1. ... DISCRETE random variables X, Y Joint Probability Mass Function y1… yc X x1
p(x1, y1)
p(x1, y2)
...
p(x1, yc) x2
p(x2, y1)
p(x2, y2)
p(x2, yc) xr
p(xr, y1)
p(xr, y2)
p(xr, yc)

2. ... DISCRETE random variables X, Y Joint Probability Mass Function y1… yc X x1
p(x1, y1)
p(x1, y2)
...
p(x1, yc)
pX (x1) x2
p(x2, y1)
p(x2, y2)
p(x2, yc)
pX (x2) xr
p(xr, y1)
p(xr, y2)
p(xr, yc)
pX (xr)
pY (y1)
pY (y2)
pY (yc) 1

3. Y =
X =

4. X and Y are not independent!
Probabilities…
X and Y are not independent!

5. X and Y are not independent!
Probabilities…
X and Y are not independent!
cdf

6. X and Y are not independent!
Probabilities…
X and Y are not independent!
cdf

7. Cumulative Probability
X =
Event
T = t
Outcomes
(AM, PM)
Probability
Cumulative Probability 2
(1, 1)
.25 3
(1, 2), (2, 1)
.45 =
.70 = 4
(1, 3), (2, 2)
.25 =
.95 = 5
(2, 3)
.05
1.00 =

8. ... DISCRETE random variables X, Y Joint Probability Mass Function y1… yc X x1
p(x1, y1)
p(x1, y2)
...
p(x1, yc)
pX (x1) x2
p(x2, y1)
p(x2, y2)
p(x2, yc)
pX (x2) xr
p(xr, y1)
p(xr, y2)
p(xr, yc)
pX (xr)
pY (y1)
pY (y2)
pY (yc) 1

9. ... CONTINUOUS random variables X, Y
Joint Probability Density Function
Joint Probability Mass Function Y y1 y2 … yc X x1
p(x1, y1)
p(x1, y2)
...
p(x1, yc)
pX (x1) x2
p(x2, y1)
p(x2, y2)
p(x2, yc)
pX (x2) xr
p(xr, y1)
p(xr, y2)
p(xr, yc)
pX (xr)
pY (y1)
pY (y2)
pY (yc) 1

10. Joint Probability Density Function
CONTINUOUS
Joint Probability Density Function
Volume under density f(x, y) over A.
“area element”
Area A

11. ... Corollary ~ CONTINUOUS random variables X, Y
Joint Probability Density Function
Joint Probability Mass Function Y y1 y2 … yc X x1
p(x1, y1)
p(x1, y2)
...
p(x1, yc)
pX (x1) x2
p(x2, y1)
p(x2, y2)
p(x2, yc)
pX (x2) xr
p(xr, y1)
p(xr, y2)
p(xr, yc)
pX (xr)
pY (y1)
pY (y2)
pY (yc) 1
Corollary ~
If X and Y are independent, then the joint cdf
satisfies
Proof: Exercise

12. Extension to multiple random variables X1, X2, X3,…, Xn
For simplicity, take n = 3:
Discrete
Continuous
(e.g, Multinomial distribution)