Question

In the reactor safety study, the failure rate of a diesel generator can be described as having a lognormal distribution with the upper and lower 90 % bounds of 3x10^-2 and 3x10^-4, respectively. a) Determine prior distribution in lognormal distribution. b) The given nuclear plant experiences 2 failures in 8,760 hours of operation. Determine the posterior distribution based on Poisson distribution. c) Determine the upper and lower 90 % bounds given this plant experience. (Consider the reactor safety study values as prior information.

Answer 1

a)

Consider a random variable X that describes the failure rate of the diesel generator

Given that X has a lognormal distribution, the random variable Y given as Y = ln(X) will have a normal distribution

For the upper 90% boundary as 0.03,

Y₁ = ln (0.03) = -3.5

For the lower 90% boundary as 0.0003,

Y₂ = ln (0.0003) = -8.11

Mean of the normal distribution E(Y) = (-3.5 – 8.11)/2 = -5.80

The corresponding value of X = e^Y = e^(-5.80) = 0.003

Y₁ = E(Y) + σ_Y z

Using z tables, for 90% confidence interval z = 1.64

Substituting in the above equation for Y₁

-3.5 = -5.8 + 1.64σ_Y

σ_Y = 1.399

The log normal distribution is given as:

ln (X) ~ N (-5.80, 1.399²)

b)

Given a Poisson Distribution, the Probability of the random variable for the number of failures in a Poisson distribution is given as:

P(X = x) = e ^{− λ} λ^x / x!

Mean of the Poisson Distribution = λ = average number of failures = 2 failures/year

Thus the Poisson’s distribution is given as

P(X = x) = e ⁻² 2^x / x!

c)

90% confidence interval for the Poisson’s distribution is given as:

λ± 1.64 * sqrt (λ)

• 2 ± 1.64* sqrt(2)
• [ -0.32 , 4.32 ]