The lifetime (in months) of a battery is modeled by a random variable X that has pdf f ( x ) = K

The lifetime (in months) of a battery is modeled by a random variable X that has pdf 

f θ ( x ) = K θ x 1 ( x > 0 ) where  K = ln ( 1 / θ )

 for an unknown parameter 

θ ∈ ( 0 , 1 ) . (Here 1 ( x > 0 ) is the indicator variable that takes value 1 when its argument is true, i.e. when x > 0 .) 

Assume that we have n independent observations X 1 , … , X n of the lifetime of n batteries. We want to use these observations to estimate θ ∈ ( 0 , 1 ) . 

Note (October 24):   Assume that the observations are of the same type of batteries. 

Input instructions:   For all problems below, when inputting the natural log function, use ln , e.g. enter ln(theta) for ln ( θ ) . 

Compute the expected value E [ X i ] and the variance Var [ X i ] of X i . 

  

(Enter your answer in terms of θ only . ) 

  

Hint: Note that the given pdf is equivalent to the pdf of a common distribution except reparametrized with a different parameter. 

  

    

E [ X i ] =

                                  unanswered                  

Var [ X i ] =