Lesson on Joint Probability Distributions in Probability and Statistics at the United States Military Academy in 2000.

2. MA206 Lesson 20
Jointly
Distributed
Random
Variables

3. Example
y
p(x,y) 0 1 2
0 .10 .04 .02
x 1 .08 .20 .06
2 .06 .14 .30
Forrest & Bubba each have 2 brain cells to use
for thinking. Let X denote the number of brain
cells that Forrest uses at a particular time, and let
Y denote the number of brain cells that Bubba
uses at a particular time. The joint pmf of X and
Y is
a. What is P(X = 1 and Y = 1) = p(1,1) = 0.20
b. P(X £1 and Y £1) = p(0,0) + p(0,1) + p(1,0) + p(1,1) = .42
c. Give a word description of the event {X ¹ 0 and Y ¹ 0} and compute the
probability of this event.
This is asking the probability that at least one brain cell is in use for each person.
P(X ¹ 0 and Y ¹ 0) = p(1,1) + p(1,2) + p(2,1) + p(2,2) = .70
d. Compute the marginal pmf of X and Y. For p(x) we must sum across the
rows and for p(y) down the columns
x 0 1 2
y 0 1 2
p(x) .16 .34 .50 p(y) .24 .38 .38
P(X £ 1) = 0.16 + 0.34 =0.50

4. Muddiest Point: #7
The joint probability distribution of the number
X of cars and the number Y of buses per cycle at
a proposed left turn lane is displayed in the
accompanying joint probability table.
a. P(X = 1 and Y = 1) = p(1,1) = 0.030
b. P(X £ 1 and Y £ 1) = p(0,0)+p(1,0)+p(0,1)+p(1,1)
= 0.120
y
p(x,y) 0 1 2
0 .025 .015 .010
1 .050 .030 .020
x 2 .125 .075 .05
3 .150 .090 .060
4 .100 .06 .04
5 .05 .030 .02
c. What is the probability that at there is exactly one car during a cycle? Exactly
one bus?.
P(X = 1) = p(1,0) + p(1,1) + p(1,2) = 0.10
P(Y = 1) = p(0,1) + p(1,1) +…+p(5,1) = 0.30
d. Suppose the left turn lane is to have a capacity of five cars and one bus is
equivalent to three cars. What is the probability of an overflow during a cycle?
P overflow P X Y P X Y
= + > = - + £
( ) ( 3 5) 1 ( 3 5)
1 [ p( 0 , 0 ) p (5 , 0) p (01) , p (11) , p (21),
] 1 0.62 0.38
= - + 
+ + + + = - =

5. Muddiest Point: #9
Each front tire on a particular type of automobile is supposed to be filled to a
pressure of 26 psi. Suppose the actual air pressure in each tire is a random variable–
X for the right tire and Y for the left tire, with pdf
( ) ( )
K x2 y2 x y f x y
î í ì
= + £ £ £ £
, 20 30, 20 30
0 otherwise
a. What is the value of K? (We know that the pdf must integrate to 1.)
ò ò ò ò 2 ò ò 2
ò ò
f x y dxdy K x y dxdy K x dydx K y dxdy
= Þ + = + =
3
380000
( , ) 1 ( )
10K x 10K y 20 19000
3
30
ò ò
20
2
30
20
2
30
20
30
20
2
30
20
30
20
30
20
30
20
2
+ = · Þ =
¥
-¥
¥
-¥
dx dy K K
26
b. What is the probability that both tires are under-filled?
P X < Y < = ò òK x2 + y dxdy = K òx dx = Kx = K =
( 26 and 26) ( ) 12 4 38304 .3024
20
26
2 2 3
20
26
20
26
20