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Probability Distributions. A random variable is a variable (letter) whose values are determined by chance It can be continuous (something that’s measured) or discrete (something that’s counted) Continuous examples: temperature, weight Discrete examples: numbers on dice

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probability distributions
Probability Distributions
  • A random variable is a variable (letter) whose values are determined by chance
    • It can be continuous (something that’s measured) or discrete (something that’s counted)
    • Continuous examples: temperature, weight
    • Discrete examples: numbers on dice
  • A discrete probability distribution consists of:
    • The values a random variable can take
    • The corresponding probabilities
probability distribution examples
Probability Distribution Examples
  • Rolling one die and looking at the number
    • Outcomes are 1, 2, 3, 4, 5, 6
    • Probabilities are 1/6, 1/6, 1/6, 1/6, 1/6, 1/6
  • Tossing 3 coins and counting number of heads
    • Outcomes are 0, 1, 2, 3
    • Probabilities are 1/8, 3/8, 3/8, 1/8
  • Selecting one card from a deck, looking for a spade
    • Outcomes are “spade,” “not a spade”
    • Probabilities are 13/52 and 39/52
probability distribution requirements
Probability Distribution Requirements
  • The sum of the probabilities of all the events in the sample space must be 1:

∑P(X)=1

  • The probability of each event in the sample space must be between 0 and 1, inclusive:

0 ≤ P(X) ≤ 1 for all X

statistics of a probability distribution
Statistics of a Probability Distribution
  • Mean: µ=∑X•P(X)
  • Expected value: Another word for the mean, written E(X)
  • Example: Find the expected value of the gain (amount of money you make) if you buy a lottery ticket
binomial distribution
Binomial Distribution
  • A binomial experiment is a special kind of probability experiment, with the following requirements:
    • A fixed number of trials
    • Each trial has two outcomes, designated success and failure
    • The outcomes of the trials are independent
    • The probability of success is the same for each trial
  • The outcomes and probabilities associated with a binomial experiment are called a binomial distribution
binomial distribution6
Binomial Distribution
  • p = Probability of success in one trial
  • q = Probability of failure in one trial
    • Note that q = 1 - p
  • n = Number of trials
  • X = Number of successes in n trials
    • Note that 0 ≤ X ≤ n
  • In a binomial experiment, the probability of exactly X successes in n trials is:
binomial distribution7
Binomial Distribution
  • You do not need to use the formula to calculate the binomial probabilities
  • You can read the probabilities from a table
  • Use Table B in Appendix C, starting on page 626
  • Specify n, X, and p
  • Add or subtract table values to get “at most” or “at least” answers
statistics for the binomial distribution
Statistics for the Binomial Distribution
  • Mean = µ
  • Standard deviation = 
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