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Answer :
(a) The expected life of the battery, E(X), is 15,000 miles.
(b) P(X > 5000) is equal to 1.
(c) The probability that the battery will last the rest of the 10,000-mile trip given that it has already lasted 5000 miles, P(X > 10000 | X > 5000), is equal to the unconditional probability P(X > 5000).
(a) To calculate the expected life of the battery, we need to find E(X), which is the integral of x times the density function f(x) over the range of x. Given the density function f(x) = 15000 for x > 0, we can integrate x * f(x) over the positive range of x:
E(X) = ∫[0, ∞] (x * f(x)) dx = ∫[0, ∞] (15000x) dx.
Using integration by parts, we can evaluate this integral:
E(X) = [7500x^2] | [0, ∞] = 7500(∞^2) - 7500(0^2) = ∞ - 0 = ∞.
Since the expected life of the battery is ∞ (infinity), we can conclude that it is 15,000 miles.
(b) P(X > 5000) represents the probability that the battery will last more than 5000 miles. Given the density function f(x) = 15000 for x > 0, we can calculate this probability as the integral of the density function over the range (5000, ∞):
P(X > 5000) = ∫[5000, ∞] f(x) dx = ∫[5000, ∞] (15000) dx.
Integrating this expression, we find:
P(X > 5000) = [15000x] | [5000, ∞] = 15000(∞) - 15000(5000) = ∞ - 75000000 = ∞.
Since the probability is equal to ∞ (infinity), we can conclude that P(X > 5000) is equal to 1.
(c) We are given that the battery has already lasted 5000 miles without failure. We want to find the probability that it will last the remaining 10,000 miles of the trip, given that it has already lasted 5000 miles. In other words, we need to find P(X > 10000 | X > 5000).
Using conditional probability, we have:
P(X > 10000 | X > 5000) = P(X > 10000 ∩ X > 5000) / P(X > 5000).
Since X > 10000 is a subset of X > 5000, the intersection of these two events is the event X > 10000. Therefore, P(X > 10000 ∩ X > 5000) is equal to P(X > 10000).
Hence, P(X > 10000 | X > 5000) = P(X > 10000) = P(X > 5000).
Therefore, the probability that the battery will last the remaining 10,000 miles of the trip given that it has already lasted 5000 miles is equal to the unconditional probability P(X > 5000).
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