# 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

##### Graph:

- 115.020.40 Reading 9 - Common Probability Distributions to 115.020.40.05 Reading 9 - 5. The Binomial Distribution