Problem 8: Suppose we have two types of lightbulbs. Let X represent the lifetime

Question

Problem 8: Suppose we have two types of lightbulbs. Let X represent the lifetime (in years) of a random lightbulb of the first type, and Y represent the lifetime (in years) of a random lightbulb of the second type. Suppose that X and Y both have exponential distributions, with lx = 1 and ly = 2.

a) Suppose in my living room, I have a light fixture that takes two bulbs. I put in one of the first type and one of the second type. Again assuming that X and Y are independent, what is the probability that they both last more than 1 year?

b) What is the probability at least one lasts more than 1 year?

Explanation / Answer

a) P(X > 1) = 1 - (1 - exp(-lx * t)) = exp(-1 * 1) = 0.367879

P(Y > 1) = 1 - (1 - exp(-ly * t)) = exp(-2 * 1) = 0.135335

As X & Y are independent

P(X & Y > 1) = P(X>1) * P(Y>1) = 0.367879*0.135335 = 0.0498 = 4.98%

b) P(atleast X or Y >1) = P(X > 1) * P(Y <= 1) + P(X <= 1) * P(Y > 1) + P(X > 1) * P(Y > 1)

P(Y<=1) = 1 - exp(-2 * 1) = 0.864665

P(X<=1) = 1 - exp(-1 * 1) = 0.632121

Therefore,

P(atleast X or Y >1) = 0.367879 * 0.864665 + 0.632121 * 0.135335 + 0.367879 * 0.135335

= 0.4534 = 45.34%

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