{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture3

# Lecture3 - IEOR 4404 Simulation Lecture 3 January 25th 2012...

This preview shows pages 1–6. Sign up to view the full content.

IEOR 4404 Simulation Lecture 3 January 25th, 2012 Mariana Olvera-Cravioto [email protected] 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Jointly discrete random variables When analyzing probabilities involving more than one random variable the relationship between them must be specified through their joint distribution . If X and Y are jointly discrete random variables, then p ( x, y ) = P ( X = x, Y = y ) is called the joint PMF of X and Y . If X and Y are independent, then p ( x, y ) = p X ( x ) p Y ( y ) where p X ( x ) = P ( X = x ) and p Y ( y ) = P ( Y = y ). IEOR 4404, Lecture 3 2
Jointly discrete random variables If X and Y are not independent, then we need to fully specify p ( x, y ) for each combination of x and y . Example: Consider tossing two identical 6-sided dice. Let X be the outcome of first die and Y be the sum of the two dice. X ∈ { 1 , 2 , 3 , 4 , 5 , 6 } and Y ∈ { 2 , 3 , 4 , . . . , 11 , 12 } The joint PMF of X and Y , p ( x, y ) is given below: X , Y 2 3 4 5 6 7 8 9 10 11 12 1 1 36 1 36 1 36 1 36 1 36 1 36 2 1 36 1 36 1 36 1 36 1 36 1 36 3 1 36 1 36 1 36 1 36 1 36 1 36 4 1 36 1 36 1 36 1 36 1 36 1 36 5 1 36 1 36 1 36 1 36 1 36 1 36 6 1 36 1 36 1 36 1 36 1 36 1 36 IEOR 4404, Lecture 3 3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Marginal and conditional PMFs The joint PMF of X and Y contains all the information about X and Y . We can recover the marginal PMFs of X and Y through the formulas p X ( x ) = X y p ( x, y ) and p Y ( y ) = X x p ( x, y ) p ( x, y ) can also be determined using the conditional PMFs : p X | Y ( x | y ) = P ( X = x | Y = y ) = P ( X = x, Y = y ) P ( Y = y ) p Y | X ( y | x ) = P ( Y = y | X = x ) = P ( X = x, Y = y ) P ( X = x ) and the relation p ( x, y ) = p X | Y ( x | y ) p Y ( y ) = p Y | X ( y | x ) p X ( x ) IEOR 4404, Lecture 3 4
Example The number of customers arriving to a certain coffee shop between 8:00 am and 10:00 am is modeled as a Poisson r.v. with parameter λ = 40. The coffee shop owners have observed that about half of the customers are women and half of them are men. What is the probability that exactly 5 women will come

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 16

Lecture3 - IEOR 4404 Simulation Lecture 3 January 25th 2012...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online