Common Probability Distributions Flashcards
Discrete random variable
May be at most a countable number of outcomes. Uniform if each outcome equally likely.
continuous random variable
cannot count number of outcomes
Interpret cumulative distribution function
Probability that random variable less than or equal to particular value. Sum up all outcomes less than or equal to value.
Bernoulli random variable
Random variable w/value of 1 is success, 0 is failure
Binomial random variable
number of successes in n Bernoulli trials
Construct binomial tree for stock price
Each period have a new split of two nodes for up and down transition probability.
Calculate and interpret tracking error
Total return on portfolio (gross of fees) minus total return on benchmark index. Use binomial random distribution to figure out probability that certain tracking error met certain percent of time.
continuous uniform distribution
All outcomes equally likely
Calculate and interpret probabilities given continuous uniform distribution
Probability that continuous random variable is any particular fixed value is 0, but it’s possible to estimate probability that continuous random variable is more or less than certain amount
Explain key properties of normal distribution
- Mean and variance completely describe it
- Skewness of zero, kurtosis of 3
- Linear combination of two or more normal random variables is also normally distributed
Univariate vs. multivariate distribution
Univariate describes single random variable
Multivariate specifies probabilities for group of related random variables
Role of correlation in multivariate distribution
All distinct pairwise correlations is third parameter that describes multivariate normal distribution
Determine probability normally distributed random variable lies inside given interval
68% within 1 s
95% within 2 s
99% within 3 s
Define standard normal distribution
Normal density with mean of 0 and standard deviation of 1
How to standardize random variable
Z = (X - μ) / σ
Calculate and interpret probabilities using standard normal distribution
N(x) is notation - use table for positive x
N(-x) = 1 - N(x)
Shortfall risk
Risk that portfolio will fall below a minimum acceptable level over some time horizon
Calculate safety-first ratio
SFRatio = [E(Rp) - RL)] / σ
RL = threshold return level
Select optimal portfolio using Roy’s safety first criterion
Determine SFRatios and Choose portfolio with highest SFRatio
Relationship between normal and lognormal distributions.
The log is normal
Variable follows lognormal distribution if its natural logarithm is normally distributed. Distribution bounded by zero with long right tail.
Distinguish between discretely and continuously compounded rates of returns
Finite intervals vs. unbroken compounding
Calculate and interpret continuously compounded rate of return
ln(1 + r)
i.e. natural logarithm of 1 plus holding period return
Explain monte carlo simulation
generate large number of random samples from specified distribution to represent role of risk in system.
Compare monte carlo simulation and historical simulation
Historical info provides the most direct evidence on distributions, but excludes any info on risks not shown in data and can’t be used for “what if” analysis
Binomial distribution function
p(x) = n!/(n-x)!x! * p^x * (1-p) ^ n-x
x is number of successes, n is number of trials, p is probability of the event
Why lognormal use to model asset prices?
Because it is bounded by zero and is skewed to right (long tail). Asset prices can’t be below zero.
Major uses of monte carlo simulation
- Planning
- Risk management
- Value complex securities
