Question

6. Suppose that the lifetime of an Ollivander's magic wand is exponentially distributed with a mean of 20 years. Luna Lovegood has had her wand for 25 years. What is the probability that it will last an additional 10 years?

Answer 1

Let x be the random variable denote the lifetime of magic w since x is exponentially distributed then the probabilities density function is f(x) = lambda e^- lambda x, 0 lessthanorequalto x < infinity given that mean = 20 years or, lambda = 20 or, lambda = 1/20 therefore f(x) = 1/20 e^-x/20, 0 lessthanorequalto x < infinity the required probability p(25 lessthanorequalto x lessthanorequalto 25 + 10) = p(x lessthanorequalto 35) - p(x lessthanorequalto 25) = integral^35_0 1/20 e^-x/20 dx - integral^25_0 1/20 e^-x/20 dx = 1/20 {[-20 e^-x/20]^35_0 - [-20 e^-x/20]^25_0} = 1/20 {20 (1 - e^-35/20) - 20 (1 - e^-25/20)} = 1/20 times 20(e^-5/4 -e^-1/4) = e^-5/4 - e^-1/4 = 0.1127