JOINT PROBABILITY DISTRIBUTION
Let x and y be two different discrete random variables.
f(x, y) -
joint probability distribution of x and y
probability distribution of the simultaneous occurrence of x and y; i.e.,
f(x, y) = P(X = x, Y = y)
gives the probability distribution that outcomes x and y can occur at the same
time
For example,
Let x - age to the nearest year of a TV set that is to be repaired
y - number of defective tubes in the set
f(x, y) = f(5, 3) = probability that the TV set is 5 years old and needs 3
new tubes
-
Characteristics of a Joint Probability Distribution
1. f(x, y)  0
for all (x, y)
2.   f(x, y) = 1
add up the probabilities of all possible combinations of x
x
y
and y within the range
3. f(x, y) = P(X = x, Y = y)
4. For any region A in the x y plane, P [(x, y)  A] =   f(x, y)
Example 1:
Two refills for a ballpoint pen are selected at random from a box containing 3 blue refills,
2 red refills and 3 green refills. If X is the number of blue refills and Y is the number of
red refills selected, find
a. the joint probability distribution function f(x, y)
b. P [(X, Y)  A] , where A is the region { (x, y)  x + y  1 }
JOINT DENSITY FUNCTION
Joint Density Function – joint distribution of continuous random variables
Characteristics of a Joint Density Function
1. f(x, y)  0
2.
U
U
L
L
 
f(x, y) dx dy = 1
3. P [ (X, Y)  A] =
 
f(x, y) dx dy
for any region A in the x y plane
A
Note: f(x, y) - surface lying above the x y plane
Probability - volume of the right cylinder bounded by the base A and the surface
QUAMETH Notes: Joint Probability Distribution
Page 1 of 4
Example 2:
A candy company distributes boxes of chocolates with a mixture of creams, toffees and
nuts coated in both light and dark chocolate. For a randomly selected box, let X and Y,
respectively be the proportion of the light and dark chocolates that are creams and
suppose that the joint density function is given by:
f(x, y) =
k(2x + 3y)
0x1
,
0y1
0
elsewhere
Find P [ (X, Y)  A] where A is the region { (x, y) 0 < x < ½ , ¼ < y < ½ }
NOTE:
For the discrete case,
ex.
For the continuous case,
P(X = x, Y = y) = f(x, y)
P(x = 2, y = 1) = f(2, 1)
P(X = x, Y = y)  f(x, y)
MARGINAL DISTRIBUTIONS
Given the joint probability distribution f(x, y) of the discrete random variable X and Y,
the probability distribution g(x) of X along is obtained by summing f(x, y) over the
values of y. Similarly, the probability distribution h(y) of Y alone is obtained by
summing f(x, y) over the values of x. g(x) and h(y) are defined to be the marginal
distributions of x and y respectively.
g(x) =  f(x, y)
h(y) =  f(x, y)
y
x
g(x) =

Uy
Ly
f(x, y) dy
h(y) =

for the discrete case
Ux
Lx
f(x, y) dx
for the continuous case
Example 3:
Derive g(x) and h(y) for Example 1.
Example 4:
Derive g(x) and h(y) for the joint density function in Example 2.
CONDITIONAL DISTRIBUTIONS
Recall:
Conditional Probability Formula
P ( B / A) = P(A  B)
P(A)
Consider 2 random variables X and Y:
If we let A be the event defined by X = x and B be the event that Y = y, we have,
P ( Y= y) / X = x )
= P (X = x, Y = y)
P (X = x)
= f(x, y)
g(x)
g(x) > 0
where X and Y are discrete random variables
QUAMETH Notes: Joint Probability Distribution
Page 2 of 4
P (Y = y / X = x ) may actually be expressed as a probability distribution denoted by
f( y / x). Therefore, f (y / x) is called by conditional distribution of the random variable Y
given that X = x.
Generalization
Let X and Y be two random variables, discrete or continuous. The conditional
probability distribution of the random variable Y given that X = x, is given by
f (y / x) = f(x, y)
g(x)
(pure function of y)
g(x) > 0
Similarly, the conditional probability distribution of the random variable X given
that Y = y, is given by
f (x / y) = f(x, y)
h(y)
(pure function of x)
h(y) > 0
Note: f (x / y) only gives P ( X = x / Y = y). If one wishes to find the probability that
the discrete random variable x falls between a and b when it is known that the discrete
variable Y = y, then we evaluate
P (a < x < b / Y = y) =
 f (x / y)
x
P (a < y < b / X = x) =
 f (x / y)
y
Similarly,
For the continuous case:
P (a < x < b / Y = y) =
P (a < y < b / X = x) =

b
a

f (x / y) dx
b
a
f (y / x) dy
Example 5:
Find the conditional probability distribution of X, given that Y = 1 for Example 1 and use
it to evaluate P (x = 0 / y = 1).
STATISTICAL INDEPENDENCE
P (B / A) = P(A  B)
P(A)
P(A  B) = P(A) * P (B / A)
P(A  B) = P(A) * P (B)
if A and B are statistically independent
QUAMETH Notes: Joint Probability Distribution
Page 3 of 4
Recall:
Similarly,
f (y / x) = f(x, y)
g(x)
f(x, y) = g(x) * f (y / x)
f(x, y) = g(x) * h(y)
OR:
if X and Y are statistically independent
f (y / x) = f(x, y)
g(x)
f(x, y) = g(x) * f (y / x)
h(y) =

Ux
Lx
f(x, y) dx
=

Ux
Lx
g(x) * f(y / x) dx
pure function of y
if x and y are independent
h(y) = f (y / x)


Ux
Lx
g(x) dx
h(y) = f(x, y) / g(x)
f(x, y) = g(x) * h(y)
Let X and Y be two random variables, discrete or continuous, with joint probability
distribution f(x, y) and marginal distributions g(x) and h(y), respectively. The random
variable X and Y are said to be statistically independent if and only if
f(x, y) = g(x) * h(y)
for all (x, y) within their range
QUAMETH Notes: Joint Probability Distribution
Page 4 of 4