1. Random Variable X, with pmf p(x) or pdf f(x)
POPULATION
Random Variable X, with pmf p(x) or pdf f(x)
Recall…
PARAMETERS
“population characteristics”
Mean:
X 2 measures how much X varies about its mean .
Variance:
Proof: See PowerPoint section , slides 56, 57 for discrete X.

2. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
PARAMETERS
Means:
Variances:

3. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
PARAMETERS
Means:
Variances:
Covariance:
Proof:

4. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
PARAMETERS
Means:
Variances:
Covariance:
Properties:
Covariance of two random variables measures how they vary together about their respective means.
Variance is  0, but covariance is unrestricted in sign.
Cov(X, X) =
Other properties based on expected value…
Var(X)

5. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Covariance:
Is there an association between X and Y, and if so, how is it measured?

6. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Covariance:
Is there an association between X and Y, and if so, how is it measured?

7. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Covariance:
Is there an association between X and Y, and if so, how is it measured?

8. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
Covariance:

9. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
Covariance:

10. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
Covariance:
… but what does it mean????

11. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured? Y X y1 y2 y3 y4 y5 x1
p(x1, y1)
p(x1, y2)
p(x1, y3)
p(x1, y4)
p(x1, y5)
pX(x1) x2
p(x2, y1)
p(x2, y2)
p(x2, y3)
p(x2, y4)
p(x2, y5)
pX(x2) x3
p(x3, y1)
p(x3, y2)
p(x3, y3)
p(x3, y4)
p(x3, y5)
pX(x3) x4
p(x4, y1)
p(x4, y2)
p(x4, y3)
p(x4, y4)
p(x4, y5)
pX(x4) x5
p(x5, y1)
p(x5, y2)
p(x5, y3)
p(x5, y4)
p(x5, y5)
pX(x5)
pY(y1)
pY(y2)
pY(y3)
pY(y4)
pY(y5) 1

12. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
Example: Y X 1 2 3 4 5
.04
.20
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, i.e., X and Y are statistically independent!
It is easy to see that Cov(X, Y) = 0.

13. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
Exercise: Y X 1 2 3 4 5
.04
.12
.20
.28
.36
.10
.15
.25
.30
Fill in the table so that X and Y are statistically independent. Then show that Cov(X, Y) = 0.
THEOREM. If X and Y are statistically independent, then Cov(X, Y) = 0. However, the converse does not necessarily hold!
Exception: The Bivariate Normal Distribution

14. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
Example: Y X 1 2 3 4 5
.08
.04
.03
.02
.01
.18
.21
.22

15. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
Example: Y X 1 2 3 4 5
.08
.04
.03
.02
.01
.18
.21
.22

16. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
Example: Y X 1 2 3 4 5
.08
.04
.03
.02
.01
.18
.21
.22
X large  high prob Y large
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.
X small  high prob Y small

17. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
PARAMETERS
Means:
Variances:
Covariance:
Linear Correlation Coefficient:
Always between –1 and +1
(“rho”)

18. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
PARAMETERS
Linear Correlation Coefficient:
JAMA. 2003;290:
Scatterplot
ρ measures the strength of linear association between X and Y.
Always between –1 and +1.

19. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
PARAMETERS
Linear Correlation Coefficient:
IQ vs. Head circumference
strong
moderate
weak
moderate
strong -1 +1
-0.75
-0.5
+0.5
+0.75
positive linear correlation
negative linear correlation

20. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
PARAMETERS
Linear Correlation Coefficient:
Body Temp vs. Age
strong
moderate
weak
moderate
strong -1 +1
-0.75
-0.5
+0.5
+0.75
positive linear correlation
negative linear correlation

21. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
A strong positive correlation exists between ice cream sales and drowning. Cause & Effect?
A strong positive correlation exists between ice cream sales and drowning. Cause & Effect? NOT LIKELY…
“Temp (F)” is a confounding variable.
PARAMETERS
Linear Correlation Coefficient:
Linear
Profit vs. Price
strong
moderate
weak
moderate
strong -1 +1
-0.75
-0.5
+0.5
+0.75
positive linear correlation
negative linear correlation

22. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
PARAMETERS
Means:
Variances:
Covariance:
Definition
Theorem
Special case: Y = constant c
Theorem

23. Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
PARAMETERS
Means:
Variances:
Covariance:
Theorem
Proof:
(WLOG)

24. Converse is not necessarily true!!!
POPULATION(S)
Random Variables X, Y with joint pmf p(x,y) or pdf f(x, y)
Is there an association between X and Y, and if so, how is it measured?
PARAMETERS
Means:
Variances:
Covariance:
Theorem
(WLOG)
Theorem
If X and Y are independent, then Cov(X, Y) = 0.
Proof: Exercise… Hint: See slide 4 above.
Converse is not necessarily true!!!
If X and Y are independent, then
Corollary

25. BIVARIATE NORMAL DISTRIBUTION
Important Exception
BIVARIATE NORMAL DISTRIBUTION