An illustrative example of the binomial distribution
Highlighting an important distinction between the two sample sizes in a binomial sample
If you flip a fair coin independently 3 times, you can view this process in two equivalent ways:
3 independent Bernoulli random variables, each with a probability of heads being 0.50.
1 binomial random variable of 3 independent Bernoulli trials, each with a probability of heads being 0.50.
Let’s choose the second approach. You can denote this binomial random variable as
Now, let’s suppose further that you repeat this process 4 times. This leads to 4 binomial random variables.
Here is a very important point: There are two different sample sizes here.
n = 3, the number of Bernoulli trials within each binomial variable
k = 4, the number of binomial random variables in this sample
When working with the binomial distribution, make sure that you distinguish between these two sample sizes. (Strictly speaking, n is a parameter of the binomial distribution, and k is the sample size.)
When describing a sample of binomial random variables with mathematical notation, it is common to use a letter such as i as the index for the random variables in the sample. In this case, I can write this sample as
It took me a while to fully absorb these details - especially the two different sample sizes. I encourage you to inspect this example carefully and write the probability mass function (PMF) explicitly to fully grasp this important distribution.