Todays Goals

1
Todays Goals

• Calculate and apply covariance and correlation
• Calculate linear functions of multiple variables
• HW 9 (due Wed. April 15) Ch 4 63. Ch 5 22
  30.
• Article will be due April 24
• Office hours this week Tu 2-350.

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Independence

• Two discrete R.V. are independent if
• p(x,y) p(x)p(y)
• Two continuous random variables X and Y are said
  to be independent if for every pair of x and y
  values,
• f(x,y) fX(x) fY(y).

3
Expected Value
If X and Y are independent random variables,
then EXY EXEY.
4
Different degrees of covariance
5
Covariance

• Covariance is a measure of how related two
  variables are specifically in a linear
  relationship
• Cov(X,Y) E(X-mx )(Y-my )
• short cut
• If X and Y are independent

6
Correlation

• The correlation Corr(X,Y) between two random
  variables X and Y is
• This number is always between -1 and 1
• If X and Y are independent, Corr(X,Y) 0
• However, the converse is not true

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Correlation
Are these variables correlated? Are they
independent?
8
Example find correlation between HW and midterm
9
Example find correlation

• cov EXY EXEY

Sd .13 .24
10
The correlation coefficient between midterm
grades and HW scores is 0.14
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Independence

• Two discrete R.V. are independent if
• p(x,y) p(x)p(y)
• Two continuous random variables X and Y are said
  to be independent if for every pair of x and y
  values,
• f(x,y) fX(x) fY(y).
• If two RV are independent, then COVX,Y 0 and
  EXY EXEY

12
Expected Value of a sum

• Regardless of covariance
• EXYEXEY

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Variance of a sum or difference

• Var(XY) Var(X)Var(Y)Cov(X,Y)
• Var(X-Y) Var(X)Var(Y)Cov(X,Y)
• If X and Y are independent then
• Var(XY) Var(X)Var(Y)
• Var(X-Y) Var(X)Var(Y)

14
Are X and Y independent? T yes, Fno
p(0,0) .02 ? .2 .07 .014 p(0)p(0)
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What is EXY?
EX 5.55 EY 8.55
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What is EXY? EXY EXEY
EX 5.55 EY 8.55
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True or False EMax(X,Y) 8.55
EX 5.55 EY 8.55
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True or False EMax(X,Y) 8.55
EX 5.55 EY 8.55
Max is not a linear function, so the flaw of
averages applies
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Correlation Proposition

• If a and c are either both positive or both
  negative, Corr(aX b, cY d) Corr(X, Y)
• For any two rvs X and Y,

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Correlation Proposition

• If X and Y are independent, then
  but does not imply independence.
• 
  for some numbers a and b with

21
Expected Value and Variance of a sum of random
variables
22
Example building design

• The vertical load on the ground-floor columns of
  an n-story building is the sum of the indivual
  loads
• Load surveys indicate that the mean and var are
  the same on all floors. What is mean and variance
  of Y?

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Example building design

• The vertical load on the ground-floor columns of
  an n-story building is the sum of the individual
  loads
• Load surveys indicate that the mean and var are
  the same on all floors, so
• EY nmx

24
Example building design

• We design the building to be safe, that is we
  design it to take a load L so that P(YgtL) is very
  small, say .001.
• What happens if we assume independence between
  the loads on different floors? What if the loads
  are not independent?

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Statistics and Statistical Inference