2. Chapter 5 Joint Probability Distributions and Random Samples  5.1 - Jointly Distributed Random Variables.2 - Expected Values, Covariance, and Correlation.3 - Statistics and Their Distributions.4 - The Distribution of the Sample Mean.5 - The Distribution of a Linear Combination

3.  Variance: PARAMETERS  Mean: Proof: See PowerPoint section 3.3-cont’d, slide 18, for discrete X. REVIEW POWERPOINT SECTION “3.3-CONT’D” FOR PROPERTIES OF EXPECTED VALUE Variance of a random variable measures how it varies about its mean.

4.  Variances: PARAMETERS  Means: Is there an association between X and Y, and if so, how is it measured?

5.  Variances:  Covariance: PARAMETERS  Means: Proof: Is there an association between X and Y, and if so, how is it measured?

6.  Variances:  Covariance: PARAMETERS  Means: Var(X) Is there an association between X and Y, and if so, how is it measured?

7.  Covariance: Is there an association between X and Y, and if so, how is it measured?

8.  Covariance: Is there an association between X and Y, and if so, how is it measured?

9.  Covariance: Is there an association between X and Y, and if so, how is it measured?

10.  Covariance: Is there an association between X and Y, and if so, how is it measured?

11.  Covariance: Is there an association between X and Y, and if so, how is it measured?

12.  Covariance: Is there an association between X and Y, and if so, how is it measured?

13.  Covariance: … but what does it mean???? Is there an association between X and Y, and if so, how is it measured?

15. In a uniform population, each of the points {(1,1), (1, 2),…, (5, 5)} has the same density. A scatterplot would reveal no particular association between X and Y. In fact, X and Y are statistically independent! Example: It is easy to see that Cov(X, Y) = 0. Is there an association between X and Y, and if so, how is it measured?

16. Fill in the table so that X and Y are statistically independent. Then show that Cov(X, Y) = 0. Exercise: THEOREM. THEOREM. If X and Y are statistically independent, then Cov(X, Y) = 0. However, the converse does not necessarily hold! Is there an association between X and Y, and if so, how is it measured? Exception: The Bivariate Normal Distribution

17. Example: Is there an association between X and Y, and if so, how is it measured?

18. Example: As X increases, Y also has a tendency to increase; thus, X and Y are said to be positively correlated. Likewise, two negatively correlated variables have a tendency for Y to decrease as X increases. The simplest mathematical object to have this property is a straight line. Is there an association between X and Y, and if so, how is it measured?

19.  Variances:  Covariance: PARAMETERS  Means: Always between –1 and +1 (“rho”) Is there an association between X and Y, and if so, how is it measured?  Linear Correlation Coefficient:

20. PARAMETERS  Linear Correlation Coefficient: Is there an association between X and Y, and if so, how is it measured? linear ρ measures the strength of linear association between X and Y. Always between –1 and +1.

21.  Linear Correlation Coefficient: positive linear correlation negative linear correlation Is there an association between X and Y, and if so, how is it measured? PARAMETERS +1 0 -0.5+0.5 weak -0.75+0.75 moderate strong IQ vs. Head circumference

22.  Linear Correlation Coefficient: positive linear correlation negative linear correlation Is there an association between X and Y, and if so, how is it measured? PARAMETERS +1 0 -0.5+0.5 weak -0.75+0.75 moderate strong Body Temp vs. Age

23. A strong positive correlation exists between ice cream sales and drowning. Cause & Effect?  Linear Correlation Coefficient: positive linear correlation negative linear correlation Is there an association between X and Y, and if so, how is it measured? PARAMETERS +1 0 -0.5+0.5 weak -0.75+0.75 moderate strong Profit vs. Price A strong positive correlation exists between ice cream sales and drowning. Cause & Effect? NOT LIKELY… “Temp (  F)” is a confounding variable.

24.  Variances:  Covariance: PARAMETERS  Means: Is there an association between X and Y, and if so, how is it measured? Proof: See text, p. 240

25.  Variances:  Covariance: PARAMETERS  Means: Is there an association between X and Y, and if so, how is it measured? Proof: (WLOG)

26.  Variances:  Covariance: PARAMETERS  Means: Is there an association between X and Y, and if so, how is it measured? (WLOG) If X and Y are independent, then Cov(X, Y) = 0. Proof: Exercise (HW problem) If X and Y are independent, then