This is a submission to a college assignment.

Binomial Distribution

We are repeatedly doing trials of an experiment. Each trial can result in two states, either a success or a failure. We have \(p\) chance that a trial will succeed and hence \(1-p\) chance that that trial may fail. What is the probability that if we do \(n\) trials, we will get \(k\) successes? That is given by \[{n \choose k} p^k (1-p)^{n-k}\]

Let’s look at some plots now.

Plots

If we take \(p = \frac{1}{2}\), do \(n = 100\) trials, and calculate and plot the probability of \(k\) successes \(\forall k \in [1, n]\) we get the following:

Convergence to the Normal Distribution

If we keep on increasing the number of trials then the bar plot starts to resemble the normal distribution more closely. With big enough \(n\) normal approximation becomes increasingly accurate.

Expected Number of Successes

Expected number of successes would be the number of trials multiplied by the chance of each trial succeeding, ie \(n * p\). This is illustrated below:

Skewness at Edges

The plots are found to be skewed at edges. This is because the number of successful trials cannot be less than zero or greater than the total number of trials. If \(p \sim 0\) then lower values of \(k\) have higher probabilities, whereas vice-versa if \(p \sim 1\).