Todays Goals

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Todays Goals

• Calcualte with joint continuous distributions
• Calculate and apply covariance and correlation
• HW 9 (due Wed. April 15) Ch 4 63. Ch 5 22
  30.
• Article will be due April 24
• Office hours next week Tu 2-350.

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Joint Probability Mass Function
Let X and Y be two discrete rvs defined on the
sample space of an experiment. The joint
probability mass function p(x, y) is defined for
each pair of numbers (x, y) by
Let A be a set consisting of pairs of (x, y)
values, then
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Example of a joint pmf

• hours per day worked and productivity
• What do the numbers inside the square add to?
• What is Pp70 and h8?
• What is Ph8?

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Joint Probability Density Function
Let X and Y be continuous rvs. Then f (x, y) is
a joint probability density function for X and Y
if for any two-dimensional set A
If A is the two-dimensional rectangle
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A shaded rectangle
Volume under density surface above A
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Marginal Probability Density Functions
The marginal probability density functions of X
and Y, denoted fX(x) and fY(y), are given by
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Example 3

• Let the joint probability density function of rvs
  X and Y be
• Draw the two-dimensional sample space.
• Find the marginal probability distributions of
  random variable X and Y.

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Example 3

• Let the joint probability density function of rvs
  X and Y be
• then

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Example 3

• Let the joint probability density function of rvs
  X and Y be

• Find the marginal distribution for Y
• 2 2y
• 2
• 2y

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Example 3

• Let the joint probability density function of rvs
  X and Y be

• Find the marginal distribution for Y
• 2 2y
• 2
• 2y

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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).

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Example

• f(x,y) x y for 0x,y1
• Is this a joint pdf?
• yes
• What is the marginal pdf of x?

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Example

• f(x,y) x y for 0x,y1
• Is this a joint pdf?
• yes
• What is the marginal pdf of x?
• True or False X and Y are independent.

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Example

• f(x,y) x y for 0x,y1
• Is this a joint pdf?
• yes
• What is the marginal pdf of x?
• True or False X and Y are independent.

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Expected Value
If X and Y are independent random variables,
then EXY EXEY.
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Expected Value
If X and Y are independent random variables,
then EXY EXEY. Note that the converse
is not true It is not true that if EXY
EXEY then X and Y are independent.
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Covariance

• Covariance is a measure of how related two
  variables are.
• Cov(X,Y) E(X-mx )(Y-my )
• short cut

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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

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Different degrees of covariance
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Different degrees of covariance
Cov 0
Weak positive cov
strong negative cov
stronger positive cov
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Relationship of homework scores to midterm grades
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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)

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Are X and Y independent? T yes, Fno
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Are X and Y independent? T yes, Fno
p(0,0) .02 ? .2 .07 .014 p(0)p(0)