QCE Vault / Specialist Maths Exponential random variables — Question 213 Original QCE Vault practice · 6 marks
Browse all questions Original practice exam Report an issue Q213 · Practice question Technology-active Complex unfamiliar 6 marks
QUESTION 213 (6 marks) Independent components A and B have exponential lifetimes with rates 0.2 and 0.3 per hour. Let T be the first failure time.
a) Determine the distribution and mean of T.
[3 marks] b) Find the probability A fails first. You may integrate the density of A times survival of B.
[3 marks] WORKED SOLUTION
Practice marking scheme 6 marks ANSWER (a) Exponential rate 0.5 0.5 0.5 ; mean 2 2 2 h. (b) 0.4 0.4 0.4 . Worked solution
(a) For nonnegative t, P ( T > t ) = P ( A > t , B > t ) = e − 0.2 t e − 0.3 t = e − 0.5 t P(T>t)=P(A>t,B>t)=e^{-0.2t}e^{-0.3t}=e^{-0.5t} P ( T > t ) = P ( A > t , B > t ) = e − 0.2 t e − 0.3 t = e − 0.5 t . Thus T has exponential rate 0.5 and mean 1/0.5=2. (b) The probability is ∫ 0 ∞ 0.2 e − 0.2 t e − 0.3 t d t = ∫ 0 ∞ 0.2 e − 0.5 t d t = 0.2 / 0.5 = 0.4 \int_0^\infty0.2e^{-0.2t}e^{-0.3t}dt=\int_0^\infty0.2e^{-0.5t}dt=0.2/0.5=0.4 ∫ 0 ∞ 0.2 e − 0.2 t e − 0.3 t d t = ∫ 0 ∞ 0.2 e − 0.5 t d t = 0.2/0.5 = 0.4 . The exponential tends to zero at the limiting endpoint. A tie has probability zero. Equivalent mathematically justified methods accepted.
Exact answers unless a decimal accuracy is specified.
Part a: Use independent joint survival.
[1 mark] Part a: Identify rate 0.5.
[1 mark] Part b: Set up density times survival.
[1 mark] Part b: Evaluate the convergent integral.
[1 mark] Part b: Obtain probability 0.4.
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