Transcript Statistics 400 - Lecture 2

Statistics 270 - Lecture 19
• Continue Chapter 5 today
Definition
• The marginal probability density function for continuous random
variables X and Y, denoted by fX(x) and fY(y), respectively, are given
by

f X ( x) 
 f ( x, y) dy and


f Y ( y) 
 f ( x, y) dx

Example:
• The front tire on a particular type of car is suppose 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 side; Y for the left side) with joint pdf
f ( x, y)  K ( x 2  y 2 ) for 20  x  30 and 20  y  30
• Find the marginal distribution of X
More Generally
Independence
• Two random variables X and Y are said to be independent if:
• Discrete:
• Continuous:
Example
• Consider 3 continuous random variables X,Y, and Z with joint pdf:
f ( x, y, z ) 


3
1
2  X  Y  Z
• X, Y and Z independent?
e
1 x 2 y 2 z 2
(
 
)
2  X Y  Z
   x, y, z  
Conditional Probability
• Let X and Y be rv’s with joint pdf f(x,y) and marginal distribution of
X, f(x). The the conditional probability density of Y, given X=x is
defined as
f ( x, y)
f Y | X ( y | x) 
f X ( x)
• Substitute the pmf’s if X and Y are discrete
Example
• Consider 2 continuous random variables X, and Y with joint pdf:
f ( x, y ) 
2
( 2 x  3 y ) for 0  x  1 and 0  y  1
5
• Find the marginal distribution of X
• Find the conditional distribution of Y given X=0.2