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Math 180C MIDTERM 2 DUE 05/22/20, 11:00pm 
Write your name and PID on the top of EVERY PAGE. 
Write the solutions to each problem on separate pages. 
CLEARLY INDICATE on the top of each page the number of 
the corresponding problem. 
Remember this exam is graded by a human being. Write your 
solutions NEATLY AND COHERENTLY, or they risk not re- 
ceiving full credit. 
You may assume that all transition probability functions are 
STATIONARY. 
You are allowed to use the textbook, lecture notes and your 
personal notes. You are not allowed to use the electronic devices 
(except for accessing the online version of the textbook) or outside 
assistance. Outside assistance includes but is not limited to other 
people, the internet and unauthorized notes. 
Math 180C MIDTERM 2, Page 2 of 5 DUE 05/22/20, 11:00pm 
1. (25 points) Let Y > 0 be a random variable having Gamma distribution with parameters 
2 and λ, i.e., the p.d.f. of Y is given by 
fY (y) = λ 
2ye−λy, y > 0, (1) 
and let X ∼ Unif[0, Y ] be a random variable uniformly distributed on [0, Y ]. 
It is given that E(X) = 1. 
(a) (10 points) Determine the unknown parameter λ. [Hint. Compute E(X) with 
unknown parameter λ.] 
We see that X has exponential distribution with rate 1. 
2. (25 points) Certain device consists of two components, A and B. Whenever one of the 
components fails, the whole device is immideately replaced by a new one. Components 
A and B are the only components that can fail. 
Suppose that the lifetimes of components A and B (in days) are independent random 
variables both having exponential distributions with rate λ. Let N(t) be the renewal 
process counting the number of the replacements of the device on the time interval [0, t]. 
(a) (10 points) Express the interrenewal times in terms of the lifetimes of components 
A and B (hint: this is not a sum) and compute the distribution of the interrenewal 
times. 
(b) (10 points) Determine an asymptotic expression for the mean age of the device at 
time t in the long run. 
(c) (5 points) What is the long run probability that the device will fail within next 24 
hours? 
Solution. Denote by Xi and Yi, i ≥ 1, the random variables describing the lifetimes 
of the components A and B correspondingly. Then all Xi and Yi, i ≥ 1, are i.i.d. with 
exponential distribution with rate λ. 
(a) Denote by Zi the lifetime of the i-th device. Then Zi = min{Xi, Yi} is the in- 
terrenewal time of the process that counts the number of the replacement of the 
device. Since Xi and Yi are independent exponentially distributed random vari- 
ables, Zi ∼ Exp(2λ). 
(b) Let δt denote the age of the device at time t. Then, using the limit theorem for the 
average age, we have 
 
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