1. Joint Distributions, Marginal Distributions, and Conditional Distributions
Note 7

2. Examples
Let W be a random variable giving the number of heads minus the number of tails in three tosses of a coin. Find the probability distribution of the random variable W assuming that the coin is biased so that a head is twice as likely to occur as a tail.
A continuous r.v. X that can assume value between x=2 and x=5 has a density function given by f(x)=2(1+x)/27. Find (a) P(X<4) (b) P(3<X<4).
IEEM151

3. Joint Discrete Distribution
px,y = P(X = x, Y = y), for all real x and y. Y
px,y 1 2 3 4
0.05
0.15 X
0.1
0.2
IEEM151

4. Joint Distribution of Discrete R. V.
three white and two black balls in a bin
white balls: numbered 1, 2 and 3
black balls: numbered 4 and 5
X = 1 if the ball is black; X = 0 o.w.
Y: number of a ball randomly picked
Find the joint distribution of X and Y
IEEM151

5. Joint Distribution of Discrete R. V.Y X
px,y 1 2 3 4 5
0.2
0.2
0.2
0.2
0.2
IEEM151

6. Example
Two refills for a ballpoint pen are selected at random from a box that contains 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 function of (X,Y) (b) P{(X,Y) belongs to A}, A={(x,y)|x+y<1}
IEEM151

7. Marginal Distribution
px = P(X = x) = ∑y P(X = x, Y = y) = ∑y px,y ,
FX(x) = P(X  x) = P(X  x, Y < ) = F(x, ).
IEEM151

8. Joint Distribution of Discrete R. V.
three white and two black balls in a bin
white balls: numbered 1, 2 and 3
black balls: numbered 4 and 5
X = 1 if the ball is black; X = 0 o.w.
Y: number of a ball randomly picked
Find the joint distribution of X and Y
Find the distribution for X alone and Y alone based on the joint distribution of (X,Y).
IEEM151

9. Example
Two refills for a ballpoint pen are selected at random from a box that contains 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 function of (X,Y) (b) P{(X,Y) belongs to A}, A={(x,y)|x+y<1}
Find the marginal distribution for X and Y respectively based on the joint distribution of (X,Y).
IEEM151

10. Conditional Distribution
Conditional p.m.f. of Y given X = x is
IEEM151

11. Joint Distribution of Discrete R. V.
three white and two black balls in a bin
white balls: numbered 1, 2 and 3
black balls: numbered 4 and 5
X = 1 if the ball is black; X = 0 o.w.
Y: number of a ball randomly picked
Find the joint distribution of X and Y
Find the distribution for X alone and Y alone based on the joint distribution of (X,Y).
Find the conditional probability of P(X=0|Y=1) and P(Y=1|X=0)
IEEM151

12. Example
Two refills for a ballpoint pen are selected at random from a box that contains 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 function of (X,Y) (b) P{(X,Y) belongs to A}, A={(x,y)|x+y<1}
Find the marginal distribution for X and Y respectively based on the joint distribution of (X,Y).
Find the conditional probability of P(X=0|Y=1).
IEEM151

13. Independence
We say X and Y are independent, if and only if P(X=x, Y=y)=P(X=x) P(Y=y)
Show that for both examples, X and Y are not independent
IEEM151

14. Joint Distribution Discrete ∑x ∑y px,y = 1, px,y  0 for all x, y
for every event A, P((X,Y) A) = ∑(x,y)A px,y
Continuous
f (x,y) ≥ 0, for all x, y.
f (x,y)dxdy = 1,
region A in the xy plane, P((X,Y) A) = A f (x,y)dxdy
IEEM151

15. Joint Density Function of Continuous R.V.
The joint density function of X and Y is
f(x,y) = (3-x-y)/2, 0 < x,y < 1.
Check that f (x,y)dxdy = 1,
and find P(X < Y < 0.8)
What is the value of c if
f(x,y) = (3-x-y)/c
for 0 < x <y < 1 is a density function?
Example 3.9
IEEM151

16. Joint Cumulative Distribution
F(x, y) = P(X ≤ x, Y ≤ y), for all real x and y.
The joint density function
f(x,y) = ∂2F(x,y)/∂x∂y = ∂2P(X>x, Y>y)/∂x∂y.
IEEM151

17. Marginal Distribution
and
respectively
FX(x) = P(X  x) = P(X  x, Y < ) = F(x, ).
Example 3.11
IEEM151

18. Conditional Distribution
Conditional p.m.f. of Y given X = x is
Conditional density function Y given X = x is
Example 3.13
Example 3.14
IEEM151

19. Independence
X and Y are independent, if and only if f(X,Y)(x,y)=fX,(x) fY(y)
Similarly, if f(X,Y,Z)(x,y,z)=fX,(x) fY(y) fz(z), we say X, Y, Z are independent
Example 3.16
IEEM151