Question

Experience shows that errors in DNA typing do occur. Although the proportion of errors occurring in forensic DNA laboratories is perhaps low due to regular proficiency testing, for practical reasons, it may be desirable to have some estimate of error rates for a specific laboratory.

Consider, then, a forensic DNA laboratory with n = 300 internal tests.

a) What is the distribution of the number of errors occurring in forensic DNA laboratories? Provide name, formula, and parameter(s) of distribution, explicitly.

b) Let λ be the true mean number of errors occurring in forensic DNA laboratories. Use a Gamma prior with mean 1 and variance 9. Find the posterior distribution of λ given data. Provide its parameters, explicitly.

c) Summarize the posterior distribution by its first two moments (i.e. mean and variance).

Answer 1

@ Let x = The no. of errors occuring in forensic DNA, laboratories Here, X~ Poisson (~) where 1 = Parameter of the disth x follows poisson disth with parameter 1. fx (x) = ett , x>0, 770 lo, O.W (6) ~ Poison (1) an*(x) = Gamma (0,8), where de mean = oß = 1 & variance = xp = 9 5 d = 12, B = 9 To find posterior dist of a q (0)x) Posterior dist of a given X=x is - q (2/2) = f(x) 2). IT (2/6, 8) s plécla)-7 (ala, p)da 1 * * See Nada patol-1, e-7 (148-1) o peta-! e-> (178) eta-1 ed (148-1) = Gamma (ot, ) r (ata). (48-1)(ata) with oct=xtol? p* = 48-4)! =

(C) Here q (2x) a Gamma (at=xtog B&= (148-)). Now, Mean of Posterior disth = E(^\x) = x*.pt - xto AB-1 T1+ß 4 This is the Bayes estimator under SELF Variance of posterior dista lal - 2* Bx2 = B² (Xto) = v(alx) = 2# B (178) 2