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The Central Limit Theorem has something to say about your last question. Specifically, the usual way of estimating such as $P\left(X \leq k \right)$ consists of using the empirical distribution function (ECDF) and computing

$$F_n(k) = \sum_{i=1}^n \frac{1}{n} \mathbf{1} \left(X_i \leq k \right)$$

where we just count the number of observations $X_1, \ldots, X_n$ that fall short of $k$ and divide by the total number observations. That is, our most intuitive estimator for this probability is the corresponding sample proportion.

It turns out that this estimator has fantastic statistical properties. It can be shown that it converges, as $n \to \infty$, to $P\left(X\leq k \right)$ with probability one and that it does so uniformly for all values of $k$. Since $F_n(k)$ is also a sum of iid terms, one can further apply the CLT to make probabilistic statements about the distance from the true number. It's not very difficult to show that

$$F_n(k) \xrightarrow{D} \mathcal{N} \left( P(X\leq k), \frac{ P(X\leq k)\left(1-P(X\leq k\right)} {n} \right) $$

By using the properties of the Normal distribution you can then make probabilistic statements about the quantity $|F_n(k)-P(X\leq k)|$ after plugging in $F_n(k)$ in the variance of the asymptotic distribution.

The Central Limit Theorem has something to say about your last question. Specifically, the usual way of estimating such as $P\left(X \leq k \right)$ consists of using the empirical distribution function (ECDF) and computing

$$F_n(k) = \sum_{i=1}^n \frac{1}{n} \mathbf{1} \left(X_i \leq k \right)$$

where we just count the number of observations $X_1, \ldots, X_n$ that fall short of $k$ and divide by the total number observations. That is, our most intuitive estimator for this probability is the corresponding sample proportion.

It turns out that this estimator has fantastic statistical properties. It can be shown that it converges, as $n \to \infty$, to $P\left(X\leq k \right)$ with probability one and that it does so uniformly for all values of $k$. Since $F_n(k)$ is also a sum of iid terms, one can further apply the CLT to make probabilistic statements about the distance from the true number. It's not very difficult to show that

$$F_n(k) \xrightarrow{D} \mathcal{N} \left( P(X\leq k), \frac{ P(X\leq k)\left(1-P(X\leq k\right)} {n} \right) $$

By using the properties of the Normal distribution you can make probabilistic statements about the quantity $|F_n(k)-P(X\leq k)|$ after plugging in $F_n(k)$ in the variance of the asymptotic distribution.

The Central Limit Theorem has something to say about your last question. Specifically, the usual way of estimating such as $P\left(X \leq k \right)$ consists of using the empirical distribution function (ECDF) and computing

$$F_n(k) = \sum_{i=1}^n \frac{1}{n} \mathbf{1} \left(X_i \leq k \right)$$

where we just count the number of observations $X_1, \ldots, X_n$ that fall short of $k$ and divide by the total number observations. That is, our best estimate for this probability is the corresponding sample proportion.

It turns out that this estimator has fantastic statistical properties. It can be shown that it converges, as $n \to \infty$, to $P\left(X\leq k \right)$ with probability one and that it does so uniformly for all values of $k$. Since $F_n(k)$ is also a sum of iid terms, one can further apply the CLT to make probabilistic statements about the distance from the true number. It's not very difficult to show that

$$F_n(k) \xrightarrow{D} \mathcal{N} \left( P(X\leq k), \frac{ P(X\leq k)\left(1-P(X\leq k\right)} {n} \right) $$

By using the properties of the Normal distribution you can then make probabilistic statements about the quantity $|F_n(k)-P(X\leq k)|$ after plugging in $F_n(k)$ in the variance of the asymptotic distribution.

The Central Limit Theorem has something to say about your last question. Specifically, the usual way of estimating such as $P\left(X \leq k \right)$ consists of using the empirical distribution function (ECDF) and computing

$$F_n(k) = \sum_{i=1}^n \frac{1}{n} \mathbf{1} \left(X_i \leq k \right)$$

where we just count the number of observations $X_1, \ldots, X_n$ that fall short of $k$ and divide by the total number observations. That is, our most intuitive estimator for this probability is the corresponding sample proportion.

It turns out that this estimator has fantastic statistical properties. It can be shown that it converges, as $n \to \infty$, to $P\left(X\leq k \right)$ with probability one and that it does so uniformly for all values of $k$. Since $F_n(k)$ is also a sum of iid terms, one can further apply the CLT to make probabilistic statements about the distance from the true number. It's not very difficult to show that

$$F_n(k) \xrightarrow{D} \mathcal{N} \left( P(X\leq k), \frac{ P(X\leq k)\left(1-P(X\leq k\right)} {n} \right) $$

By using the properties of the Normal distribution you can then make probabilistic statements about the quantity $|F_n(k)-P(X\leq k)|$ after plugging in $F_n(k)$ in the variance of the asymptotic distribution.