4. 3-2 Random Variables denoted by a variable such as X.
In an experiment, a measurement is usually
denoted by a variable such as X.
In a random experiment, a variable whose
measured value can change (from one replicate of
the experiment to another) is referred to as a
random variable.

5. 3-2 Random Variables

6. 3-3 Probability Used to represent risk or uncertainty in engineering
Used to quantify likelihood or chance
Used to represent risk or uncertainty in engineering
applications
Can be interpreted as our degree of belief or
relative frequency

7. 3-3 Probability particular values occur.
Probability statements describe the likelihood that
particular values occur.
The likelihood is quantified by assigning a number
from the interval [0, 1] to the set of values (or a
percentage from 0 to 100%).
Higher numbers indicate that the set of values is
more likely.

8. 3-3 Probability random variable.
A probability is usually expressed in terms of a
random variable.
For the part length example, X denotes the part
length and the probability statement can be written
in either of the following forms
Both equations state that the probability that the
random variable X assumes a value in [10.8, 11.2] is
0.25.

9. 3-3 Probability Complement of an Event
Given a set E, the complement of E is the set of
elements that are not in E. The complement is
denoted as E’.
Mutually Exclusive Events
The sets E1 , E2 ,...,Ek are mutually exclusive if the
intersection of any pair is empty. That is, each
element is in one and only one of the sets E1 , E2
,...,Ek .

10. 3-3 Probability
Probability Properties

11. 3-3 Probability Events A measured value is not always obtained from an
experiment. Sometimes, the result is only classified
(into one of several possible categories).
These categories are often referred to as events.
Illustrations
The current measurement might only be
recorded as low, medium, or high; a manufactured
electronic component might be classified only as
defective or not; and either a message is sent through a
network or not.

12. 3-4 Continuous Random Variables
3-4.1 Probability Density Function

13. 3-4 Continuous Random Variables
3-4.1 Probability Density Function
The probability distribution or simply distribution of a random variable X is a description of the set of the probabilities associated with the possible values for X.

14. 3-4 Continuous Random Variables
3-4.1 Probability Density Function

15. 3-4 Continuous Random Variables
3-4.1 Probability Density Function

16. 3-4 Continuous Random Variables

17. 3-4 Continuous Random Variables

18. 3-4 Continuous Random Variables
3-4.2 Cumulative Distribution Function

19. 3-4 Continuous Random Variables

20. 3-4 Continuous Random Variables

21. 3-4 Continuous Random Variables

22. 3-4 Continuous Random Variables
3-4.3 Mean and Variance

23. 3-4 Continuous Random Variables

24. 3-5 Important Continuous Distributions
3-5.1 Normal Distribution
Undoubtedly, the most widely used model for the distribution of a random variable is a normal distribution.
Central limit theorem
Gaussian distribution

25. 3-5 Important Continuous Distributions
3-5.1 Normal Distribution

26. 3-5 Important Continuous Distributions
3-5.1 Normal Distribution

27. 3-5 Important Continuous Distributions

28. 3-5 Important Continuous Distributions
3-5.1 Normal Distribution

29. 3-5 Important Continuous Distributions
3-5.1 Normal Distribution

30. 3-5 Important Continuous Distributions
3-5.1 Normal Distribution

31. 3-5 Important Continuous Distributions

32. 3-5 Important Continuous Distributions
3-5.1 Normal Distribution

33. 3-5 Important Continuous Distributions
3-5.1 Normal Distribution

34. 3-5 Important Continuous Distributions

35. 3-5 Important Continuous Distributions

36. 3-5 Important Continuous Distributions
OPTIONS NOPAGE NODATE LS=80;
DATA PAGE82;
MEAN=10; SD=2;X=13;P=0.98;
Z1=(X-MEAN)/SD;
P1=PROBNORM(Z1);
Z2=PROBIT(P);
X1=MEAN+Z2*SD;
PROC PRINT;
VAR X P1;
VAR P X1;
TITLE 'EXAMPLE IN PAGE 82-83';
RUN;
QUIT;
EXAMPLE IN PAGE 82-83
OBS X P P X1

37. 3-5 Important Continuous Distributions
DATA P342B;
Z=PROBIT(0.05);
/* The PROBIT function returns the pth quantile from the standard normal distribution. The probability that an observation from the standard normal distribution is less than or equal to the returned quantile is p. */
MU=20; STD=2;
X= MU+Z*STD;
PROC PRINT;
VAR Z X;
TITLE 'PROB 3-42 (B) IN PAGE 90';
DATA P343D;
MU=27; SIGMA=2;XU=29;XL=22;
ZU=(XU-MU)/SIGMA; ZL=(XL-MU)/SIGMA;
P1=PROBNORM(ZU); P2=PROBNORM(ZL);
/* The PROBNORM function returns the probability that an observation from the standard normal distribution is less than or equal to x. */
  ANS=P1-P2;
VAR ZU ZL P1 P2 ANS;
TITLE 'PROB 3-43 (D) IN PAGE 90';
RUN; QUIT;
  PROB 3-42 (B) IN PAGE 90
OBS X Z   
PROB 3-43 (D) IN PAGE 90
OBS ZU ZL P P ANS

