1. MULTIPLE RANDOM VARIABLES A vector random variable X is a function that assigns a vector of real numbers to each outcome of a random experiment. e.g. Random experiment = measure vital signs of a patient X = [systolic blood pressure diastolic blood pressure heart rate] so X could be [120 80 75], [165 100 85] etc. e.g. Random experiment = Football season X = [ #Bengals wins #Colts wins #Cowboys wins..... ]

2. Event for n – dimensional R.V. ↔ region in n-dimensional real space e.g. X = (A,B) is a 2-d random variable { 0 ≤ A,B ≤ 1} { A 2 + B 2  1} 01 a b 1 1 a b

3. Product form for X = { x 1,...., x n } Event A = { x 1  A 1 } ∩ { x 2  A 2 } ∩..... ∩ { x n  A n } P(A) ≡ P (x 1  A 1,..., x n  A n ) = P ( region of A in n-dimensional S ) e.g. { 3 ≤ A ≤ 5} ∩ { 6 ≤ B ≤ 8 } Almost all events can be approximated as unions of product forms b a 6 5 8 3

4. a b Approximating a 2-dimensional event as a union of product forms

5. Independence: R.V. ‘s X 1, X 2,..., X n are independent if P( X 1  A 1, X 2  A 2,..., X n  A n ) = P (X 1  A 1 ) P (X 2  A 2 )... P (X n  A n ) e.g. If X 1 and X 2 are 2 coin tosses P(X 1 = heads, X 2 = tails) = P( X 1 = heads ) P(X 2 = tails ) but for two pressure gauges measuring pressures in the same boiler, the measurements are not independent

6. For pairs of R.V.’s Discrete Case: - Joint pmf P X,Y (x,y) = P ( {X = x} ∩ {Y = y} ) = P ( X = x,Y = y ) - Marginal pmf P X (x) = P ( X = x) = P ( X = x and Y = anything ) = Similarly, P Y (y) = e.g. P( Team A reaches playoffs) = P ( A reaches playoffs, B makes playoffs) + P (A reaches playoffs, B does not)

7. Continuous case: - Joint CDF F X,Y (x,y) = P ( X ≤ x, Y ≤ y) 1) F X,Y (x 1,y 1 ) ≤ F X,Y (x 2,y 2 ) if x 1 ≤ x 2, y 1 ≤ y 2 2) F X,Y (-∞,y) = F X,Y (x, - ∞) = 0 3) F X,Y (∞, ∞) = 1 - Marginal CDF F X (x) = F X,Y (x, ∞) = P ( X ≤ x, Y ≤ ∞) = P ( X ≤ x ) F Y (y) = F X,Y (∞,y)

8. - Joint PDF If f X,Y (x,y) exists, X,Y are said to be jointly continuous

9. - Marginal PDF

10. Independent Random Variables Discrete : P X,Y (x,y) = P X (x) P Y (y) x, y  X, Y are independent F X,Y (x,y) = F X (x) F Y (y) Continuous : f X,Y (x,y) = f X (x) f Y (y) F X,Y (x,y) = F X (x) F Y (y) If X, Y are independent  g(X) and h(Y) are also independent, g( ) and h( ) are functions.

11. Conditional Probability - if X is discrete If X, Y are independent

12. - if X and Y are discrete If X, Y are independent

13. - If X and Y are continuous Note

14. If X, Y are independent

15. Total Probability: - Discrete X - Continuous X This approach yields CDF’s when conditional probabilities are easier to calculate.

16. Ex 5.33: # of arriving customers in time t ~ Poisson w/ parameter βt Service time / customer ~ Exponential w/ parameter α Find pmf of number of customers N that arrive during the service time t of a customer. Assume arrivals are independent of service times.