The lifetime (in months) of a battery is modeled by a random variable X that has pdf
Question:
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 ] =
Probability & Statistics for Engineers & Scientists
ISBN: 978-0130415295
7th Edition
Authors: Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Keying