1.4 common probability distributions: Intro

Question 1

Discrete random variables

Answer 1

have a countable number of possible values, which could still be unlimited (e.g., all integers)

Question 2

Continuous random variables

Answer 2

do not have a countable number of outcomes.

Question 3

true or false

Sometimes, continuous distributions are used to model a discrete random variable (e.g., quoted stock price) just for convenience

Answer 3

true

Question 4

The probability function

Answer 4

specifies the probability that a random variable takes on a specific value

P(X = x) is the probability the random variable X equals a specific value x

Sometimes the notation is shortened to p(x)

The probability function is between 0 and 1, and all the probabilities sum to 1

Question 5

how is the the probability function called for continuous distributions?

Answer 5

the probability density function, f(x)

Question 6

The cumulative distribution function

Answer 6

gives the probability the random variable X is less than or equal to a specific value x

It is represented by F(x) = P( X ≤ x)

Question 7

The discrete uniform distribution

Answer 7

a simple probability distribution because all possible outcomes are equally likely

For example, rolling a six-sided die represents a discrete uniform distribution.

–> There are six possible outcomes (1, 2, 3, 4, 5, 6) that are equally likely.

–> The probability of each outcome is 1/6, or 16.7%

Question 8

The continuous uniform distribution

Answer 8

very simple, yet still useful for generating random numbers and representing uncertainty when all outcomes are equally likely.

For example, when a = 0 and b = 10, it means the random variable has an equal chance of being any value between 0 and 10

Question 9

A random variable with an infinite number of possible values can be classified as:

A: a lognormal random variable only.

B: a continuous random variable only.

C: either a continuous or a discrete random variable.

Answer 9

C: either a continuous or a discrete random variable.

Question 10

A random variable Y has a continuous uniform distribution over the interval [0, 1]. The probability of Y having a value between 0.1 and 0.5 is closest to:

A. 30%.

B. 40%.

C. 50%.

Answer 10

B. 40%.

Question 11

Which of the following is a continuous random variable?

A. The value of a futures contract quoted in increments of $0.05

B. The total number of heads recorded in 1 million tosses of a coin

C. The rate of return on a diversified portfolio of stocks over a three-month period

Answer 11

C. The rate of return on a diversified portfolio of stocks over a three-month period

Question 12

The binomial distribution

Answer 12

useful when modeling a random variable with only two outcomes (e.g., success or failure)

It is commonly used to model asset prices underlying options

assumes p is constant for all trials and the trials are independent

Question 13

A Bernoulli trial

Answer 13

only has two outcomes. Often p is used to represent the probability of success.

This approach can be used to model stock price movements

–> If the stock has an initial price of S, it can move to uS(with a probability of p) or dS (with a probability of 1−p) over one period

Question 14

A binomial random variable X

Answer 14

the number of successes in n Bernoulli trials

The binomial random variable can be described by two parameters, n and p

Question 15

The binomial distribution is only symmetric if what?

what is the mean and variance

Answer 15

if p = 0,5

mean = np

variance = np(1 - p)

Question 16

The binomial tree

Answer 16

used to model stock price movements

can be extended for multiple periods

This approach is useful for pricing American options that can be exercised early because the value can be calculated at each node