Download Joint and Marginal Distributions

Joint and Marginal Distributions
Joint probability mass function: an example
A box containing 9 balls, among them 4 are red, 3 are white
and 2 are blue. A random sample of three balls is chosen
without replacement. Let
R = Number of red balls in the sample
W = Number of white balls in the sample
4
2
3 2
P({R = 2} ∩ {W = 1}) = P(R = 2, W = 1) =
P({R = 2}) =
4
2
5
1
9
3
30
=
,
84
P({W = 1}) =
3
1
1 0
9
3
6
2 =
9
3
P(R = r , W = w) 6= P(R = r )P(W = w).
=
45
84
18
84
Example continued
w
r
0
1
2
3
fR (r )
0
0
3/84
6/84
1/84
10/84
1
4/84
24/84
12/84
0
40/84
2
12/84
18/84
0
0
30/84
3
4/84
0
0
0
4/84
fW (w)
20/84
45/84
18/84
1/84
1
In general, for 0 < r + w ≤ 3,
4
r
f (r , w) = P(R = r , W = w) =
f (r , w) 6= fR (r )fW (w)
3
w
2
3−r −w
9
3
.
Independence of two random variables
I
Two discrete random variables X and Y are independent if
and only if
f (j, k ) = fX (j)fY (k ) for all (j, k).
I
If X and Y independent, then g(X ) and h(Y ) are also
independent where g and h are functions.
Checking independence
Consider the discrete bivariate random vector (X , Y ) with joint
pmf given by
f (10, 1) = f (20, 1) = f (20, 2) =
1
10
and
f (10, 2) = f (10, 3) =
Are X and Y independent?
1
, f (20, 3) = 3/10.
5
Independence of n random variables
I
If fi (x) is the pmf of Xi and f is the joint probability mass
function of X1 , · · · , Xn , then the independence of
X1 , · · · , Xn is equivalent to
f (x1 , x2 , · · · , xn ) = f1 (x1 )f2 (x2 ) · · · fn (xn )
for every (x1 , · · · , xn ) ∈ R n .
Example
A group of students is enrolled in courses a, b and c. From the
previous years it is known that
(a): For course a, 20% of students get A, 45% of them get B ,
and 30% of them get C;
(b): For course b, these proportions are 15%, 48% and 33%
respectively;
(c): For course c, these proportions are 18%, 50% and 30%
respectively.
Assume that the grades a student receives in the three courses
are independent. Find the probability that a student randomly
selected from this group receives grade B in all 3 courses.
Joint density for bivariate random variable
Definition: A function f (x, y ) from R 2 → R is called a joint
probability density function of continuous bivariate vector (X , Y )
if for any A,
Z Z
P((X , Y ) ∈ A) =
f (x, y )dxdy .
A
Example 1
Suppose that (X , Y ) has a joint pdf

 6xy 2 0 < x, y < 1
f (x, y ) =
 0
otherwise
What is P(X + Y ≥ 1)?
Marginal density
The marginal density of X and Y can be obtained by
Z
∞
fX (x) =
f (x, y)dy
Z−∞
∞
fY (y ) =
f (x, y)dx.
−∞
Joint CDF
Definition: The pair of random variables (X , Y ) has joint
cumulative distribution function
F (u, v ) = P(X ≤ u, Y ≤ v ).
(1) FX (u) = P(X ≤ u) = P(X ≤ u, Y ≤ ∞) = F (u, ∞),
FY (v ) = F (∞, v ) and F (∞, ∞) = 1.
(2) F (−∞, v ) = F (u, ∞) = 0.
(3) P(a1 < X ≤ b1 , a2 < Y ≤ b2 ) =
F (b1 , b2 ) − F (a1 , b2 ) − F (b1 , a2 ) + F (a1 , a2 ).
Joint density and joint CDF
If (X , Y ) has joint density f (x, y ),
Z
u
Z
v
F (u, v ) =
f (x, y )dxdy .
−∞
−∞
If f (x, y) is continuous at point (u, v ), then
∂2
F (u, v ) = f (u, v ).
∂u∂v