Chapter 6: Discrete Probability Distributions

Section 6.1: Discrete Random Variables

Knowledge Prerequisites

declarative knowledge (definitions)
1. Section 1.1:
  - discrete variable
2. Section 2.1:
  - frequency distribution
  - relative frequency
3. Section 2.2:
  - histogram
4. Section 3.1:
  - mean
5. Section 3.2:
  - standard deviation
  - variance
6. Section 5.1:
  - probability
  - outcome
  - random event
  - probability rules
  - probability model
  - unusual event
  - equally-likely outcomes
  - classical [a.k.a., classicist] probability
7. Section 5.2:
  - disjoint [a.k.a., mutually-exclusive] events
  - addition rule for probability
  - complement of an event, E^c
  - complement rule for probabilities
  - keywords for probability:
    - or
    - and [sometimes you will need to identify "and" even though word is not used]
    - not
8. Section 5.3:
  - independent events
  - multiplication rule for probability
procedural knowledge
1. Section 2.1:
  - calculate relative frequency distribution
  - how to construct relative frequency distribution
2. Section 2.2:
  - how to construct a histogram
  - how to interpret a histogram
3. Section 5.1:
  - verify probability models
  - write complete sample space for a given situation
  - calculate probabilities using classicist approach
  - construct a probability model using given data
4. Section 5.2:
  - how to calculate the probability of an event using the addition rule
  - how to calculate the probability of an event using the complement rule
5. Section 5.3:
  - how to calculate the probability of an event using the multiplication rule
conditional knowledge
1. Section 2.1:
  - how to check your work in relative frequency distributions
  - how to identify the difference between frequency and relative frequency distributions
2. Section 2.2:
  - how to interpret the bars in a relative frequency histogram
  - identify and explain why a distribution is symmetric, skewed left, or skewed right
3. Section 3.1:
  - know the importance of using the correct symbol for population mean and sample mean
  - interpret the mean and median of a distribution, i.e., explain what they describe about a distribution
4. Section 3.2:
  - know the importance of using the correct symbol for population standard deviation and sample standard deviation
  - know the importance of using the correct symbol for population variance and sample variance
  - explain what measures of dispersion describes about a distribution
5. Section 5.1:
  - know the importance of the concept of randomness or chance in probability
  - interpret value of probability
  - identify when an event is considered unusual
6. Section 5.2:
  - how to determine which probability rule to use for a given problem
  - how to identify the complement of an event
7. Section 5.3:
  - conceptually identify if two events are independent
  - how to use the multiplication rule for a given problem

Learning Goals

declarative knowledge (definitions)
1. random variable
2. discrete random variable
3. continuous random variable
4. probability distribution
5. probability histogram
6. mean of discrete random variable (a.k.a., expected value)
7. variance of discrete random variable
8. standard deviation of discrete random variable
procedural knowledge
1. construct a probability histogram [on graph paper]
2. compute the mean of a discrete ramdom variable, using TI83/84
3. compute the variance of a discrete ramdom variable, using TI83/84
4. compute the standard deviation of a discrete ramdom variable, using TI83/84
5. calculate the probabilities from a frequency distribution
conditional knowledge
1. how to identify possible values of a discrete random variable
2. determine if a distribution is a discrete probability distribution
3. identify the difference between a discrete random variable and a continuous random variable
4. interpret the mean of a discrete ramdom variable
important notes
1. answers must be in complete sentences using the correct symbols, e.g., P(x ≥ 6) = 0.032
2. answers to probability questions can be in any of the following form: unreduced fraction, reduced fration, decimal, or percentage
3. you are required to use the correct symbols for mean, μ_X, and standard deviation, σ_X, of a binomial variable

