9.1: Uniform and Binomial Distributions Flashcards
Define probability distribution and provide an example.
Probability distribution describes the probabilities of all the possible outcomes for a random variable.
The probabilities of all possible outcomes must sum to 1.
Example: each possible outcome on a die has a probability of 1/6, so they sum to 1.
What are the 2 conditions of a discrete random variable? Provide an example of a discrete random variable.
A discrete random variable is one that:
- Has a countable number of possible outcomes.
- For each possible outcome, there is a measurable and positive probability.
Example: the number of days it will rain in a given month, because there is a countable number of possible outcomes ranging from zero to the number of days in the month.
What is a probability function? What are its 2 key properties?
A probability function specifies the probability that a random variable is equal to a specific value.
Two key properties:
- Between the values (can be equal to) 0 and 1
- The sum of the probabilities for all possible outcomes x equals to 1
What is a continuous random variable? Provide an example.
A continuous random variable is one for which the number of possible outcomes is infinite, even if upper and lower bounds exist.
Example: the actual amount of daily rainfall between 0 and 100 inches is an example of a continuous random variable because the actual amount of rainfall can take on an infinite number of values.
What is the difference between discrete probability distributions and continuous probability distributions?
Discrete distribution, probability = 0 when x cannot occur or p(x) > 0 if it can.
Continuous distribution, probability = 0 even when x can occur.
What is a cumulative distribution function? How is it expressed? Explain with an example.
Cumulative distribution function defines the probability that a random variable, X, takes on a value equal to or less than a specific value x.
It represents the sum of the probabilities for the outcomes up to and including a specified outcome. The cumulative distribution function for a random variable X may be expressed as F(x) = P(X <= x).
Example:
X = {1, 2, 3, 4}, p(x) = x/10
so that F(3) = 0.1 + 0.2 + 0.3
meaning that F(3) is the cumulative probability that outcomes 1, 2, or 3 occur.
What is a discrete uniform random variable? Explain with an example.
A discrete uniform random variable is one for which the probabilities for all possible outcomes for a discrete random variable are equal.
Example:
F(6) = 0.2 when P(2 <= X <= 8)
where X = {2, 4, 6, 8}, p(x) = 0.2
What is a binomial random variable?
A) What are its 2 conditions?
B) What type of distribution is it?
A binomial random variable may be defined as the number of successes in a given number of trials, where the outcome can be either success or failure.
2 conditions:
- The probability of success, p, is constant for each trial.
- The trials are independent.
Binomial distribution is a discrete distribution.
What is a Bernoulli random variable? How is it expressed?
A binomial random variable for which the number of trials is 1 is called a Bernoulli random variable.
The binomial probability function defines the probability of x successes in n trials.
** see equation
How is the expected value of a binomial random variable expressed?
Expected value of binomial random variable:
E(X) = np
The intuition is that if n trials are performed, the probability of success on each trial is p such that np successes are expected.
How is the variance of a binomial random variable expressed?
Variance of a binomial random variable:
variance of X = np(1 - p)
Describe stock price movement using a binomial tree.
There are 2 possible outcomes to stock price movements:
- Up transition probability is the probability of an up-move, is p
- Down transition probability is the probability of a down-move, is 1 - p
A binomial tree is constructed by showing all the possible combinations of up-moves and down-moves over a number of successive periods.
What is a node in a binomial tree?
A node is each of the possible values along a binomial tree.
What is continuous uniform distribution? What are its 3 properties?
Continuous uniform distribution is defined over a range that spans between some lower limit, a, and some upper limit, b. These limits serve as the parameters of the distribution, such that outcomes can only occur within these limits.
Properties:
- All values of x are between limits a and b; otherwise the probability of x outside the boundaries is zero.
- P(x1 <= X <= x2) = (x2 - x1)/(b - a) defines the probability of outcomes between x1 and x2.
- Outcomes are equal over equal-size possible intervals such that the distribution is linear over the variable’s range.
