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Statistics 3502/6304
Prof. Eric A. Suess
Chapter 4
Introduction to Probability
• Random Variables – Discrete and Continuous
• Probability Distributions for Discrete Random Variables
• Binomial Distribution
• Probability Distribution for Continuous Random Variables
• Normal Distribution
Random Variables – Discrete and Continuous
• Qualitative Random Variables – Categories No - Yes, 0 - 1
• Quantitative Random Variables – Counts 0,1,2,…. X > 0
• When observation on a quantitative random variable can assume only
a countable number of values, the variable is called a discrete
random variable.
• When observations on quantitative random variables can assume one
of the uncountable number of values on a line interval, the variables
is called a continuous random variable.
Probability Distributions for Discrete Random
Variables
• The probability distribution for a discrete random variable displays
the probability 𝑃 𝑦 associated with each value of y
• See Table 4.6 and Table 4.7 page 157
Binomial Distribution
• Many random experiments count the number of successes in a
certain number of trials. When the following properties hold the
experiment is call a Binomial Experiment
1. The experiment consists of n identical trials.
2. Each trial results in one of two outcomes. Success or Failure.
3. The probability of success on a single trial remains the same.
4. The trials are independent.
5. The random variable Y is the number of success observed during
the n trials.
Binomial Distribution
The probability of observing y success in n trials of a binomial
experiment is
𝑛!
𝑃 𝑦 =
𝜋 𝑦 (1 − 𝜋)𝑛−𝑦
𝑦! 𝑛 − 𝑦 !
𝑦 = 0,1, … , 𝑛
Binomial Distribution
• Binomial probabilities can be computed using:
• The formula on the previous slide.
• Using Minitab Calc > Probability Distributions > Binomial…
• Using MS Excel
• See Example 4.8
Binomial Distribution
• Mean of the Binomial Distribution
𝜇 = 𝑛𝜋
• Standard Deviation of the Binomial Distribution
𝜎=
𝑛𝜋(1 − 𝜋)
Probability Distribution for Continuous
Random Variables
• Total area is 1
• For continuous random variables we compute areas as probabilities
𝑃 𝑎<𝑦<𝑏
• Normal curve
• Area under a normal curve.
• Standard Normal Distribution. Table 1
Probability Distribution for Continuous
Random Variables
• Z-score
𝑦 − 𝜇
𝑧=
𝜎
• See example 4.17 page 175
Probability Distribution for Continuous
Random Variables
• 100p%
𝑦𝑝 = 𝜇 + 𝑧𝑝 𝜎
• For p = .975 𝑧𝑝 = 1.96 or 2