1. IE 360: Design and Control of Industrial Systems I
Lecture 10
Joint Random Variables Part 2
IE 360: Design and Control of Industrial Systems I
References
Montgomery and Runger
Section 5-2
Copyright  2010 by Joel Greenstein

2. Expected Value of Joint RVs
The expected value of joint rvs follows the same ideas as “regular” (aka “univariate”= “one variable”) rvs but you need to use the joint pmf/pdf
Let X and Y be discrete random variables with joint pmf f(x, y). The expected value of some function of X and Y, denoted g(X, Y), is
Let X and Y be continuous random variables with joint pdf f(x, y). The expected value of some function of X and Y, denoted g(X, Y), is
g(X, Y) could be something like XY, or X/Y (as in “X divided by Y”) or other non-linear functions, or even just X or 3Y + 2
Rules of Expectation apply in this case still
If two rvs are independent (their joint pmf/pdf equals the product of the marginals), then E[XY] = E[X]E[Y]

3. Discrete Joint RV Expectation Example
I am notoriously clumsy. Let X be the number of times I drop the chalk and Y be the number of times I drop the eraser in a one-hour class. Observant students in the past have developed the following joint pmf to describe how often I drop the chalk and eraser when I teach.
X=0
X=1
X=2
X=3
Y=0
0.1
0.3
0.26
Y=1
0.03
Y=2
0.02
Y=3
0.01
What is the expected number of times I drop the chalk?
This is E[X], so we have g(X, Y) = X
It is really the same as computing the expected value of X using the marginal distribution of X, since the sum of the joint pmf over the Y-values is the marginal pmf of X.

4. Continuous Joint RVs Example
Consider joint continuous random variables X and Y, with pdf f(x, y) = e-x-y, x>0, y>0
Find E[X]; we have g(X, Y)=X
More integration by parts!

5. Continuous Joint RVs Example 2
Consider joint continuous random variables X and Y, with pdf f(x, y) = e-x-y, x>0, y>0
Find E[Y]; we have g(X, Y)=Y
This is just the same as E[X] but swapping X and Y, so we have E[Y] = 1
Find E[XY]; we have g(X, Y)=XY
Yikes – a double dose of integration by parts!

6. Covariance Interpreting the covariance
If two rvs are related through a joint pmf/pdf, we may ask:
If one goes up, does the other go down?
Do large changes in one correspond to large changes in the other?
These questions are answered by the covariance, Cov(X, Y) = σXY
Describes nature of linear relationship between two rvs
Same idea for discrete or continuous joint rvs
Definition: Cov(X, Y) = σXY = E[(X-μX)(Y-μY)] may be positive or negative!!!
Easier computation: Cov(X, Y) = σXY = E[XY] – μXμY
Interpreting the covariance
If two rvs are independent, then Cov(X, Y) = 0
The reverse is not true. If the covariance between two random variables is zero, we cannot conclude that the two random variables are independent
independence is a result of the pmf/pdf relationship described in the previous lecture
An important factor is the sign of the covariance
Positive indicates an increase in one rv is associated with an increase in the other
Negative indicates a decrease in one rv is associated with an increase in the other

7. Covariance Example
If E[X] = 1, E[Y] = -1, and E[XY] = 4, what is the covariance of X and Y?
Cov(X, Y) = E[XY] – μXμY = E[XY] – E[X] E[Y] = 4 – (1)(-1) = 5
X and Y have positive covariance
If E[W] = 10, E[Z] = -10, and E[WZ] = 400, what is the covariance of W and Z?
Cov(W, Z) = E[WZ] – μWμZ = E[WZ] – E[W] E[Z] = 400 – (10)(-10) = 500
W and Z have positive covariance
Are the relationships between X and Y and W and Z very different?
Maybe , maybe not. The covariance is not unitless, so it is not meaningful to compare the covariances of different pairs of rvs

8. Correlation
The correlation, a unitless extension of the covariance, can also be interpreted as a description of the linear relationship between two random variables
Let X and Y be rvs with Cov(X, Y)= σXY and standard deviations σX and σY. The correlation, ρXY (pronounced “rho of X and Y”) is defined as follows:
Now, we have a unitless number that ranges from -1 to +1: -1 ≤ ρXY ≤ 1
If the correlation is -1, we have perfectly negative correlation
If the correlation is +1, we have perfectly positive correlation
If the correlation is 0, we have no linear correlation x y
ρXY = -1 x y
ρXY = +1 x y
ρXY = 0 x y
ρXY = 0

9. Correlation Example Assume
E[X] = ½ , E[Y] = 1/3, E[XY] = ¼, Var[X] = ¼ and Var[Y] = 1/12
What is the correlation of X and Y?
First, compute the covariance
Cov(X, Y) = E[XY] – μXμY = E[XY] – E[X] E[Y] = ¼ - ½(1/3) = 1/12
Now, compute the correlation
X and Y are somewhat positively correlated

10. Related reading Montgomery and Runger, section 5-2
The captions of Figures 5-14, 5-15, and 5-16 should refer to Examples 5-21, 5-22, and 5-23, respectively.
Some links that might help (maybe)
Nice pictures of correlations are at
Now you are ready to do HW 10