1. Chapter 2. Random Variables
2.1 Discrete Random Variables
2.2 Continuous Random Variables
2.3 The Expectation of a Random Variable
2.4 The Variance of a Random Variable
2.5 Jointly Distributed Random Variables
2.6 Combinations and Functions of Random Variables

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

3. 2.1.1 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

4. 2.1.2 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 :

5. 2.1.2 Probability Mass Function (1/2)
Example 1 : Machine Breakdowns
P (cost=50)=0.3, P (cost=200)=0.2, P (cost=350)=0.5 =1 50
200
350
0.3
0.2
0.5
0.5
0.3 50
200
350
0.2

6. 2.1.3 Cumulative Distribution Function (1/2)
Abbreviation : c.d.f
1.0
0.5
0.3 50
200
350

7. 2.1.3 Cumulative Distribution Function (2/2)
Example 1 : Machine Breakdowns

8. 2. 2 Continuous Random Variables 2. 2
2.2 Continuous Random Variables Example of Continuous Random Variables (1/1)
Example 14 : Metal Cylinder Production
Suppose that the random variable is the diameter of a randomly chosen cylinder manufactured by the company. Since this random variable can take any value between 49.5 and 50.5, it is a continuous random variable.

9. 2.2.2 Probability Density Function (1/4)
Probability Density Function (p.d.f.)
Probabilistic properties of a continuous random variable

10. 2.2.2 Probability Density Function (2/4)
Example 14
Suppose that the diameter of a metal cylinder has a p.d.f
49.5
50.5

11. 2.2.2 Probability Density Function (3/4)
This is a valid p.d.f.

12. 2.2.2 Probability Density Function (4/4)
The probability that a metal cylinder has a diameter between 49.8 and 50.1 mm can be calculated to be
49.5
50.5
49.8
50.1

13. 2.2.3 Cumulative Distribution Function (1/3)

14. 2.2.2 Probability Density Function (2/3)
Example 14

15. 2.2.2 Probability Density Function (3/3)1
49.5
49.7
50.0
50.5

16. 2. 3 The Expectation of a Random Variable 2. 3
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
Expectation of a continuous random variable with p.d.f f(x)
The expected value of a random variable is also called the mean of the random variable

17. 2.3.1 Expectations of Discrete Random Variables (2/2)
Example 1 (discrete random variable)
The expected repair cost is

18. 2.3.2 Expectations of Continuous Random Variables (1/2)
Example 14 (continuous random variable)
The expected diameter of a metal cylinder is
Change of variable: y=x-50

19. 2.3.2 Expectations of Continuous Random Variables (2/2)
Symmetric Random Variables
If has a p.d.f that is symmetric about a point
so that
Then, (why?)
So that the expectation of the random variable is equal to the point of symmetry

21. 2.3.3 Medians of Random Variables (1/2)
Information about the “middle” value of the random variable
Symmetric Random Variable
If a continuous random variable is symmetric about a point , then both the median and the expectation of the random variable are equal to

22. 2.3.3 Medians of Random Variables (2/2)
Example 14

23. 2. 4 The variance of a Random Variable 2. 4
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

24. 2.4.1 Definition and Interpretation of Variance (2/2)
Two distribution with identical mean values but different variances

25. 2.4.2 Examples of Variance Calculations (1/1)

26. 2.4.3 Chebyshev’s Inequality (1/1)
If a random variable has a mean and a variance , then
For example, taking gives

27. Proof

28. 2.4.4 Quantiles of Random Variables (1/2)
The th quantile of a random variable X
A probability of that the random variable takes a value less than the th quantile
Upper quartile
The 75th percentile of the distribution
Lower quartile
The 25th percentile of the distribution
Interquartile range
The distance between the two quartiles

29. 2.4.4 Quantiles of Random Variables (2/2)
Example 14
Upper quartile :
Lower quartile :
Interquartile range :

30. 2. 5 Jointly Distributed Random Variables 2. 5
2.5 Jointly Distributed Random Variables Jointly Distributed Random Variables (1/4)
Joint Probability Distributions
Discrete
Continuous

31. 2.5.1 Jointly Distributed Random Variables (2/4)
Joint Cumulative Distribution Function
Discrete
Continuous

