Random Variables

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 from an experiment • For each element of an experiment’s sample space, the random 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 any value along a continuum (but may be reported “discretely” • Random Variables are denoted by upper case letters (Y) • Individual outcomes for RV are denoted by lower case letters (y)

3. Probability Distributions • Probability Distribution: Table, Graph, or Formula that describes values a random variable can take on, and its corresponding probability (discrete RV) or density (continuous RV) • Discrete Probability Distribution: Assigns probabilities (masses) to the individual outcomes • Continuous Probability Distribution: Assigns density at individual points, probability of ranges can be obtained by integrating 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) Y = Sum of the up faces of the two die. Table gives value of y for all elements in S

6. Rolling 2 Dice – Probability Mass Function & CDF

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 the data) • Notation: • Mean: E(Y) = m • Variance: V(Y) = s2 • Standard Deviation: s

10. Expected Values of Discrete RV’s

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

12. Example – Rolling 2 Dice

13. -3 -2 -1 1 0 2 3 2.1 Discrete Random VariableDefinition of a Random Variable (1/2) • Random variable • A numerical value to each outcome of a particular experiment

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. 0.5 0.3 0.2 50 200 350 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

17. 1.0 0.5 0.3 0 50 200 350 Cumulative Distribution Function (1/2) • Cumulative Distribution Function • Function : • Abbreviation : c.d.f

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 VariableExpectations 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 VariableDefinition 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 VariablesJointly 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

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 • 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)