(10) 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.
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.
A continuous random variable as positive probabilities associated with a range of outcome values-the probability of any single value is zero.
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).
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 random variable, a Bernoulli random variable, and a binomial random variable.
A discrete uniform distribution is one where there are n discrete, equally likely outcomes.
LOS 10. e: Define a discrete uniform random variable, a Bernoulli random variable, and a binomial random variable.
The binomial distribution is a probability distribution for a binomial (discrete) random variable that has two possible outcomes.
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
LOS 10. h: 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.
LOS 10. i: Define the continuous uniform distribution and calculate and interpret probabilities, given a continuous uniform distribution.
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, then for:
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.
LOS 10. k: Distinguish between a univariate and a multivariate distribution and explain the role of correlation in the multivariate normal distribution.
Multivariate distributions describe the probabilities for more than one random variable, whereas a univariate distribution is for a single random variable.
LOS 10. l: 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 tits 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. m: 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. n: 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.
