Description
Mini Project #
Name
Names of group members (if applicable)
Contribution of each group member
Section 1. Answers to the specific questions asked
Section 2: R code. Your code must be annotated. No points may be given if a brief
look at the code does not tell us what it is doing.
1. (10 points) Consider Exercise 4.11 from the textbook. In this exercise, let XA be the
lifetime of block A, XB be the lifetime of block B, and T be the lifetime of the satellite.
The lifetimes are in years. It is given that XA and XB follow independent exponential
distributions with mean 10 years. One can follow the solution of Exercise 4.6 to show
that the probability density function of T is
fT (t) = (
0.2 exp(−0.1t) − 0.2 exp(−0.2t), 0 ≤ t < ∞,
0, otherwise,
and E(T) = 15 years.
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(a) Use the above density function to analytically compute the probability that the
lifetime of the satellite exceeds 15 years.
(b) Use the following steps to take a Monte Carlo approach to compute E(T) and
P(T > 15).
i. Simulate one draw of the block lifetimes XA and XB. Use these draws to
simulate one draw of the satellite lifetime T.
ii. Repeat the previous step 10,000 times. This will give you 10,000 draws
from the distribution of T. Try to avoid ‘for’ loop. Use ‘replicate’ function
instead. Save these draws for reuse in later steps. [Bonus: 1 bonus point
for not taking more than 1 line of code for steps (i) and (ii).]
iii. Make a histogram of the draws of T using ‘hist’ function. Superimpose the
density function given above. Try using ‘curve’ function for drawing the
density. Note what you see.
iv. Use the saved draws to estimate E(T). Compare your answer with the exact
answer given above.
v. Use the saved draws to estimate the probability that the satellite lasts more
than 15 years. Compare with the exact answer computed in part (a).
vi. Repeat the above process of obtaining an estimate of E(T) and an estimate
of the probability four more times. Note what you see.
(c) Repeat part (vi) five times using 1,000 and 100,000 Monte Carlo replications
instead of 10,000. Make a table of results. Comment on what you see and
provide an explanation
2. (10 points) Use a Monte Carlo approach estimate the value of π based on 10, 000
replications. [Ignorable hint: First, get a relation between π and the probability
that a randomly selected point in a unit square with coordinates — (0, 0), (0, 1),
(1, 0), and (1, 1) — falls in a circle with center (0.5, 0.5) inscribed in the square.
Then, estimate this probability, and go from there.]
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