1. Random Variables 2.1 Discrete Random Variables 2.2 The Expectation of a Random Variable 2.3 The Variance of a Random Variable 2.4 Jointly Distributed Random Variables 2.5 Combinations and Functions of Random Variables

2. Random Variables Random Variable (RV): A numeric outcome that results f rom an experiment For each element of an experiment’s sample space, the r andom variable can take on exactly one value Discrete Random Variable: An RV that can take on only a finite or countably infinite set of outcomes Continuous Random Variable: An RV that can take on a ny value along a continuum (but may be reported “discre tely” Random Variables are denoted by upper case letters (Y) Individual outcomes for RV are denoted by lower case le tters (y)

3. Probability Distributions Probability Distribution: Table, Graph, or Formula that descr ibes values a random variable can take on, and its correspo nding probability (discrete RV) or density (continuous RV) Discrete Probability Distribution: Assigns probabilities (mas ses) to the individual outcomes Continuous Probability Distribution: Assigns density at indivi dual points, probability of ranges can be obtained by integra ting density function Discrete Probabilities denoted by: p(y) = P(Y=y) Continuous Densities denoted by: f(y) Cumulative Distribution Function: F(y) = P(Y≤y)

4. Discrete Probability Distributions

5. Example – Rolling 2 Dice (Red/Green) Red\Green 123456 1 234567 2 345678 3 456789 4 5678910 5 6789 11 6 789101112 Y = Sum of the up faces of the two die. Table gives value of y for all eleme nts in S

6. Rolling 2 Dice – Probability Mass Function & CDF yp(y)F(y) 21/36 32/363/36 4 6/36 54/3610/36 65/3615/36 76/3621/36 85/3626/36 94/3630/36 103/3633/36 112/3635/36 121/3636/36

7. Rolling 2 Dice – Probability Mass Function

8. Rolling 2 Dice – Cumulative Distribution Function

9. Expected Values of Discrete RV’s Mean (aka Expected Value) – Long-Run average value an RV (or function of RV) will take on Variance – Average squared deviation between a realization of an RV (or function of RV) and its mean Standard Deviation – Positive Square Root of Variance (in same units as t he data) Notation: –Mean: E(Y) =  –Variance: V(Y) =  2 –Standard Deviation: 

10. Expected Values of Discrete RV’s

11. Expected Values of Linear Functions of Discrete RV’s

12. Example – Rolling 2 Dice yp(y)yp(y)y 2 p(y) 21/362/364/36 32/366/3618/36 43/3612/3648/36 54/3620/36100/36 65/3630/36180/36 76/3642/36294/36 85/3640/36320/36 94/3636/36324/36 103/3630/36300/36 112/3622/36242/36 121/3612/36144/36 Sum36/36 =1.00 252/36 =7.00 1974/36= 54.833

13. 2.1 Discrete Random Variable Definition of a Random Variable (1/2) Random variable –A numerical value to each outcome of a particular experiment 02 1 3 -2-3

14. Definition of a Random Variable (2/2) Example 1 : Machine Breakdowns –Sample space : –Each of these failures may be associated with a repair cost –State space : –Cost is a random variable : 50, 200, and 350

15. Probability Mass Function (1/2) Probability Mass Function (p.m.f.) –A set of probability value assigned to each of the values taken by the discrete random variable – and –Probability :

16. Probability Mass Function (1/2) Example 1 : Machine Breakdowns –P (cost=50)=0.3, P (cost=200)=0.2, P (cost=350)=0.5 –0.3 + 0.2 + 0.5 =1 50200350 0.30.20.5 0.3 50200350 0.2

17. Cumulative Distribution Function (1/2) Cumulative Distribution Function –Function : –Abbreviation : c.d.f 1.0 0.5 0.3 50200350 0

18. Cumulative Distribution Function (2/2) Example 1 : Machine Breakdowns

19. Cumulative Distribution Function (1/3) Cumulative Distribution Function

20. 2.3 The Expectation of a Random Variable Expectations of Discrete Random Variables (1/2) Expectation of a discrete random variable with p.m.f The expected value of a random variable is also called the mean of the random variable

21. Expectations of Discrete Random Variables (2/2) Example 1 (discrete random variable) –The expected repair cost is

22. 2.4 The variance of a Random Variable Definition and Interpretation of Variance (1/2) Variance( ) –A positive quantity that measures the spread of the distribution of the random variable about its mean value –Larger values of the variance indicate that the distribution is more spread out –Definition: Standard Deviation –The positive square root of the variance –Denoted by

23. Definition and Interpretation of Variance (2/2) Two distribution with identical mean values but different variances

24. Examples of Variance Calculations (1/1) Example 1

25. 2.5 Jointly Distributed Random Variables Jointly Distributed Random Variables (1/4) Joint Probability Distributions –Discrete

26. Jointly Distributed Random Variables (2/4) Joint Cumulative Distribution Function –Discrete

27. Jointly Distributed Random Variables (3/4) Example 19 : Air Conditioner Maintenance –A company that services air conditioner units in residences and office blocks is interested in how to schedule its technicians in the most efficient manner –The random variable X, taking the values 1,2,3 and 4, is the service time in hours –The random variable Y, taking the values 1,2 and 3, is the number of air conditioner units

28. Jointly Distributed Random Variables (4/4) Joint p.m.f Joint cumulative distribution function Y= number of units X=service time 1234 10.120.080.070.05 20.080.150.210.13 30.01 0.020.07

29. Conditional Probability Distributions (1/2) Conditional probability distributions –The probabilistic properties of the random variable X under the knowledge provided by the value of Y –Discrete

30. Conditional Probability Distributions (2/2) Example 19 –Marginal probability distribution of Y –Conditional distribution of X

31. Covariance –May take any positive or negative numbers. –Independent random variables have a covariance of zero –What if the covariance is zero?

32. Independence and Covariance Example 19 (Air conditioner maintenance)