Symbol in context

$\mathbb{E}$

This entry links every published equation using this exact notation. Its meaning may change between equations.

Meanings in context

Standard notation

The expected value operator: the probability-weighted average of the quantity inside its brackets.

\mathbb{E}[\text{tokens per round}] = \frac{1-\alpha^{\gamma+1}}{1-\alpha}.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

\mathbb{E}[C_{\mathrm{find}}] = \frac{c}{p}.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

\mathbb{E}\!\left[(x - \hat{x})^2\right] \approx \frac{s^2}{12}

This equation gives an approximation: it relates the quantities while allowing an approximation.

\mathbb{E}[\text{duplicates}] \;\approx\; n \cdot p \cdot (1 - i).

This equation gives an approximation: it relates the quantities while allowing an approximation.

\mathbb{E}[C_n] \approx \sum_{i=1}^{n} \frac{c_i}{1-q_i}

This equation gives an approximation: it relates the quantities while allowing an approximation.

\mathbb{E}\big[r_\theta(x,y)\big] - \beta\,\mathbb{E}\big[D_{\mathrm{KL}}\!\left(\pi^{\mathrm{RL}} \,\|\, \pi^{\mathrm{SFT}}\right)\big] + \gamma\,\mathbb{E}\big[\log \pi^{\mathrm{RL}}(x)\big],

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

\mathbb{E}[\text{cost to first success}] = \frac{\bar{c}}{p},

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

p^{*} \;=\; \mathbb{E}\big[\, v(q) \;\big|\; c(q) \le p^{*} \,\big],

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

\lim_{m \to \infty} \operatorname{var}\!\left( \frac{d_m(P_m, Q_m)^p}{\mathbb{E}\left[ d_m(P_m, Q_m)^p \right]} \right) = 0 \;\;\Longrightarrow\;\; \lim_{m \to \infty} \Pr\!\left[ D_{\max}^{(m)} \le (1 + \varepsilon)\, D_{\min}^{(m)} \right] = 1

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

\lim_{d \to \infty} \mathbb{E}\!\left[ \frac{D_{\max}^{k} - D_{\min}^{k}}{d^{1/k - 1/2}} \right] = C \cdot \frac{1}{(k+1)^{1/k}} \sqrt{\frac{1}{2k+1}},

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.