32. 2.5.1 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

33. 2.5.1 Jointly Distributed Random Variables (4/4)
number of units
X=service time 1 2 3 4
0.12
0.08
0.07
0.05
0.15
0.21
0.13
0.01
0.02
Joint p.m.f
Joint cumulative distribution function

34. 2.5.2 Marginal Probability Distributions (1/2)
Obtained by summing or integrating the joint probability distribution over the values of the other random variable
Discrete
Continuous

35. 2.5.2 Marginal Probability Distributions (2/2)
Example 19
Marginal p.m.f of X
Marginal p.m.f of Y

36. Example 20: (a jointly continuous case)
Joint pdf:
Marginal pdf’s of X and Y:

37. 2.5.3 Conditional Probability Distributions (1/2)
The probabilistic properties of the random variable X under the knowledge provided by the value of Y
Discrete
Continuous
The conditional probability distribution is a probability distribution.

38. 2.5.3 Conditional Probability Distributions (2/2)
Example 19
Marginal probability distribution of Y
Conditional distribution of X

39. 2.5.4 Independence and Covariance (1/5)
Two random variables X and Y are said to be independent if
Discrete
Continuous
How is this independency different from the independence among events?

40. 2.5.4 Independence and Covariance (2/5)
May take any positive or negative numbers.
Independent random variables have a covariance of zero
What if the covariance is zero?

41. 2.5.4 Independence and Covariance (3/5)
Example 19 (Air conditioner maintenance)

42. 2.5.4 Independence and Covariance (4/5)
Correlation:
Values between -1 and 1, and independent random variables have a correlation of zero

43. 2.5.4 Independence and Covariance (5/5)
Example 19: (Air conditioner maintenance)

44. What if random variable X and Y have linear relationship, that is,
where
That is, Corr(X,Y)=1 if a>0; -1 if a<0.

45. 2. 6 Combinations and Functions of Random Variables 2. 6
2.6 Combinations and Functions of Random Variables Linear Functions of Random Variables (1/4)
Linear Functions of a Random Variable
If X is a random variable and for some numbers then and
Standardization
-If a random variable has an expectation of and a variance of , has an expectation of zero and a variance of one.

46. 2.6.1 Linear Functions of Random Variables (2/4)
Example 21:Test Score Standardization
Suppose that the raw score X from a particular testing procedure are distributed between -5 and 20 with an expected value of 10 and a variance 7. In order to standardize the scores so that they lie between 0 and 100, the linear transformation is applied to the scores.

47. 2.6.1 Linear Functions of Random Variables (3/4)
For example, x=12 corresponds to a standardized score of y=(4ⅹ12)+20=68

48. 2.6.1 Linear Functions of Random Variables (4/4)
Sums of Random Variables
If and are two random variables, then and
If and are independent, so that then

49. Properties of

50. 2.6.2 Linear Combinations of Random Variables (1/5)
If is a sequence of random variables and and are constants, then
If, in addition, the random variables are independent, then

51. 2.6.2 Linear Combinations of Random Variables (2/5)
Averaging Independent Random Variables
Suppose that is a sequence of independent random variables with an expectation and a variance .
Let
Then and
What happened to the variance?

52. 2.6.2 Linear Combinations of Random Variables (3/5)

53. 2.6.2 Linear Combinations of Random Variables (4/5)
Example 21
The standardized scores of the two tests are
The final score is

54. 2.6.2 Linear Combinations of Random Variables (5/5)
The expected value of the final score is
The variance of the final score is

55. 2.6.3 Nonlinear Functions of a Random Variable (1/3)
Nonlinear function of a Random Variable
A nonlinear function of a random variable X is another random variable Y=g(X) for some nonlinear function g.
There are no general results that relate the expectation and variance of the random variable Y to the expectation and variance of the random variable X

56. 2.6.3 Nonlinear Functions of a Random Variable (2/3)
For example,
Consider
What is the pdf of Y? 1 x
f(x)=1
E(x)=0.5
1.0
2.718 y
f(y)=1/y
E(y)=1.718

57. 2.6.3 Nonlinear Functions of a Random Variable (3/3)
CDF methd
The p.d.f of Y is obtained by differentiation to be
Notice that

58. Determining the p.d.f. of nonlinear relationship between r.v.s:
Given and ,what is ?
If are all its real roots, that is,
then
where

59. Example: determine
-> one root is possible in