# 5. The Binomial Distribution

## e. define a discrete uniform random variable, a Bernoulli random variable, and a binomial random variable;

## f. calculate and interpret probabilities given the discrete uniform and the binomial distribution functions;

## g. construct a binomial tree to describe stock price movement;

## h. calculate and interpret tracking error;

## i. define the continuous uniform distribution and calculate and interpret probabilities, given a continuous uniform distribution;

What is a Bernoulli trial?

- A Bernoulli trial is an experiment with two outcomes, which can represent success or failure, up move or down move, or another binary outcome.

- As one of these two outcomes must definitely occur, that is, they are exhaustive, and also mutually exclusive, it follows immediately that the sum of the probabilities of a "success" and a "failure" is 1.

What is the mathematical formula for the binomial probability of obtaining `r`

successes in `n`

trials?

$$p(r) = _n C_r p^r(1 - p)^{n - r} = \frac{n!}{(n - r)! r!} p^r(1-p)^{n-r}$$

- `p(r)`

= probability of exactly `r`

successes

- `n`

= number of events

- `p`

= probability of success on any one trial

What are the four rules for events in the binomial probability formula?

1. Dichotomous - they can only fall into two categories

2. Mutually exclusive

3. Independent

4. Randomly selected

What is a tracking error?

Tracking error is a measure of how closely a portfolio follows the index to which it is benchmarked: `total return on the portfolio - total return on the benchmark index`

.

##### Source:

- CFA