Question

The lifetimes of A's dog and cat are independent exponential random variables with respective rates \(\lambda_{d}\) and \(\lambda_{c} .\) One of them has just died. Find the expected additional lifetime of the other pet.

Short answer

The expected additional lifetime of the other pet after one of them has died is \(\frac{1}{\lambda_{d} + \lambda_{c}}\).

Step-by-step solution

Find the conditional probability for a pet's death

Given that one of the pets has just died, we want to find the conditional probability that the pet that died was the dog or the cat. We can find this probability as follows: Let A be the event that the dog died and B be the event that the cat died. Using the definition of conditional probability, we have: \(P(A|A \cup B) = \frac{P(A \cap (A \cup B))}{P(A \cup B)}\) Similarly, \(P(B|A \cup B) = \frac{P(B \cap (A \cup B))}{P(A \cup B)}\)

Find the joint probabilities and probability of the union

Since the lifetimes of the dog and cat are independent, the joint probabilities are just the product of the individual probabilities. Therefore, \(P(A \cap (A \cup B)) = P(A)P(A \cup B)\) and \(P(B \cap (A \cup B)) = P(B)P(A \cup B)\) Additionally, since A and B are mutually exclusive events, we can write the probability of the union as the sum of the individual probabilities: \(P(A \cup B) = P(A) + P(B)\)

Find the individual probabilities

The individual probabilities can be expressed in terms of the rates of the respective exponential distributions: \(P(A) = \lambda_{d}\) and \(P(B) = \lambda_{c}\)

Substitute the values and find the conditional probabilities

Now, we can substitute the values of the individual probabilities into the expressions for the conditional probabilities from Step 1: \(P(A|A \cup B) = \frac{\lambda_{d}}{\lambda_{d} + \lambda_{c}}\) and \(P(B|A \cup B) = \frac{\lambda_{c}}{\lambda_{d} + \lambda_{c}}\)

Find the expected additional lifetime of the remaining pet

By using the memoryless property of exponential distributions, we know that the expected additional lifetime of the dog or the cat is equal to the expected value of their respective lifetimes, which is the reciprocal of their rates: Expected additional lifetime of dog = \(\frac{1}{\lambda_{d}}\) Expected additional lifetime of cat = \(\frac{1}{\lambda_{c}}\) Taking into account the probabilities of which animal has died, we can now find the expected additional lifetime of the remaining pet: Expected additional lifetime of the remaining pet = \(P(A|A \cup B) \times \frac{1}{\lambda_{c}} + P(B|A \cup B) \times \frac{1}{\lambda_{d}}\) Substituting the conditional probabilities, we get: Expected additional lifetime of the remaining pet = \(\frac{\lambda_{d}}{\lambda_{d} + \lambda_{c}} \times \frac{1}{\lambda_{c}} + \frac{\lambda_{c}}{\lambda_{d} + \lambda_{c}} \times \frac{1}{\lambda_{d}}\) Simplifying the expression: Expected additional lifetime of the remaining pet = \(\frac{1}{\lambda_{d} + \lambda_{c}}\) Hence, the expected additional lifetime of the other pet after one of them has died is \(\frac{1}{\lambda_{d} + \lambda_{c}}\).