(R9) Common Probability Distributions Flashcards
Probability distribution
Lists all the possible outcomes of an experiment, along with their associated probabilities. A probability distribution completely describes a random variable.
AKA: probability function
Discrete random variable
Has positive probabilities associated with a finite number of outcomes (countable, non zero probabilities for each outcome)
Continuous random variable
Has positive probabilities associated with a range of outcome values. The probability of any single value is zero (Uncountable). Described by probability density function instead of a probability function or probability distribution
Probability Function
All possible outcomes plus associated probabilities. Completely describes the variable
Cumulative Distribution Function
A function giving the probability that a random variable is less than or equal to a specified value. Either increases or remains constant over each possible outcome (Used for discrete and continuous random variables)
Discrete Uniform Distribution and 3 characteristics
The probability of every finite possible outcome is equally likely.
- Outcomes are countable
- probability between 0 and 1
- Sum of all probabilities equal 1
Binomial Random Variable
The number of successes in n Bernoulli trials for which the probability of success is constant for all trials and trials are independent
Bernoulli random Variable
A random variable having outcomes 0 and 1
Bernoulli trial
An experiment that can produce one of two outcomes (success or failure)
Probability of bernoulli trial
1/n where n is the number of trials
Probability of binomial distribution
Mean and variance formula for Bernoulli random variable and binomial random variable
Bernoulli Mean = p
Bernoulli Variance = p (1 - p)
Binomial Mean = np
Binomial Variance = np (1-p)
Binomial Tree
The graphical representation of a model of asset price dynamics in which at each period, the asset moves up with probability p or down with probability (1-p)
Continuous Uniform Distribution
P(X1 <= X <= X2)
Described by a lower limit a and an upper limit b (a and b are parameters of the distribution)
Probability = X2 - X1 / b - a
Normal Distribution
Completely described by mean and variance with a skew of 0 and kurtosis of 3. This distribution is only used for continuous random variables
Central Limit Theorem
The sum of a large number of independent random variables are approximatelly normally dsitributed
Univariate Normal Distribution
A distribution that specifies the probabilities for a single random variable
Multivariate Normal Distribution
A probability distribution for a group of random variables that is completely defined by the means and variances of the variables plus all the correlations between pairs of variables
Confidence Intervals (90, 95, and 99%)
90%: sample mean + or - 1.65s
95%: sample mean + or - 1.96s
99%: sample mean + or - 2.58s
Standardizing definition and formula
Refers to the number of standard deviations away from the mean an observation lies
z = (observed value - mean) divdided by standard deviation
Roy’s safety first criterion (Safety first ratio) formula
(Expected return on portfolio - return on risk-free return) divided by standard deviation
The larger the ratio the better
Shortfall Risk
The risk that portfolio value will fall below some minimum acceptable level over some time horizon
Define lognormal distribution and list 3 characteristics
A random variable x follows a lognormal distribution if its natural log (LN Y) is normally distributed. Used to model asset prices, while normal distribution model returns.
- Bounded by zero on the lower end
- Upper end is unbounded
- Positvely skewed (skewed to right)
Formula for lognormal distribution
Lognormal distribution = (VT/V0), where VT is ending value asset price and V0 is beginning value asset price
Normal distribution = Ln (VT/V0)
Continuously Compounded Return Formula
The natural logarithm of 1 plus the holding period return, or equivalently, the natural logarithm of the ending price over the beginning price
ln(1+HPR) or ln(Vt/Vo)
