(9) Common Probability Distributions Flashcards
LOS 10. a: Define a probability distribution and distinguish between discrete and continuous random variables and their probability functions.
A probability distribution lists all the possible outcomes of an experiment, along with their associated probabilities. A probability distribution completely describes a random variable.
LOS 10. a: Define a probability distribution and distinguish between discrete and continuous random variables and their probability functions.
A discrete random variable has positive probabilities associated with a finite number of outcomes (countable, non zero probabilities for each outcome)
A continuous random variable has positive probabilities associated with a range of outcome values-the probability of any single value is zero (Uncountable)
LOS 10. b: Describe the set of possible outcomes of a specified discrete random variable.
The set of possible outcomes of a specified discrete random variable is a finite set of values. An example is the number of days last week on which the value of a particular portfolio increased. For a discrete distribution, p(x) = 0 when x cannot occur, or P(x) > 0 if it can.
LOS 10. c: Interpret a cumulative distribution function.
A cumulative distribution function (cdf) gives the probability that a random variable will be less than or equal to specific values. The cumulative distribution function for a random variable X may be expressed as F(x) = P(X =< x). The CDF is more relevant than discrete and continuous probability functions.
LOS 10. d: Calculate and interpret probabilities for a random variable, given its cumulative distribution function.
Given the cumulative distribution function for a random variable, the probability that an outcome will be less than or equal to a specific value is represented by the area under the probability distribution to the left of that value.
LOS 10. e: Define a discrete uniform distribution, a Bernoulli random variable, and a binomial random variable.
A discrete uniform distribution is one where there are n discrete, equally likely outcomes (each probability is the same for the distribution).
LOS 10. e: Define a discrete uniform random variable, a Bernoulli random variable, and a binomial random variable.
A Bernoulli random variable is one where there are only 2 outcomes (each outcome is called a trial)
A Binomial random variable measures the number of successes in N bernoulli trials
LOS 10. f: Calculate and interpret probabilities given the discrete uniform and the binomial distribution functions.
For a discrete uniform distribution with n possible outcomes, the probability for each outcome equals 1/n.
LOS 10. f: Calculate and interpret probabilities given the discrete uniform and the binomial distribution functions.
For a binomial distribution, if the probability of success is p, the probability of x successes in n trials is:
Explain the Bernoulli equation logic
LOS 10. g: Construct a binomial tree to describe stock price movement.
A binomial tree illustrates the probabilities of all the possible values that a variable (such as a stock price) can take on, given the probability of an up-move and the magnitude of an up-move (the up-move factor).
With an initial stock price S = 50, U = 1.01, D = 1/1.01, and prob(U) = 0.6, the possible stock prices after two periods are:
uuS = 1.012 x 50 = 51.01 with probability of (0.6)2 = 0.36
udS = 1.01(1/1.01) x 50 = 50 with probability of (0.6)(0.4) = 0.24
duS = (1/1.01)1.01 x 50 = 50 with probability of (0.4)(0.6) = 0.24
uuS = (1/1.01)2 x 50 = 49.01 with probability of (0.4)2 = 0.16
Calculate and interpret tracking error.
Tracking error is calculated as the total return on a portfolio minus the total return on a benchmark or index portfolio.
Tracking error measures how closely a portfolio follows its benchmark
LOS 10. h: Define the continuous uniform distribution and calculate and interpret probabilities, given a continuous uniform distribution.
A continuous uniform distribution is described by a lower limit ‘a’ and an upper limit ‘b’
A continuous uniform distribution is one where the probability of X occurring in a possible range is the length of the range relative to the total of all possible values. Letting a and b be the lower and upper limit of the uniform distribution, respectively. See image below.
If its asking for P(X =< A), then formula is
x - a / b - a
LOS 10. j: Explain the key properties of the normal distribution.
The normal probability distribution and normal curve have the following characteristics:
- The normal curve is symmetrical and bell-shaped with a single peak at the exact center of the distribution.
- Mean = median = mode, and all are in the exact center of the distribution.
- The normal distribution can be completely defined by its mean and standard deviation because the skew is always zero and kurtosis is always 3.
- A linear combination of normally distributed variables is also normally distributed
LOS 10. j: Distinguish between a univariate and a multivariate distribution and explain the role of correlation in the multivariate normal distribution.
Also what parameters are needed for a univariate and a multivariate distribution?
Multivariate distributions describe the probabilities for more than one random variable, whereas a univariate distribution is for a single random variable.
Univariate parameters: mean and variance
Multivariate parameters: mean, variance, and corrleations between each pair of random variables
LOS 10. j: Distinguish between a univariate and a multivariate distribution and explain the role of correlation in the multivariate normal distribution.
The correlation(s) of a multivariate distribution describes the relation between the outcomes of its variables relative to their expected values.
LOS 10. k: Determine the probability that a normally distributed random variable lies inside a given interval.
A confidence interval is a range within which we have a given level of confidence of finding a point estimate (e.g., the 90% confidence interval for X is Xbar – 1.65s to Xbar + 1.65s).
The probability that a normall distributed random variable X will be within A standard deviations of its mean, µ, [i.e., P(µ - Aσ <= X <= µ + Aσ)], may be calculated as F(A) – F(-A), where F(A) is the cumulative standard normal probability of A, or as 1 – 2[F(-A)].
LOS 10. l: Define the standard normal distribution, explain how to standardize a random variable, and calculate and interpret probabilities using the standard normal distribution.
