DISCRETE PROBABILITY DISTRIBUTIONS

DISCRETE PROBABILITY DISTRIBUTIONS

1. BINOMIAL DISTRIBUTIONS

   • PROPERTIES
     • The experiment consists of n repeated trials.
     • Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
     • The probability of success, denoted by P, is the same on every trial.
     • The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.
   • EXPERIMENT
     • Flip a coin 2 times and count the number of times the coin lands on heads. This is a binomial experiment because:
       • 1) The experiment consists of repeated trials. We flip a coin 2 times.
       • 2) Each trial can result in just two possible outcomes - heads or tails.
       • 3) The probability of success is constant - 0.5 on every trial.
       • 4) The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials.
   • CALCULATE
     • PROBABILITIES
       • P(k out of n) = {n!/[k!(n-k)!]} (p^k)(1-p)^(n-k)
     • MEAN (μx)
       • n * P
     • VARIANCE (σ2x)
       • n * P * ( 1 - P )

2. GEOMETRIC DISTRIBUTIONS

   • PROPERTIES
     • The experiment consists of a sequence of independent trials.
     • Each trial results in one of two outcomes: success, S, or failure, F.
     • The geometric random variable X is defined as the number of trials until the first S is observed.
     • The probability p(x) is the same for each trial.
   • EXPERIMENT
     • You ask people outside a polling station who they voted for until you find someone that voted for the independent candidate in a local election. The geometric distribution would represent the number of people who you had to poll before you found someone who voted independent. You would need to get a certain number of failures before you got your first success.
   • CALCULATE
     • MEAN (μx)
       • 1/p
     • PROBABILITIES
       • (1-p)^(x-1)p
     • VARIANCE (σ2x)
       • (1-p)/(p^2)

3. NEGATIVE-BINOMIAL DISTRIBUTIONS

   • PROPERTIES
     • The experiment consists of x repeated trials.
     • Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
     • The probability of success, denoted by P, is the same on every trial.
     • The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.
     • The experiment continues until r successes are observed, where r is specified in advance.
   • EXPERIMENT
     • The properties are the same as those listed for a binomial distribution, with the exception that the trials will be repeated until a fixed number of successes occur.
   • CALCULATE
     • MEAN (μx)
       • r / P
     • PROBABILITIES
       • b*(x; r, P) = x-1Cr-1 * Pr * (1 - P)x - r
     • VARIANCE (σ2x)
       • rq / p^2

4. HYPERGEOMETRY DISTRIBUTIONS

   • PROPERTIES
     • A sample of size n is randomly selected without replacement from a population of N items.
     • In the population, k items can be classified as successes, and N - k items can be classified as failures.
   • EXPERIMENT
     • You have an urn of 10 marbles - 5 red and 5 green. You randomly select 2 marbles without replacement and count the number of red marbles you have selected.
   • CALCULATE
     • MEAN (μx)
       • n * k / N
     • PROBABILITIES
       • h(x; N, n, k) = [ kCx ] [ N-kCn-x ] / [ NCn ]
     • VARIANCE (σ2x)
       • n * k * ( N - k ) * ( N - n ) / [ N2 * ( N - 1 ) ]