1. Chapter 3 Discrete Random Variables and Probability Distributions  3.1 - Random Variables.2 - Probability Distributions for Discrete Random Variables.3 - Expected Values (cont’d).4 - The Binomial Probability Distribution.5 - Hypergeometric and Negative Binomial Distributions.6 - The Poisson Probability Distribution

2. POPULATION Pop values x Probabilities f (x) x1x1 f (x 1 ) x2x2 f (x 2 ) x3x3 f (x 3 ) ⋮⋮ Total1 Example: X = Cholesterol level (mg/dL) random variable X Probability Histogram X Total Area = 1 f(x) = Probability that the random variable X is equal to a specific value x, i.e., |x|x “probability mass function” (pmf) f(x) = P(X = x)

3. X POPULATION Pop values x Probabilities f (x) x1x1 f (x 1 ) x2x2 f (x 2 ) x3x3 f (x 3 ) ⋮⋮ Total1 Example: X = Cholesterol level (mg/dL) random variable X Probability Histogram Total Area = 1 F(x) = Probability that the random variable X is less than or equal to a specific value x, i.e., “cumulative distribution function” (cdf) F(x) = P(X  x) |x|x

4. Hey!!! What about the population mean  and the population variance  2 ??? POPULATION Pop vals xpmf f (x) cdf F(x) = P(X  x) x1x1 f (x 1 ) F(x 1 ) = f(x 1 ) x2x2 f (x 2 ) F(x 2 ) = f(x 1 ) + f(x 2 ) x3x3 f (x 3 ) F(x 3 ) = f(x 1 ) + f(x 2 ) + f(x 3 ) ⋮⋮⋮ Total1 increases from 0 to 1 Example: X = Cholesterol level (mg/dL) random variable X X = P( a  X  b) F( b ) – F( a – ) = Calculating “interval probabilities”… f (x) F(b) = P(X  b) P(X  b) – P(X  a – ) F( a – ) = P(X  a – ) |a–|a– |a|a |b|b

5. 5 POPULATION Pop values x Probabilities f (x) x1x1 f (x 1 ) x2x2 f (x 2 ) x3x3 f (x 3 ) ⋮⋮ Total1 Population mean Also denoted by E[X], the “expected value” of the variable X. Population variance Example: X = Cholesterol level (mg/dL) random variable X Just as the sample mean and sample variance s 2 were used to characterize “measure of center” and “measure of spread” of a dataset, we can now define the “true” population mean  and population variance  2, using probabilities.

6. 6 POPULATION Pop values x Probabilities f (x) x1x1 f (x 1 ) x2x2 f (x 2 ) x3x3 f (x 3 ) ⋮⋮ Total1 Example: X = Cholesterol level (mg/dL) random variable X General Properties of “Expectation” of X Suppose X is transformed to another random variable, say h(X). Then by def,

7. 7 POPULATION Pop values x Probabilities f (x) x1x1 f (x 1 ) x2x2 f (x 2 ) x3x3 f (x 3 ) ⋮⋮ Total1 Example: X = Cholesterol level (mg/dL) random variable X General Properties of “Expectation” of X Suppose X is constant, say b, throughout entire population… Then by def, b

8. 8 POPULATION Pop values x Probabilities f (x) x1x1 f (x 1 ) x2x2 f (x 2 ) x3x3 f (x 3 ) ⋮⋮ Total1 Example: X = Cholesterol level (mg/dL) random variable X General Properties of “Expectation” of X Suppose X is constant, say b, throughout entire population… Then…

9. 9 POPULATION Pop values x Probabilities f (x) x1x1 f (x 1 ) x2x2 f (x 2 ) x3x3 f (x 3 ) ⋮⋮ Total1 Example: X = Cholesterol level (mg/dL) random variable X General Properties of “Expectation” of X Multiply X by any constant a… Then by def, a

10. 10 POPULATION Pop values x Probabilities f (x) x1x1 f (x 1 ) x2x2 f (x 2 ) x3x3 f (x 3 ) ⋮⋮ Total1 Example: X = Cholesterol level (mg/dL) random variable X General Properties of “Expectation” of X Multiply X by any constant a… Then… i.e.,…

11. 11 POPULATION Pop values x Probabilities f (x) x1x1 f (x 1 ) x2x2 f (x 2 ) x3x3 f (x 3 ) ⋮⋮ Total1 Example: X = Cholesterol level (mg/dL) random variable X General Properties of “Expectation” of X Multiply X by any constant a… Then… i.e.,… Add any constant b to X…

12. 12 POPULATION Pop values x Probabilities f (x) x1x1 f (x 1 ) x2x2 f (x 2 ) x3x3 f (x 3 ) ⋮⋮ Total1 Example: X = Cholesterol level (mg/dL) random variable X General Properties of “Expectation” of X Multiply X by any constant a… Then… i.e.,… Add any constant b to X…

13. 13 POPULATION Pop values x Probabilities f (x) x1x1 f (x 1 ) x2x2 f (x 2 ) x3x3 f (x 3 ) ⋮⋮ Total1 Example: X = Cholesterol level (mg/dL) random variable X General Properties of “Expectation” of X

14. 14 POPULATION Pop values x Probabilities f (x) x1x1 f (x 1 ) x2x2 f (x 2 ) x3x3 f (x 3 ) ⋮⋮ Total1 Example: X = Cholesterol level (mg/dL) random variable X General Properties of “Expectation” of X Multiply X by any constant a… then  X is also multiplied by a.

15. 15 POPULATION Pop values x Probabilities f (x) x1x1 f (x 1 ) x2x2 f (x 2 ) x3x3 f (x 3 ) ⋮⋮ Total1 Example: X = Cholesterol level (mg/dL) random variable X General Properties of “Expectation” of X Multiply X by any constant a… then  X is also multiplied by a. i.e.,…

16. 16 POPULATION Pop values x Probabilities f (x) x1x1 f (x 1 ) x2x2 f (x 2 ) x3x3 f (x 3 ) ⋮⋮ Total1 Example: X = Cholesterol level (mg/dL) random variable X General Properties of “Expectation” of X Add any constant b to X… then b is also added to  X.

17. 17 POPULATION Pop values x Probabilities f (x) x1x1 f (x 1 ) x2x2 f (x 2 ) x3x3 f (x 3 ) ⋮⋮ Total1 Example: X = Cholesterol level (mg/dL) random variable X General Properties of “Expectation” of X Add any constant b to X… then b is also added to  X. i.e.,…

18. 18 POPULATION Pop values x Probabilities f (x) x1x1 f (x 1 ) x2x2 f (x 2 ) x3x3 f (x 3 ) ⋮⋮ Total1 Example: X = Cholesterol level (mg/dL) random variable X General Properties of “Expectation” of X

19. 19 POPULATION Pop values x Probabilities f (x) x1x1 f (x 1 ) x2x2 f (x 2 ) x3x3 f (x 3 ) ⋮⋮ Total1 Example: X = Cholesterol level (mg/dL) random variable X General Properties of “Expectation” of X sample This is the equivalent of the “alternate computational formula” for the sample variance s 2.