Approximations

Continuity Correction

The below corrections are used to adjust for the difference between discrete and continuous distributions.

When	Use
$X > x$	$x + 0.5$
$X \leq x$	$x + 0.5$
$X < x$	$x - 0.5$
$X \geq x$	$x - 0.5$

Bin(n,p) to Poisson(p)

If $n$ is large and $p$ is small, then finding a defective item in the sample is called a rare event. In practice $n>50$ and $np<5$ means the event is considered rare. In above case, the Poisson distribution gives a very close approximation to the binomial distribution.

Generally the approximation between Poisson and Binomial distribution is good when $p< 0.1$ and $np < 5$.

$$\text{Bin}(n,p) \approx \text{Poisson}(np)$$

Normal approximation for Bin(n,p)

When $n$ is large and $np > 5$:

$$\text{Bin}(n,p) \approx \text{N}(np, np(1-p))$$

Normal approximatiox for Poisson(p)

When $p > 10$: $$

$$\text{Poisson}(p) \approx \text{N}(p, p)$$

Normal approximation for Student’s t-distribution

When $v$ is large (typically $v \gt 30$), or when the sample is large:

$$t \sim t(v) \approx \text{N}(0,1)$$