1. UNR, MATH/STAT 352, Spring 2007

2. Random variable (rv) – central concept of probability theory Numerical description of experiment

3. UNR, MATH/STAT 352, Spring 2007 S Sample space x Definition: A random variable X is a function whose domain is the sample space and range is the real line. Outcome

4. UNR, MATH/STAT 352, Spring 2007 Experiment: tossing a die Sample space 1/6 Probabilities Random variable X = face value

5. UNR, MATH/STAT 352, Spring 2007 Experiment: tossing a die Sample space 1/6 Probabilities Random variable X = 1 for odd, 0 for even face value P( X =1) = P( X =0) = 1/2

6. UNR, MATH/STAT 352, Spring 2007 Experiment: tossing a coin Sample space 1/2 Probabilities Random variable X = 1 for T, 0 for H P( X =1) = P( X =0) = 1/2 Head Tail 1/2

7. UNR, MATH/STAT 352, Spring 2007 P( X =1) = P( X =0) = 1/2

8. UNR, MATH/STAT 352, Spring 2007 Experiment: tossing two dice Sample space (1,1) (2,1) (1,2) (2,2) (1,3) (2,3) (1,4) (2,4) (1,5) (2,5) (1,6) (2,6) (3,1)(3,2)(3,3)(3,4)(3,5)(3,6) (4,1)(4,2)(4,3)(4,4)(4,5)(4,6) (5,1)(5,2)(5,3)(5,4)(5,5)(5,6) (6,1)(6,2)(6,3)(6,4)(6,5)(6,6) Second die First die Each outcome (n,m) has probability 1/36

9. UNR, MATH/STAT 352, Spring 2007 Experiment: tossing two dice (1,1) (2,1) (1,2) (2,2) (1,3) (2,3) (1,4) (2,4) (1,5) (2,5) (1,6) (2,6) (3,1)(3,2)(3,3)(3,4)(3,5)(3,6) (4,1)(4,2)(4,3)(4,4)(4,5)(4,6) (5,1)(5,2)(5,3)(5,4)(5,5)(5,6) (6,1)(6,2)(6,3)(6,4)(6,5)(6,6) 234567 789101112 X = sum of faces 34 4 5 5 5 6 6 6 67 7 7 7 8 8 8 8 9 9 9 10 11 Sample space

10. UNR, MATH/STAT 352, Spring 2007 Experiment: tossing two dice 234567 789101112 X = sum of faces 34 4 5 5 5 6 6 6 67 7 7 7 8 8 8 8 9 9 9 10 11 2 3 4 5 6 7 8 9 10 11 12 Possible values Probabilities 1 36 2 3 4 5 6 5 4 3 2 1

11. UNR, MATH/STAT 352, Spring 2007 Experiment: old car with four cylinder engine goes to inspection http://www.geocities.com/aruba692003/Old-car.jpg http://www.seanhylandmotorsport.com/frppcrate/4cyleng.jpg + + http://www.autoinspection.com/back/aselogo.gif X = number of cylinders with low compression X = {0, 1, 2, 3, 4} Old car Four-cylinder engine

12. UNR, MATH/STAT 352, Spring 2007 Experiment: measuring the life time of a device (computer chip) Time Failure X = time until failure X > 0

13. UNR, MATH/STAT 352, Spring 2007 The same sample space may be associated with different random variables In general, it is impossible to reconstruct the original sample space that was used to construct a random variable from the knowledge of this random variable We work with random variables and do not think of original sample spaces There are differences in working with discrete and continuous random variables