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.

3-2 Random Variables

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

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.

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.

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 .

3-3 Probability Probability Properties

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.

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

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.

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

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

3-4 Continuous Random Variables

3-4 Continuous Random Variables

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

3-4 Continuous Random Variables

3-4 Continuous Random Variables

3-4 Continuous Random Variables

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

3-4 Continuous Random Variables

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

3-5 Important Continuous Distributions 3-5.1 Normal Distribution

3-5 Important Continuous Distributions 3-5.1 Normal Distribution

3-5 Important Continuous Distributions

3-5 Important Continuous Distributions 3-5.1 Normal Distribution

3-5 Important Continuous Distributions 3-5.1 Normal Distribution

3-5 Important Continuous Distributions 3-5.1 Normal Distribution

3-5 Important Continuous Distributions

3-5 Important Continuous Distributions 3-5.1 Normal Distribution

3-5 Important Continuous Distributions 3-5.1 Normal Distribution

3-5 Important Continuous Distributions

3-5 Important Continuous Distributions

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 P1 P X1 1 13 0.93319 0.98 14.1075

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 1 16.7103 -1.64485    PROB 3-43 (D) IN PAGE 90 OBS ZU ZL P1 P2 ANS 1 1 -2.5 0.84134 .006209665 0.83514

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.

3-6 Probability Plots 3-6.1 Normal Probability Plots

3-6 Probability Plots 3-6.1 Normal Probability Plots

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

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

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

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

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

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

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

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

3-11 More Than One Random Variable and Independence 3-11.1 Joint Distributions

3-11 More Than One Random Variable and Independence 3-11.1 Joint Distributions

3-11 More Than One Random Variable and Independence 3-11.1 Joint Distributions

3-11 More Than One Random Variable and Independence 3-11.1 Joint Distributions

3-11 More Than One Random Variable and Independence

3-11 More Than One Random Variable and Independence

3-11 More Than One Random Variable and Independence

3-12 Functions of Random Variables

3-12 Functions of Random Variables 3-12.1 Linear Functions of Independent Random Variables

3-12 Functions of Random Variables 3-12.1 Linear Functions of Independent Random Variables

3-12 Functions of Random Variables 3-12.1 Linear Functions of Independent Random Variables

3-12 Functions of Random Variables 3-12.2 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 - μ22 - 2μ1μ2 = [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

3-12 Functions of Random Variables 3-12.2 Linear Functions of Random Variables That Are Not Independent

3-12 Functions of Random Variables 3-12.2 Linear Functions of Random Variables That Are Not Independent

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

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

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

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

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

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 3-195 IN PAGE 140’; RUN; QUIT;   PROB 3-195 IN PAGE 140 OBS MU SDBAR ZL ZU P1 P2 ANS 1 200 2.25 -1.77778 0.88889 0.81297 0.037720 0.77525