\big)}{300\big(1+...
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  • 296
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How to get a good estimate of a calculation full of prime numbers and decimals?

For this type of question, you said you tried one significant digit, but what about more than one? For example, in your example, we can round it to two significant digits - like this: $$\frac{1930\...
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1 vote
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Construct function with special decay

If such a function $f$ existed on the reals, that would mean $f(R) = 1$ and $f(2R) = 0$. Since $f$ is twice differentiable, the first derivative of $f$ is continuous. We also know by continuity of the ...
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1 vote
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Comparing the truncated $\ell^{1}$-norm of polynomial coefficients with the supremum norm on the unit disc.

The answer to the question is negative. Assume for a contradiction that $$\|p-a_0\|_{\ell^1}\le C\|p\|_\infty$$ As $|a_o|=|p(0)|\le \|p\|_\infty$ we get $$\|p\|_{\ell^1}\le D\|p\|_\infty,\qquad D=C+1$$...
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4 votes

Estimating the value of this Integral

Your solution is nice and simple. On purpose, I shall make a complex one. Consider $$y=x^{\sin (x)+\cos (x)}\quad \implies \quad \log(y)=[\sin (x)+\cos (x)]\log(x)$$ that is to say $$\log(y)=\log(x) \...
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3 votes
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Dynamic dice game, how to reasonably estimate answer by hand without laboriously calculating

You can estimate it as follows: There are two possibilities - Either you roll a six, or you don't. If you don't, the expected gain is $\frac{1+2+3+4+5}{5}=3$ (Note that this is the expected gain GIVEN ...
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  • 424
1 vote

Fitting distribution to histogram (the only data)

If we have $$X_1,...,X_n\sim \text{Lognormal}(\mu,\sigma^2)$$ then $$\ln(X_1),...,\ln(X_n)\sim\mathcal{N}(\mu,\sigma^2)$$ which means $$\hat{u}=\frac{1}{n}\sum_{j=1}^n\ln(X_j)$$ $$\hat{\sigma}^2=\frac{...
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0 votes
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Estimation of the min sample size from the population to make the proportion within certain range with least probability $~0.99$

$n\hat{p}$ is the number of customers in your sample who use the detergent. It is a $\text{Binomial}(n, p)$ random variable which has mean $np$ and variance $npq$. By the normal approximation to the ...
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  • 82.1k
1 vote

Estimation of the min sample size from the population to make the proportion within certain range with least probability $~0.99$

The first equation marked in red comes from the Central Limit Theorem. The sample proportion $\hat{p}$ is a sample mean of the variable indicating whether a given person falls in the required category,...
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  • 14.7k
2 votes
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Estimation for operators

$$\|B\|^2A^*A-A^*B^*BA=A^*(\|B\|^2I-B^*B)A\ge 0$$
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