Statistics / Unit 7: Random Variables
7.4 The Binomial Setting
Notes
A binomial random variable $X$ counts the number of successes in a fixed number of trials. The setting needs (BINS):
- Binary: each trial is a success or failure.
- Independent trials.
- Number of trials $n$ is fixed in advance.
- Same probability of success $p$ on every trial.
- Mean: $\mu_X = np$. Standard deviation: $\displaystyle \sigma_X = \sqrt{np(1 - p)}$.
- Example: The number of heads in 10 flips is binomial with $n = 10$ and $p = 0.5$, so $\mu = 5$.
Practice
Which variable is binomial?
The coin flips have a fixed $n$, independent trials and constant $p$. The others have no fixed $n$, dependent trials or more than two outcomes.
A binomial variable has $n = 20$ and $p = 0.3$. What is its mean?
$np = 20(0.3) = 6$.
A binomial variable has $n = 100$ and $p = 0.2$. Using $\displaystyle \sigma = \sqrt{np(1 - p)}$, its standard deviation is:
$\displaystyle \sqrt{100(0.2)(0.8)} = \sqrt{16} = 4$.
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