Probability Distributions

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# Probability Distributions - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Probability Distributions' - calder

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Presentation Transcript
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
• 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
• Probabilities are 13/52 and 39/52
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
• 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
• 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 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 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
• Mean = µ
• Standard deviation = 