38. 3-6 Probability Plots 3-6.1 Normal Probability Plots
How do we know if a normal distribution is a reasonable
model for data?
Probability plotting is a graphical method for determining
whether sample data conform to a hypothesized
distribution based on a subjective visual examination of the
data.
Probability plotting typically uses special graph paper, known
as probability paper, that has been designed for the
hypothesized distribution. Probability paper is widely
available for the normal, lognormal, Weibull, and various chi-
square and gamma distributions.

39. 3-6 Probability Plots
3-6.1 Normal Probability Plots

40. 3-6 Probability Plots
3-6.1 Normal Probability Plots

41. 3-7 Discrete Random Variables
Only measurements at discrete points are
possible

42. 3-7 Discrete Random Variables
3-7.1 Probability Mass Function

43. 3-7 Discrete Random Variables
3-7.1 Probability Mass Function

44. 3-7 Discrete Random Variables
3-7.2 Cumulative Distribution Function

45. 3-7 Discrete Random Variables
3-7.2 Cumulative Distribution Function

46. 3-7 Discrete Random Variables
3-7.3 Mean and Variance

47. 3-7 Discrete Random Variables
3-7.3 Mean and Variance

48. 3-7 Discrete Random Variables
3-7.3 Mean and Variance

49. 3-11 More Than One Random Variable and Independence
Joint Distributions

50. 3-11 More Than One Random Variable and Independence
Joint Distributions

51. 3-11 More Than One Random Variable and Independence
Joint Distributions

52. 3-11 More Than One Random Variable and Independence
Joint Distributions

53. 3-11 More Than One Random Variable and Independence

54. 3-11 More Than One Random Variable and Independence

55. 3-11 More Than One Random Variable and Independence

56. 3-12 Functions of Random Variables

57. 3-12 Functions of Random Variables
Linear Functions of Independent
Random Variables

58. 3-12 Functions of Random Variables
Linear Functions of Independent
Random Variables

59. 3-12 Functions of Random Variables
Linear Functions of Independent
Random Variables

60. 3-12 Functions of Random Variables
Linear Functions of Random Variables
That Are Not Independent
Y=X1+X2 (X1 and X2 are not independent)
E(Y) = E(X1+X2)= E(X1) + E(X2) = μ1 + μ2
V(Y) = E(Y2) – E(Y)2
= E[(X1 + X2)2] – [E(X1 + X2)]2
= E(X12 + X22 + 2X1X2) – (μ1 + μ2)2
= E(X12) + E(X22) + 2E(X1X2) - μ12 - μ μ1μ2
= [E(X12) - μ12]+[E(X22) - μ22 ] + 2[E(X1X2) - μ1μ2]
= σ12 + σ22 + 2[E(X1X2) - μ1μ2]
where the quantity E(X1X2) - μ1μ2 is called covariance

61. 3-12 Functions of Random Variables
Linear Functions of Random Variables
That Are Not Independent

62. 3-12 Functions of Random Variables
Linear Functions of Random Variables
That Are Not Independent

63. 3-13 Random Samples, Statistics, and
The Central Limit Theorem

64. 3-13 Random Samples, Statistics, and The Central Limit Theorem
𝜇 𝑋 =E( 𝑋 )=E( 𝑋 1 + 𝑋 2 + …. + 𝑋 𝑛 𝑛 )= 1 𝑛 ∙𝑛𝜇=𝜇
𝜎 𝑋 2 =𝑉 𝑋 =V( 𝑋 1 + 𝑋 2 + …. + 𝑋 𝑛 𝑛 )= 1 𝑛 2 ∙𝑛 𝜎 2 = 𝜎 2 𝑛

65. 3-13 Random Samples, Statistics, and
The Central Limit Theorem

66. 3-13 Random Samples, Statistics, and
The Central Limit Theorem

67. 3-13 Random Samples, Statistics, and
The Central Limit Theorem

68. 3-13 Random Samples, Statistics, and The Central Limit Theorem
OPTIONS NODATE NONUMBER;
DATA EX3195AD;
MU=200; SD=9; N=16;
SDBAR=SD/SQRT(N);
ZU=(202-MU)/SDBAR;
ZL=(196-MU)/SDBAR;
P1=PROBNORM(ZU);
P2=PROBNORM(ZL);
ANS=P1-P2;
PROC PRINT;
VAR MU SDBAR ZL ZU P1 P2 ANS;
TITLE 'PROB IN PAGE 140’;
RUN; QUIT;
PROB IN PAGE 140
OBS MU SDBAR ZL ZU P P ANS