L9. Common Probability Distributions

Question 1

Learning outcomes

Answer 1

a. Define a probability distribution and distinguish between discrete and continuous random variables and their probability functions
b. Describe the set of possible outcomes of a specified discrete random variable
c. Interpret a cumulative distribution function
d. Calculate and interpret probabilities for a random variable, given its cumulative distribution function
e. Define a discrete uniform uniform random variable, a Bernoulli random variable, and a binomial random variable
f. Calculate and interpret probabilities given the discrete uniform and the binomial distribution functions
g. Construct a binomial tree to describe stock price movement
h. Define the continuous uniform distribution and calculate and interpret probabilities given a continuous uniform distribution
i. Explain the key properties of the normal distribution
j. Distinguish between a univariate and a multivariate distribution and explain the role of correlation in the multivariate normal distribution
k. Determine the probability that a normally distributed random variable lies inside a given interval
l. Define the standard normal distribution, explain how to standardise a random variable, and calculate and interpret probabilities using the standard normal distribution
m. Define shortfall risk, calculate the safety-first ratio, and select an optimal portfolio using Roy’s safety first criterion
n. Explain the relationship between normal and lognormal distributions and why the lognormal distribution is used to model asset prices
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
p. Explain monte carlo simulation and describe its application and limitations
q. Compare monte carlo simulation and historical simulation

Question 2

Probability distribution

Answer 2

Specifies the probabilities of a random variable’s possible outcomes

4 types of distributions

1. Uniform
2. Binomial
3. Normal
4. Lognormal

Question 3

Discrete random variables

Answer 3

Random variable is a quantity whose future outcomes are uncertain

2 types of basic types of random variables
- discrete random variables
a) can take on at most a countable number of possible values
eg. variable X can take on a limited number of outcomes (n possible outcomes)
eg. variable Y can take on an unlimited number of outcomes (without end)
BUT we can count all the possible outcomes of X and Y (even if we go on forever in the case of Y)

• continuous random variables
  a) cannot count the outcomes

Question 4

Probability distribution association

Answer 4

2 ways to view probability distribution

1. Basic view = probability function
   - specifies the probability that a random variable takes on a specific value: p(X = x) is the probability that a random variable X takes on the value x

• for discrete random variable, notation for probability function is p(x) = P(X = x)
• for continuous random variable, notation is denoted f(x) and called the probability density function (pdf) or the density

2 properties

a) 0 ≲ p(x) ≲ 1 because probability is between 0 - 1
b) sum of probabilities p(x) over all values of X equals 1. if we add up the probabilities of all the distinct possible outcomes of a random variable, the sum must = 1

2. Cumulative distribution function (CDF)
   - a function giving the probability that a random variable is less than or = to a specified value
   - finding the probability of a range of outcomes rather than a specific outcome
   - notation for both discrete and continuous random variable is F(x) = P(X ≲ x)

Question 5

Discrete uniform distribution

Answer 5

Probability function is uniform, meaning expected values are all the same

EG.
X=x Probability function cumulative distribution fn
1 0.125 0.125
2 0.125 0.250
3 0.125 0.375
4 0.125 0.500
5 0.125 0.625
6 0.125 0.750
7 0.125 0.875
8 0.125 1.000

What is the probability of P(X≲7)?
Ans: 87.5%

What is the probability of P(4 ≲ X ≲ 6)?
Ans: 0.750 - 0.375 = 0.375 or 37.5%

What is the probability of P(4 < X < 6)?
Ans: 0.750 - 0.500 = 0.250 or 25%

Properties of CDF

1. cdf lies between 0 and 1 for any x: 0≲F(x) ≲ 1
2. as we increase x, remain constant or increases

Question 6

Continuous uniform distribution

Answer 6

The pdf for a uniform random variable is

f(x) = 1 /b-a
for a < x < b