Econ 120A - Discrete Probability Distributions 3 Notes
Homework Question 1
Given: Random variable X with mean E[X] = 5 and variance Var[X] = 2.
Find: E[3X - 3] and Var[3X - 3].
Homework Question 2
Random variable X has the following probability distribution:
X = -6 with probability p(x) = 1/6
X = 0 with probability p(x) = 2/3
X = 6 with probability p(x) = 1/6
Let Y = X^2. Find E[X] and E[Y].
Homework Question 3
In a quiz, the mean number of points accrued is E[P] = 121 and the variance is Var[P] = 230.
The prize is the square of the accrued points: Prize = P^2.
Find the average prize that a randomly chosen participant obtains.
Binomial Distribution
Many experiments have dichotomous responses (two possible outcomes).
Examples:
Yes/No for a survey question
Pass/Fail for a smog test
Defective/Non-Defective for a product quality test
Bernoulli Random Variables
Bernoulli random variables take on two values (0 and 1) with probabilities (1 - \pi) and \pi, respectively.
Bernoulli Distribution
Code one outcome of the experiment as 0 and the other as 1.
Define a random variable X that takes on the value 0 for one outcome and 1 for the other.
E.g., Smog check: X(fail) = 0, X(pass) = 1
E.g., Coin toss: X(T) = 0, X(H) = 1
Denote the probability of the outcome for which X = 1 as \pi.
Then, the probability of the outcome for which X = 0 is (1 - \pi).
p(0) = (1 - \pi)
p(1) = \pi
The random variable X so defined has a Bernoulli distribution (or is a Bernoulli random variable) with parameter or probability \pi.
Example
Flip a coin and define:
X(H) = 1
X(T) = 0
If the coin is fair, the probability of H and T are both 0.5.
X is a Bernoulli random variable with probability (or parameter) p(0) = p(1) = 0.5
Characteristics of a Binomial Experiment
A binomial experiment consists of n identical trials.
There are only two possible outcomes in each trial: success and failure.
\pi represents the probability of success in a single trial.
1 - \pi is the probability of failure in a single trial. These probabilities do not change from trial to trial.
The trials are independent.
Let X = number of successes in n trials.
If the characteristics of a binomial experiment are present, then the random variable X follows a binomial distribution. X is a binomial random variable, and x can take values of 0, 1, 2, 3, 4, …, n.