Consider a satellite whose work is based on a certain block A. This block has an independent backup B. The satellite performs its task until both A and B fail. The lifetimes of A and B are exponentially distributed with the mean lifetime of 10 years. (a) What is the probability that the satellite will work for more than 10 years? (b) Compute the expected lifetime of the satellite.

a) 0.6

b) 15 years

Step-by-step explanation:

a) Since the two blocks have the same probability function, the exponential probability that one of the blocks will fail, is given as

P(X>x) = 1 - e^(-λ.x)

λ = rate parameter = (1/10) = 0.1

x = variable whose probability is required = 10 years.

λ.x = 0.1 × 10 = 1

P(X>10) = (1 - e⁻¹) = 0.632

Probability that both blocks fail = 0.632 × 0.632 = 0.3999

Probability that the satellite will work for more than 10 years = Probability that at least one of the blocks is working = 1 - (probability that both blocks are not working) = 1 - 0.3999 = 0.60

b) Expected lifetime of the satellite for the two blocks in a sort of parallel combination will be given as

E(X) = (1/λ) + (1/λ) - [1/(λ + λ)]

E(X) = 10 + 10 - 5 = 15 years

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