115 CFA
File: Reading 9 - 5. The Binomial Distribution

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.

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