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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