Since many discrete [[Probability Distribution|probability distributions]] are representative of [[Random Sample|random samples]] of other distributions, for large [[Count|sample sizes]], they can be approximated using continuous distributions. $ \begin{align*} P(X = x) &\approx P(x - 0.5 \le X \le x + 0.5) \\ P(a \le X \le b) &\approx P(a - 0.5 \le X \le b + 0.5) \\ P(a < X < b) &\approx P(a + 0.5 \le X \le b - 0.5) \end{align*} $ **Continuity Adjustment**: When approximating a discrete distribution using a continuous one, represent each discrete whole number $x$ by the interval $[x - 0.5, x + 0.5]$. # Binomial-Normal $ B(n, p) = n\bar{X} \approx N(n\mu, n\sigma^2) = N(np, npq) $ Since the [[Binomial Random Variable|binomial distribution]] $B(n, p)$ is the sum of $X_1 + X_2 + \cdots X_n$ of a [[Random Sample|random sample]] of size $n$ for the [[Bernoulli Random Variable]] $B(1, p)$. For large values[^1] of $n$, a binomial random variable has roughly the same distribution as a normal distribution, with the same [[Mean|mean]] and [[Variance|variance]]. [^1]: If $np > 10$, $nq > 10$.