The standard normal probability distribution has a mean of 0 and a standard deviation of 1. Standardizing finds the number of standard deviations away from the mean X lies
LOS 10. l: Define the standard normal distribution, explain how to standardize a random variable, and calculate and interpret probabilities using the standard normal distribution.
A normally distributed random variable X can be standardized as Z = (x - µ) / σ and Z will be normally distributed with mean = 0 and standard deviation 1. This formula calculated the number of standard deviations away from the mean X lies
µ = population mean
σ = standard deviation
LOS 10. l: Define the standard normal distribution, explain how to standardize a random variable, and calculate and interpret probabilities using the standard normal distribution.
The z-table is used to find the probability that X will be less than or equal to a given value.
LOS 10. m: Define shortfall risk, calculate the safety-first ratio, and select an optimal portfolio using Roy’s safety-first criterion.
The safety-first ratio for portfolio P, based on a target return RT, is:
[Equation]
Shortfall risk is the probability that a portfolio’s value (or return) will fall below a specific value over a given period of time.
Greater safety-first ratios are preferred and indicate a smaller shortfall probability.
Roy’s safety-first criterion states that the optimal portfolio minimizes shortfall risk.
E(Rp) - expected portfolio return
RT - target/threshold return
LOS 10. n: Explain the relationship between normal and lognormal distributions and why the lognormal distribution is used to model asset prices.
If x is normally distributed, ex follows a lognormal distribution. A lognormal distribution is often used to model asset prices, since a lognormal random variable cannot be negative and can take on any positive value.
A normal distribution is unbounded on both ends, while a lognormal distribution is bounded by zero and unbounded on the upper end.
Normal distribution is used to model returns (used as an approximation of returns)
Lognormal distribution = (VT/V0), where VT is ending value asset price and V0 is beginning value asset price
Normal distribution = Ln (VT/V0)
If an assets continuously compounded return is normally distributed then the assets future prices must be lognormally distributed
LOS 10. o: Distinguish between discretely and continuously compounded rates of return and calculate and interpret a continuously compounded rate of return, given a specific holding period return.
As we decrease the length of discrete compounding periods (e.g., from quarterly to monthly) the effective annual rate increases. As the length of the compounding period in discrete compounding gets shorter and shorter, the compounding becomes continuous, where the effective annual rate = ei – 1.
For a holding period return (HPR) over any period, the equivalent continuously compounded rate over the period is ln(1 + HPR).
Holding period return = (VT/V0) - 1; where VT is the ending asset price and V0 is the beginning asset price
Continuously compounded return (rT) = (VT/V0)
LOS 10. p: Explain Monte Carlo simulation and describe its applications and limitations.
Monte Carlo simulation uses randomly generated values for risk factors, based on their assumed distributions, to produce a distribution of possible security values. Its limitations are that it is fairly complex and will provide answers that are no better than the assumptions used.
LOS 10. q: Compare Monte Carlo simulation and historical simulation.
Historical simulation uses randomly selected past changes in risk factors to generate a distribution of possible security values, in contrast to Monte Carlo simulation, which uses randomly generated values. A limitation of historical simulation is that it cannot consider the effects of significant events that did not occur in the same period. It also does not factor in cause/effect relationships
Two types of probability distributions:
- Probability Function (Possible outcomes + associated probabilities)
- Probability that the random variable takes on a specific value
- P(X=x)
- This completely describes the variable
- Used for discrete variables
- Cumulative Distribution Function (probability that a random variable X is less than or equal to a particular value x, P (X=
Calculate the Mean and variance of a bernoulli and binomial random variable
- Bernoulli mean = p
- Binomial mean = nxp
- Bernoulli variance = p (1-p)
- Binomial variance = nxp (1-p)
Where p = probability and n = number of trials
Two types of continuous distributions
- Normal Distribution
- Lognormal distribution
Probability density function is solving for…
The area under the curve
Centril Limit Theorem Definition
the sum of a large number of independent random variables is approximately normally distributed
Univariate vs. Multivariate distributions
- Univariate: describes the distributions of a single random variable
- Multivariate: specifies probabilities associated with a group of random variables
- Takes into account the interrelationships between the two variables
Parameters needed to describe a univariate (2) and multivariate (3) normal distribution:
Multivariate parameters:
- The means of the individual random variables
- The variances of the individual random variables
- The correlation between each possible pair of random variables
Univariate parameters:
- Mean
- Standard deviation
Confidence interval formulas
- 90% CI = sample mean + or - 1.65 (standard deviation)
- 95% CI = sample mean + or - 1.96 (standard deviation)
- 99% CI = sample mean + or - 2.58 (standard deviation)
What does E(Rp) and RT stand for?
Rp - expected portfolio return
RT - target/threshold return
3 Features of a lognormal distribution
- Bounded by zero on the lower end
- Upper end is unbounded
- skewed to the right (positive skew)
Continuously compounded rate of return from HPR (holding period return) =
Rcc = ln (1 + hpr)
Volatility definition and formula. How do you annualize volatility?
Volatility is expressed by taking the standard deviation of continuously compounded returns
Continuosly compounded return = erT
Where rT is the compounded return
rT = (VT/Vo)
VT is the ending asset price
Vo is the beginning asset price
Take the standard deviation of continously compounded returns x square root of 250 = annualized volatility
Two major assumptions that are necessary for the probability function of a binomial random variable
- The probability of an earnings increase is constant from year to year
- Earnings increases are independent trials
Calculate mean/expected value and variance of a cumulative distribution function for a continuous uniform distribution
Mean/expected value = (a+b)/2
Variance = (b-a)/12
