Binomial Distribution

Definitions

a series of independent trials
each trial can be a failure or success, and the probability of success is the same for each trial
there are a finite number of trials

P (X = r) =^{n} C_{r} p^{r} q^{n - r}

where

^{n} C_{r} = \frac{n!}{r! (n - r)!}

where

$X$ to represent the number of successful outcomes out of $n$ trials
$r$ is the number of successes

If a variable $X$ follows a binomial distribution where the probability of success in a trial is $p$ and the number of trials is $n$ , this can be written as
$X \sim B (n, p)$

Expectation

$E (X) = n p$

Variance

$V a r (X) = n p q$

Approximate Binomial Distribution

Normal Distribution

If $X \sim B (n, p)$ and $n p > 5$ and $n q > 5$ , you can use $X \sim N (n p, n p q)$ to approximate it.
If you’re approximating the binomial distribution with the normal distribution, then you need to apply a continuity correction to make sure your results are accurate.

Finding ≤ probabilities
- if you’re using the normal distribution to find $P (X \leq a)$ , you actually need to calculate $P (X < a + 0.5)$
Finding ≥ probabilities
- The value $b$ extends down to $(b - 0.5)$ on a continuous scale so you need to use a range of $P (X > b - 0.5)$ to make sure that you include it.
Finding “between” probabilities
- $P (a - 0.5 < X < b + 0.5)$

Poisson Distribution

$X \sim B (n, p)$ can be approximated by $X \sim P o (n p)$ if $n$ is large and $p$ is small

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