Common Probability Distributions Flashcards
Describes the probabilities of all possible outcomes for a random variable
Probability Distribution
The number of possible outcomes that can be counted , and for each possible outcome, there is a measureable and positive probability
Discrete Random Variable
A variable which the number of possible outcomes is infinite, even if lower and upper bounds exist.
Continuous Random Variable
A function that defines the probability that a random variable, X, takes on a value equal to or less than a specific value, X.
Cumulative Distribution Function
*F(x) == P(X<=x)
A variable for which the probabilities of all possible outcomes for a discrete random variable are equal
Discrete Uniform Random Variable
The number of “successes” in a given number of trials, where the outcome is either a success or a failure. (The trials must be independent.)
Binomial Random Variable
Calculation for the probability of x successes in n trials (in a Bernoulli random variable)
p(x) = (n!)/((n-x)!x! * (p^x*(1-p)^(n-x))
Expected Value for Bernoulli Random Variable
E(x) = n * p
Variance of a Binomial Random Variable
Var(X) = (n*p) * (1-p)
A distribution defined over a range that spans between a lower limit, a, and upper limit, b, which serve as parameters of the distribution.
Continuous Uniform Distribution
Properties of Continuous Uniform Distribution
*a<= x1<= x2<= b
**P(x<a> b) = 0
*** P(x1 < x < x2) = (x2-x1) / (b-a)
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Properties of a Normal Distribution
- X is normally distributed with mean,u, and variance, sigma^2
- Skewness = 0, Mean = Median = Mode
- ** Kurtosis = 3
- *** A linear combination of normally distributed random variables is also normally distributed.
- ** The probabilities of outcomes further above and below the mean get smaller and smaller, but do not go to zero.
A distribution the specifies the probabilities associated with a group of random variables and is meaningful only when the behavior of each random variable in the group is in some way dependent on the behavior of the others.
Multivariate Distribution
Confidence Interval Ranges
90% C.I. is x +/- 1.65 sd
95% C.I. is x +/- 1.96 sd
99% C.I. is x +/- 2.58 sd
A normal distribution that has been standardized so that it has a u = 0, and sigma = 1.
Standard Normal Distribution
* z = (x - u) /sigma
The number of standard deviations above or below the mean
z-score
The probability that a portfolio value or return will fall below a particular value or return over a given period of time.
Shortfall Risk
The optimal portfolio minimizes the probability that the return of the portfolio falls below some minimum acceptable level.
Roy’s Safety First Ratio
* min P(Rp -Rl)
A distribution that is generated by the function e^x, where x is normally distributed
Lognormal Distribution
- Skewed to the right
- Bounded from below by zero, so that it is useful for modeling asset prices which never take a negative value
Calculation for Continuously Compounded Returns
Rcc = e^Rcc - 1
Applications of Rcc
- ln(EAR) = Rcc
- Rcc = ln(S1/S0) = ln( 1 + HPR)
- ** HPRt = e^Rcc*T - 1
A technique based on the repeated generation of one or more risk factors that affect security values, in order to generate a distribution of security values.
Monte Carlo Simulation
Monte Carlo Simulations are used to:
- Value complex securities
- Simulate the profits/losses for a trading strategy
- ** Calculate estimates of value at risk(VAR) to determine the riskiness of a portfolio of assets and liabilities
- *** Simulate pension funds assets and liabilities overtime to examine the variability of the difference
- ** Value portfolios of assets that have non-normal returns distributions
Limitations of Monte Carlo Simulations
- Fairly Complex
- Will provide answers that are no better than the assumptions about the distributions of the risk factors and the pricing/valuation model that is used.
- ** Not an analytic method, only statistical method.
A simulation based on actual changes in value or risk factors over some prior period.
Historical simulation
Advantage to historical simulations
distribution of changes in risk factors don’t have to be estimated.
Disadvantages of historical simulations
- risk factors of past may not be good predictors of the future
- Cannot address “what-if” scenarios like MC simulations can. (i.e. What if we increase variance of risk B by 20%. How much does portfolio change? and in what direction?
