4.97 Suppose that the positive random variables X₁,..., X, are independent of each other and have respective failure-rate functions r₁(x),...,n(x). Prove that the random variable V = min(X₁,..., Xn) has the failure-rate function k-1 rk (x). In particular, conclude that V has an exponential distribution with parameter k-1 Ak when each of the X; is exponen- tially distributed with parameter ₁.
4.97 Suppose that the positive random variables X₁,..., X, are independent of each other and have respective failure-rate functions r₁(x),...,n(x). Prove that the random variable V = min(X₁,..., Xn) has the failure-rate function k-1 rk (x). In particular, conclude that V has an exponential distribution with parameter k-1 Ak when each of the X; is exponen- tially distributed with parameter ₁.
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Question number 4.97
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4.11 Probability Distributions and Entropy
205
4.97 Suppose that the positive random variables X₁,..., Xn are independent
of each other and have respective failure-rate functions r₁(x),...,n(x).
Prove that the random variable V = min(X₁,..., Xn) has the failure-rate
function 1 rk(x). In particular, conclude that V has an exponential
distribution with parameter 1 Ak when each of the X; is exponen-
tially distributed with parameter λi.
=
4.98 Suppose that the positive random variable X has the failure-rate function
(1 + x)−¹ for x > 0. Verify the remarkable result that the
reliability function of X is also given by (1+x)-¹ for x > 0.
4.99 The lifetime X of a semiconductor has the failure-rate function r(x) =
λax-¹/(1+x) for x ≥ 0, where the parameters a and λ are positive.
2016 THE
What is the reliabilitu
of the lifetime X? Verify that the failure-
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