Weibull Distribution Weibull Distribution Definition A random - PowerPoint PPT Presentation

Weibull Distribution

Weibull Distribution Definition A random variable X is said to have a Weibull distribution with parameters α and β ( α > 0 , β > 0) if the pdf of X is � β α x α − 1 e − ( x /β ) α α x ≥ 0 f ( x ; α, β ) = 0 x < 0

Weibull Distribution Definition A random variable X is said to have a Weibull distribution with parameters α and β ( α > 0 , β > 0) if the pdf of X is � β α x α − 1 e − ( x /β ) α α x ≥ 0 f ( x ; α, β ) = 0 x < 0 Remark: 1. The family of Weibull distributions was introduced by the Swedish physicist Waloddi Weibull in 1939.

Weibull Distribution Definition A random variable X is said to have a Weibull distribution with parameters α and β ( α > 0 , β > 0) if the pdf of X is � β α x α − 1 e − ( x /β ) α α x ≥ 0 f ( x ; α, β ) = 0 x < 0 Remark: 1. The family of Weibull distributions was introduced by the Swedish physicist Waloddi Weibull in 1939. 2. We use X ∼ WEB( α, β ) to denote that the rv X has a Weibull distribution with parameters α and β .

Weibull Distribution

Weibull Distribution Remark:

Weibull Distribution Remark: 3. When α = 1, the pdf becomes � β e − x /β 1 x ≥ 0 f ( x ; β ) = 0 x < 0 which is the pdf for an exponential distribution with parameter λ = 1 β . Thus we see that the exponential distribution is a special case of both the gamma and Weibull distributions.

Weibull Distribution Remark: 3. When α = 1, the pdf becomes � β e − x /β 1 x ≥ 0 f ( x ; β ) = 0 x < 0 which is the pdf for an exponential distribution with parameter λ = 1 β . Thus we see that the exponential distribution is a special case of both the gamma and Weibull distributions. 4. There are gamma distributions that are not Weibull distributios and vice versa, so one family is not a subset of the other.

Weibull Distribution

Weibull Distribution

Weibull Distribution

Weibull Distribution

Weibull Distribution

Weibull Distribution Proposition Let X be a random variable such that X ∼ WEI ( α, β ) . Then � �� 2 � � � � � � � 1 + 1 1 + 2 1 + 1 and V ( X ) = β 2 E ( X ) = β Γ Γ − Γ α α α The cdf of X is � 1 − e − ( x /β ) α x ≥ 0 F ( x ; α, β ) = 0 x < 0

Weibull Distribution

Weibull Distribution Example: The shear strength (in pounds) of a spot weld is a Weibull distributed random variable, X ∼ WEB (400 , 2 / 3) . a. Find P ( X > 410) .

Weibull Distribution Example: The shear strength (in pounds) of a spot weld is a Weibull distributed random variable, X ∼ WEB (400 , 2 / 3) . a. Find P ( X > 410) . b. Find P ( X > 410 | X > 390) .

Weibull Distribution Example: The shear strength (in pounds) of a spot weld is a Weibull distributed random variable, X ∼ WEB (400 , 2 / 3) . a. Find P ( X > 410) . b. Find P ( X > 410 | X > 390) . c. Find E ( X ) and V ( X ) .

Weibull Distribution Example: The shear strength (in pounds) of a spot weld is a Weibull distributed random variable, X ∼ WEB (400 , 2 / 3) . a. Find P ( X > 410) . b. Find P ( X > 410 | X > 390) . c. Find E ( X ) and V ( X ) . d. Find the 95 th percentile.

Weibull Distribution

Weibull Distribution In practical situations, γ = min( X ) > 0 and X − γ has a Weibull distribution.

Weibull Distribution In practical situations, γ = min( X ) > 0 and X − γ has a Weibull distribution. Example (Problem 74): Let X = the time (in 10 − 1 weeks) from shipment of a defective product until the customer returns the product . Suppose that the minimum return time is γ = 3 . 5 and that the excess X − 3 . 5 over the minimum has a Weibull distribution with parameters α = 2 and β = 1 . 5 . a. What is the cdf of X?

Weibull Distribution In practical situations, γ = min( X ) > 0 and X − γ has a Weibull distribution. Example (Problem 74): Let X = the time (in 10 − 1 weeks) from shipment of a defective product until the customer returns the product . Suppose that the minimum return time is γ = 3 . 5 and that the excess X − 3 . 5 over the minimum has a Weibull distribution with parameters α = 2 and β = 1 . 5 . a. What is the cdf of X? b. What are the expected return time and variance of return time?

