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You should know $\sin(x)$ and $\cos(x)$ for $x = 0$, $30$, $60$, $90, \ldots, 360$ degrees. Memorize $\sin$ and $\cos$ of $6$ degrees and $12$ degrees and you can use the addition formulas to calcul …

The assumptions are that in each case, the animals captured constitute a random sample of the animals on the island, that whether an animal is captured in the second sample is independent of whether …

Say there are $i_t$ inputs at time $t$, $t=0,\ldots,M$, and $N_t$ items in the system at that time (including those $i_t$). The number of items entering the system from time $0$ to $M$ is $E_M = \sum_ …

You are using $$ \int_0^1 \frac{\sinh(x)}{x}\; dx = \sum_{n=0}^\infty \int_0^1 \frac{x^{2n}}{(2n+1)!}\; dx = \sum_{n=0}^\infty \frac{1}{(2n+1)(2n+1)!}$$ If you stop at $n=m$, the error will be $$ E_m …

Since e.g. $(x^1, y^2)$ could be very close to $0$, no such bound is possible.

With $\ge$, this does not define the function uniquely. It might grow exponentially or faster, e.g. $2^x$ satisfies the inequality. For the version with $=$, see http://oeis.org/A018819 EDIT: This …

Here's one way: flip a coin (independent of the population) and take a random sample of size either $1$ (if Heads) or $2$ (if Tails). The sample mean is an unbiased estimator of the population averag …

No, it's not true. With $n \approx k^{3/2}$, $n - {n \choose k} (1 - n^{-2/3})^{k(k-1)/2} < 0$ for large $k$ (in fact for any $k \ge 4$).

Let's suppose $a > 0$ and $b > 0$. If $g(k) = k^a b^k/k!$, then $$ \dfrac{g(k+1)}{g(k)} = \left(1 + \dfrac{1}{k} \right)^a \dfrac{b}{k+1} \to 0 \ \text{as}\ k \to \infty$$ Take $0 < r < 1$ and $K$ …

For an inhomogeneous Poisson process with intensity $\lambda(t)$, $X(t)$ (representing the number of occurrences in the interval $[0, t]$) is a Poisson random variable with parameter $\Lambda(t) = \in …

$$V(\overline{X}^2) = \mathbb E[\overline{X}^4] - \mathbb E[\overline{X}^2]^2 $$ Since $X \sim \mathcal N(\mu, \sigma^2)$, $\overline{X} \sim \mathcal N(\mu, \sigma^2/n)$. You should get $$V(\overline …

Hint: $$\mathbb E[\overline{X}^2] = \text{Var}(\overline{X}) + \mathbb E[\overline{X}]^2$$ Do you know the mean and variance of $\overline{X}$?

I'm assuming $N$ is a positive integer. You can expand $\cos^N(x)$ as a linear combination of $\cos(kx)$ for $k$ from $0$ to $N$, and use $$ \int \cos(2\pi a/m)\; dm = m \cos(2\pi a/m) + 2 \pi a\; …

If you're interested in the Chebyshev series of $\exp(x)$ on $[-1,1]$, the first few terms, according to Maple, are $$1.26606587775200818\,T \left( 0,x \right) + 1.13031820798497007\,T \left( 1,x \ri …

Get the latest financial statement from the company that owns the store. The total value of inventory (as of a certain date) should be listed. Divide by the number of stores owned.