Exercise2.3 An electronicdevicecontains k identical, independentlyworkingsensitiveelements whose lifetimesareexponentiallydistributedexp(). Evenifoneelementdoesnotoperate,the entire deviceisrenderedoutoforderandmustbesenttoalaboratoryforrepair. n deviceswith lifetimes t1, ..., tn

Exercise2.3 An electronicdevicecontains k identical, independentlyworkingsensitiveelements whose lifetimesareexponentiallydistributedexp(θ). Evenifoneelementdoesnotoperate,the entire deviceisrenderedoutoforderandmustbesenttoalaboratoryforrepair. n deviceswith lifetimes t1, ..., tn hours havebeensenttoalaboratory.Basedonthisdata,findtheMLEsof θ, the expectedlifetimesofasingleelementandanentiredevice,andtheprobabilitythatadevicewill operate atleast a hours.

(Hint: showfirstthat ti ∼ exp(kθ).)