Weibull Distribution In practical situations, γ = min( X ) > 0 and X − γ has a Weibull distribution. Example (Problem 74): Let X = the time (in 10 − 1 weeks) from shipment of a defective product until the customer returns the product . Suppose that the minimum return time is γ = 3 . 5 and that the excess X − 3 . 5 over the minimum has a Weibull distribution with parameters α = 2 and β = 1 . 5 . a. What is the cdf of X? b. What are the expected return time and variance of return time? c. Compute P ( X > 5) .

Weibull Distribution In practical situations, γ = min( X ) > 0 and X − γ has a Weibull distribution. Example (Problem 74): Let X = the time (in 10 − 1 weeks) from shipment of a defective product until the customer returns the product . Suppose that the minimum return time is γ = 3 . 5 and that the excess X − 3 . 5 over the minimum has a Weibull distribution with parameters α = 2 and β = 1 . 5 . a. What is the cdf of X? b. What are the expected return time and variance of return time? c. Compute P ( X > 5) . d. Compute P (5 ≤ X ≤ 8) .

Lognormal Distribution

Lognormal Distribution Definition A nonnegative rv X is said to have a lognormal distribution if the rv Y = ln( X ) has a normal distribution. The resulting pdf of a lognormal rv when ln( X ) is normally distributed with parameters µ and σ is � 2 πσ x e − [ln( x ) − µ ] 2 / (2 σ 2 ) 1 x ≤ 0 √ f ( x ; µ, σ ) = 0 x < 0

Lognormal Distribution Definition A nonnegative rv X is said to have a lognormal distribution if the rv Y = ln( X ) has a normal distribution. The resulting pdf of a lognormal rv when ln( X ) is normally distributed with parameters µ and σ is � 2 πσ x e − [ln( x ) − µ ] 2 / (2 σ 2 ) 1 x ≤ 0 √ f ( x ; µ, σ ) = 0 x < 0 Remark: 1. We use X ∼ LOGN( µ, σ 2 ) to denote that rv X have a lognormal distribution with parameters µ and σ .

Lognormal Distribution Definition A nonnegative rv X is said to have a lognormal distribution if the rv Y = ln( X ) has a normal distribution. The resulting pdf of a lognormal rv when ln( X ) is normally distributed with parameters µ and σ is � 2 πσ x e − [ln( x ) − µ ] 2 / (2 σ 2 ) 1 x ≤ 0 √ f ( x ; µ, σ ) = 0 x < 0 Remark: 1. We use X ∼ LOGN( µ, σ 2 ) to denote that rv X have a lognormal distribution with parameters µ and σ . 2. Notice here that the parameter µ is not the mean and σ 2 is not the variance, i.e. σ 2 � = V ( X ) µ � = E ( X ) and

Lognormal Distribution

Lognormal Distribution

Lognormal Distribution

Lognormal Distribution Proposition If X ∼ LOGN ( µ, σ 2 ) , then E ( X ) = e µ + σ 2 / 2 and V ( X ) = e 2 µ + σ 2 · ( e σ 2 − 1) The cdf of X is F ( x ; µ, σ ) = P ( X ≤ x ) = P [ln( X ) ≤ ln( x )] � Z ≤ ln( x ) − µ � � ln( x ) − µ � = P = Φ x ≤ 0 σ σ where Φ( z ) is the cdf of the standard normal rv Z.

Lognormal Distribution

Lognormal Distribution Example (Problem 115) Let I i be the input current to a transistor and I 0 be the output current. Then the current gain is proportional to ln( I 0 / I i ) . Suppose the constant of proportionality is 1 (which amounts to choosing a particular unit of measurement), so that current gain = X = ln( I 0 / I i ) . Assume X is normally distributed with µ = 1 and σ = 0 . 05 .

Lognormal Distribution Example (Problem 115) Let I i be the input current to a transistor and I 0 be the output current. Then the current gain is proportional to ln( I 0 / I i ) . Suppose the constant of proportionality is 1 (which amounts to choosing a particular unit of measurement), so that current gain = X = ln( I 0 / I i ) . Assume X is normally distributed with µ = 1 and σ = 0 . 05 . a. What is the probability that the output current is more than twice the input current?

Lognormal Distribution Example (Problem 115) Let I i be the input current to a transistor and I 0 be the output current. Then the current gain is proportional to ln( I 0 / I i ) . Suppose the constant of proportionality is 1 (which amounts to choosing a particular unit of measurement), so that current gain = X = ln( I 0 / I i ) . Assume X is normally distributed with µ = 1 and σ = 0 . 05 . a. What is the probability that the output current is more than twice the input current? b. What are the expected value and variance of the ratio of output to input current?