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 >> 115.020.40.05 Reading 9 - 5. The Binomial Distribution