Common Probability Distributions
Describe the set of possible outcomes of a specified discrete random variable
A discrete random variable X can take on a limited number of outcomes (x1,x2,...xn where n possible outcomes) or a discrete random variable can take on a unlimited number of outcomes y1,y2...(without end). Because we can count all the possible outcomes of X and Y (even if we go on forever in the case of Y), both X and Y satisfy the definition of a discrete random variable.
Describe a binomial
A binominal random variable: a binominal random variable is the sum of Bernoulli random variables. (X=Y1+Y2+....+Yn)
Describe a discrete random variable
A discrete uniform random variable : the distribution has a finite number of specified outcomes and each outcomes is equally likely.
Define a probability distribution and distinguish between discrete and continuous random variables and their probability functions
A probability distribution specifies the probabilities of the possible outcomes of a random variable. Four types of probability distributions: uniform, binomial, normal and lognormal. Two basic types of random variables: discrete and continuous random variables. A discrete random variable can take on at most a countable number of possible values (limited number of outcomes) Continuous random variable: outcome cannot be counted
Distinguish between a univariate and a multivariate distribution and explain the role of correlation in the multivariate normal distribution
A univariate distribution describes a single random variable A multivariate distribution specifies the probabilities for a group of related random variables. A multivariate normal distribution for the returns on n stocks is completely defined by 3 lists of parameters; the list of the mean returns on the individual securities (n means in total), the list of securities variances of return (n variances in total) and the list of all distinct pairwise return correlations: n(n-1)/2 distinct correlation in total Which statement is most accurate when considering a multivariate normal distribution for the returns on 10 stocks from different sectors? There are: A multivariate normal distribution for the returns of n stocks is defined by n means, nvariances, and n × (n − 1)/2 distinct correlations. There should be 45 = [10 × (10 − 1)]/2 distinct correlations in this case
Determine the probability that a normally distribute random variable lies inside a given interval
Approx. 50% of all observations fall in to interval: µ +- (2/3) standard deviation(σ) 68% µ +- σ 95% µ +- 2σ 99% µ +- 3σ
Interpret a cumulative distributions function
Gives the probability that a random variable X is less than or equal to a particular value. The CDF has two characteristic properties: CDF lies between 0 and 1 for any x and as we increase x, the CDF either increases or remains constant.
Compare Monte Carlo Simulation and Historical Simulation
In historical simulation: samples from a historical record of returns to simulate a process. For instance: in one example we would draw K returns from that record to generate one simulation trail.
Explain Monte Carlo simulations
Monte Carlo simulation: in finance involves the use of a computer to represent the operation of a complex financial system. Create large number of random samples from a specified probability distribution or distributions to represent the role of risk in the system. THE ANALYST CHOOSES THE PROBABILITY DISTRIBUTIONS IN MONTE CARLO SIMULATION
Define shortfall risk, calculate the safety-first ratio, and select an optimal portfolio using Roy's safety-first criterion
Shortfall risk: the risk that portfolio value will fall below some minimum acceptable level over some time horizon. Safety first ratio: SFRatio={E(rp)-Rl}/σ (if we substitute the risk free rate for the Rl than the SF ratio becomes the Sharpe ratio) Roys safety first criterion: choose the portfolio with the highest SF ratio
Explain the key properties of the normal distribution
Symmetrical and bell-shaped The range of possible outcomes are all real numbers lying between negative infinity and infinity. The normal distribution is completely described in two parameters: its mean and variance (or standard deviation) The normal distribution has a skewness of 0 (its symmetric), a kurtosis (measure of peakedness) of 3 and its excess kurtosis equals 0. As a consequence of symmetry, the mean, median and mode are all equal for normal random variables.
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
The continuously compounded return associated with a holding period is the natural logarithm of the ending price over the beginning price. The continuously compounded return from t to t+1 is ln(1+Rt+1) Discretely rate of return is the normal rate of return?
The probability of x outcomes with N samples
The following equation allows us to find the probability of x outcomes with N samples. Where p is the probability of success and q in (1-p). If we want to find the likelihood of x or more outcomes, simply add the equation together. n!/(n-X)!X! * p^x * q^n-x
Explain the relationship between normal and lognormal distribution and why the lognormal distribution is used to model asset prices
The lognormal distribution: a random variable Y follows a lognormal distribution if its natural logarithm, ln Y, is normally distributed. If you think of the term lognormal as "the log is normal" than you will have no trouble remembering this relationship. Lognormal distribution is bounded below and is skewed to the right. If a stock's continuously compounded return is normally distributed, then the future stock price is necessarily lognormally distributed.
Calculate and interpret probabilities of a random variable given its cumulative distribution function
To find f(x) (where F(x)= P(X<x)) we sum up or cumulate values of the probability function for all outcomes less than or equal to x.
Define the standard normal distribution, explain how to standardize a random variable and calculate and interpret probabilities using the standard normal distribution
Two steps in standardizing a random variable X Subtract the mean of X from X, then divide that result by the standard deviation of X. The result is the standard normal random variable, Z Z= (X-µ)/ σ We can answer all probability questions about X using the standardized values and probability tables for Z. The 90th percentile point is Z=1.282 The 95th percentile point is Z=1.65 The 99th percentile point is Z=2.327
Describe a Bernoulli random variable
the probability of any single experiment that asks a yes or not question. If we let y equal 1 when the outcome is a success and y equal 0 when the outcome is a failure. (example will a stock move up tomorrow?)
