lightb nt types.The lifetime of the first lightbulb (in years) is described by an exponential random variable X with mean 1, and the second lightbulb's lifetime is described by an exponential random variable Y with mean 2, independent from X. Let T be the amount of time from now until either one of the lightbulbs burns out, so T min(X,Y).

A First Course in Probability (10th Edition)
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Chapter1: Combinatorial Analysis
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You have a lamp with two different lightbulbs of different types. The
lifetime of the first lightbulb (in years) is described by an exponential
random variable X with mean 1, and the second lightbulb's lifetime is
described by an exponential random variable Y with mean 2,
independent from X. Let T be the amount of time from now until either
one of the lightbulbs burns out, so T = min(X,Y).
(a) Find the cumulative distribution function of T.
(b) Find the probability density function of T.
(c) Find the expectation of T.
Transcribed Image Text:You have a lamp with two different lightbulbs of different types. The lifetime of the first lightbulb (in years) is described by an exponential random variable X with mean 1, and the second lightbulb's lifetime is described by an exponential random variable Y with mean 2, independent from X. Let T be the amount of time from now until either one of the lightbulbs burns out, so T = min(X,Y). (a) Find the cumulative distribution function of T. (b) Find the probability density function of T. (c) Find the expectation of T.
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