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.

Remember When…

• Experiment: Toss a coin three times.
• What is the number of possible outcomes?
  • HHH, HHT, HTH, HTT, THH, THT, TTH, TTT (8 possible outcomes)
• Let X = number of tails in 3 coin tosses.
• Value of x:
  • 0, 1, 1, 2, 1, 2, 2, 3

Binomial Distribution Example: Number of Tails in Three Coin Flips

• Let T = “success” in a coin flip.
• \pi = probability of success in each trial.
• X = number of successes in 3 coin flips.
• X \sim Bin(3, \pi). X is a binomial random variable with parameters n=3 and \pi.

Binomial Distribution

• \binom{n}{x} = \frac{n!}{x!(n-x)!} are called binomial coefficients and are referred to as “n choose x”.
  • This is the number of ways of choosing x elements from a set of n elements (n ≥ x).
• n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1 is “n-factorial”.
• If X \sim Bin(n, \pi), then p(x) = \binom{n}{x} \pi^x (1 - \pi)^{n-x}.

Binomial Distribution

• p(x) = \binom{n}{x} \pi^x (1 - \pi)^{n-x}
• Interpretation of the formula:
  • The probability of getting x successes (and n-x failures) is \pi^x (1 - \pi)^{n-x}.
  • The coefficients count the ways in which x successes can occur in n trials.

Application: Probability of Getting Exactly Two Tails in Three Coin Flips

• In this application, n = 3, \pi = \frac{1}{2}, and x = 2.

To Sum Up

• A random variable is binomial if:
  • It measures the number of successes in n identical trials.
  • Only two possible outcomes in each trial: success or failure.
  • The probability of success (and of failure) remains constant from one trial to another.
  • Trials are independent.