Section 6.2: The Binomial Probability Distribution

Knowledge Prerequisites

declarative knowledge (definitions)
1. Section 1.1:
  - discrete variable
2. Section 2.1:
  - frequency distribution
  - relative frequency
3. Section 2.2:
  - histogram
4. Section 3.1:
  - mean
5. Section 3.2:
  - standard deviation
  - variance
6. Section 5.1:
  - probability
  - outcome
  - random event
  - probability rules
  - probability model
  - unusual event
  - equally-likely outcomes
  - classical [a.k.a., classicist] probability
7. Section 5.2:
  - disjoint [a.k.a., mutually-exclusive] events
  - addition rule for probability
  - complement of an event, E^c
  - complement rule for probabilities
  - keywords for probability:
    - or
    - and [sometimes you will need to identify "and" even though word is not used]
    - not
8. Section 5.3:
  - independent events
  - multiplication rule for probability
9. Section 6.1:
  - all
procedural knowledge
1. Section 2.1:
  - calculate relative frequency distribution
  - how to construct relative frequency distribution
2. Section 2.2:
  - how to construct a histogram
  - how to interpret a histogram
3. Section 5.1:
  - verify probability models
  - write complete sample space for a given situation
  - calculate probabilities using classicist approach
  - construct a probability model using given data
4. Section 5.2:
  - how to calculate the probability of an event using the addition rule
  - how to calculate the probability of an event using the complement rule
5. Section 5.3:
  - how to calculate the probability of an event using the multiplication rule
6. Section 6.1:
  - all
conditional knowledge
1. Section 2.1:
  - how to check your work in relative frequency distributions
  - how to identify the difference between frequency and relative frequency distributions
2. Section 2.2:
  - how to interpret the bars in a relative frequency histogram
  - identify and explain why a distribution is symmetric, skewed left, or skewed right
3. Section 3.1:
  - know the importance of using the correct symbol for population mean and sample mean
  - interpret the mean and median of a distribution, i.e., explain what they describe about a distribution
4. Section 3.2:
  - know the importance of using the correct symbol for population standard deviation and sample standard deviation
  - know the importance of using the correct symbol for population variance and sample variance
  - explain what measures of dispersion describes about a distribution
5. Section 5.1:
  - know the importance of the concept of randomness or chance in probability
  - interpret value of probability
  - identify when an event is considered unusual
6. Section 5.2:
  - how to determine which probability rule to use for a given problem
  - how to identify the complement of an event
7. Section 5.3:
  - conceptually identify if two events are independent
  - how to use the multiplication rule for a given problem
8. Section 6.1:
  - all

Learning Goals

declarative knowledge (definitions)
1. binomial experiment
2. trial of a binomial experiment
3. four conditions necessary for a binomial experiment
4. binomial random variable
5. binomial probability function and its component parts: n, p, and x
6. mean of a binomial random variable
7. standard deviation of a binomial random variable
procedural knowledge
1. construct a binomial probability histogram [on graph paper]
2. calculate mean of a binomial probability histogram
3. calculate standard deviation of a binomial probability histogram
4. identify a probability experiment as a binomial experiment
5. how to calculate the probability of k successes, i.e., P(X = k) using binompdf on the TI83/84: http://stats.jjw3.com/math1431/ti83binProb.htm
6. how to calculate the probability of various k successes using binomcdf on the TI83/84: http://stats.jjw3.com/math1431/ti83binProb.htm
conditional knowledge
1. how to determine which probability rule to use for a given problem
2. how the n and p affect the shape of the binomial probability histogram
3. how to identify when to use the complement for binomial probability
4. how to identify which inequality (≤, <, >, ≥) is needed to calculate the probability of a binomial variable [See Table 9 on p. 313]
5. explain when a binomial distribution is approximately normal
important notes
1. you are not expected to use the table in the book; you are expected to use the TI83/84 to calculate all answers
2. you must state the conditions to a binomial experiment before using binompdf or binomcdf on the TI83/84
3. answers must be in complete sentences using the correct symbols, e.g., P(x ≥ 6) = 0.032
4. you are required to use the correct symbols for mean, μ_X, and standard deviation, σ_X, of a binomial variable
5. answers to probability questions can be in any of the following form: unreduced fraction, reduced fration, decimal, or percentage
6. P(at least 1) = 1 – P(none)

Chapter 6: Required Formulas – Need to Know for Tests

Relative Frequency:
Mean of a Binomial Random Variable: μ_X = np

Chapter 6: Required Formulas – Will be Given on Tests

Standard Deviation of a Binomial Random Variable: