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Chapter 5 - Discrete Probability Distributions
# **5-1 Probability Distributions**
* A **random variable** is a variable that has a single numerical value, determined by chance, for each outcome of a procedure
* A **probability distribution** is a description that gives the probability for each value of the random variable. Often expressed in the format a table, formula, or graph.
* A **discrete random variable** has a collection of values that is finite/countable.
* A **continuous random variable** has infinitely many values, and the collection of values is not countable.
* Probability histograms are one of the ways to graph a probability distribution
* The **expected value** of a discrete random variable x is denoted by E, and it is the mean value of the outcomes, so E = mu and E can also be found by evaluating the sum of x \* P(x)
* The expected value of a random variable x is equal to the mean mu
* If, under a given assumption, the probability of a particular outcome is very small and the outcome occurs significantly less than or significantly greater than what we expect with that assumption, we conclude that the assumption is probably not correct
# **5-2 Binomial Probability Distributions**
* A **binomial probability distribution**results from a procedure that meets these 4 requirements:
* Procedure has a fixed number of trials
* The trials must be independent, meaning that the outcome of any individual trial doesn't affect the probabilities in the other trials
* Each trial must have all outcomes classified into exactly 2 categories, commonly referred to as success and failure
* The probability of a success remains the same in all trials
* When using a binomial probability distribution, always be sure that x and p are consistent in the sense that they both refer to the same category being called a success
* When sampling without replacement and the sample size is no more than 5% of the size of the population, treat the selections as being independent even though they are actually dependent.
* Significantly low values
* A **random variable** is a variable that has a single numerical value, determined by chance, for each outcome of a procedure
* A **probability distribution** is a description that gives the probability for each value of the random variable. Often expressed in the format a table, formula, or graph.
* A **discrete random variable** has a collection of values that is finite/countable.
* A **continuous random variable** has infinitely many values, and the collection of values is not countable.
* Probability histograms are one of the ways to graph a probability distribution
* The **expected value** of a discrete random variable x is denoted by E, and it is the mean value of the outcomes, so E = mu and E can also be found by evaluating the sum of x \* P(x)
* The expected value of a random variable x is equal to the mean mu
* If, under a given assumption, the probability of a particular outcome is very small and the outcome occurs significantly less than or significantly greater than what we expect with that assumption, we conclude that the assumption is probably not correct
# **5-2 Binomial Probability Distributions**
* A **binomial probability distribution**results from a procedure that meets these 4 requirements:
* Procedure has a fixed number of trials
* The trials must be independent, meaning that the outcome of any individual trial doesn't affect the probabilities in the other trials
* Each trial must have all outcomes classified into exactly 2 categories, commonly referred to as success and failure
* The probability of a success remains the same in all trials
* When using a binomial probability distribution, always be sure that x and p are consistent in the sense that they both refer to the same category being called a success
* When sampling without replacement and the sample size is no more than 5% of the size of the population, treat the selections as being independent even though they are actually dependent.
* Significantly low values
