ISYE 6644 UPDATED ACTUAL EXAM
QUESTIONS AND CORRECT ANSWERS FULL
SOLUTION GRADED A+
◉ True or False: Simulation can be used to analyze supply-chain
models too complicated to solve analytically. Answer: True.
◉ For bisection, what equation form do you need first? Answer:
Move everything to one side and define g(x)=0.
◉ For e^x = x^2, what g(x) should you use for bisection? Answer:
g(x)=e^x - x^2.
◉ What does a sign change between g(a) and g(b) tell you in
bisection? Answer: A root lies somewhere in [a,b], assuming g is
continuous.
◉ In bisection, after checking the midpoint, which interval do you
keep? Answer: Keep the subinterval where g(x) changes sign.
◉ What is Newton's method update formula? Answer: x_{n+1}=x_n -
g(x_n)/g'(x_n).
,◉ For conditional probability, what is P(A | B)? Answer: P(A |
B)=P(A ∩ B)/P(B).
◉ For continuous conditional probability, how do you compute P(A ∩
B)? Answer: Integrate the PDF over the interval where both events are
true.
◉ In the practice problem P(X<1 | X>1/2), what is the numerator
event? Answer: 1/2 < X < 1.
◉ In the practice problem P(X<1 | X>1/2) with support 0<X<2, what
is the denominator event? Answer: 1/2 < X < 2.
◉ Common mistake: Can you cancel x/2 across numerator and
denominator integrals? Answer: No. Compute each integral
separately; do not cancel variable terms inside integrals.
◉ If f(x)=x/2, what is the antiderivative? Answer: ∫(x/2)dx = x^2/4.
◉ For the practice conditional probability with f(x)=x/2, what is
P(X<1 | X>1/2)? Answer: 0.2.
◉ What distribution models the trial number of the first success?
Answer: Geometric distribution.
, ◉ For Geom(p) counting the successful trial itself, what is P(X=k)?
Answer: (1-p)^{k-1}p.
◉ For Geom(p) counting the successful trial itself, what is E[X]?
Answer: 1/p.
◉ If each trial succeeds with probability 0.6, what is P(first success
on trial 3)? Answer: (0.4)^2(0.6)=0.096.
◉ In a geometric problem, what does q usually mean? Answer: q=1-p,
the failure probability.
◉ For a 10-sided die, what is the success probability of rolling a 5?
Answer: p=1/10.
◉ Expected tosses until rolling a 5 on a 10-sided die? Answer:
E[X]=1/p=10.
◉ What are the possible values of a Bernoulli random variable?
Answer: 0 and 1.
◉ For X~Bern(p), what are P(X=1) and P(X=0)? Answer: P(X=1)=p
and P(X=0)=1-p.
◉ What does LOTUS stand for? Answer: Law of the Unconscious
Statistician.
QUESTIONS AND CORRECT ANSWERS FULL
SOLUTION GRADED A+
◉ True or False: Simulation can be used to analyze supply-chain
models too complicated to solve analytically. Answer: True.
◉ For bisection, what equation form do you need first? Answer:
Move everything to one side and define g(x)=0.
◉ For e^x = x^2, what g(x) should you use for bisection? Answer:
g(x)=e^x - x^2.
◉ What does a sign change between g(a) and g(b) tell you in
bisection? Answer: A root lies somewhere in [a,b], assuming g is
continuous.
◉ In bisection, after checking the midpoint, which interval do you
keep? Answer: Keep the subinterval where g(x) changes sign.
◉ What is Newton's method update formula? Answer: x_{n+1}=x_n -
g(x_n)/g'(x_n).
,◉ For conditional probability, what is P(A | B)? Answer: P(A |
B)=P(A ∩ B)/P(B).
◉ For continuous conditional probability, how do you compute P(A ∩
B)? Answer: Integrate the PDF over the interval where both events are
true.
◉ In the practice problem P(X<1 | X>1/2), what is the numerator
event? Answer: 1/2 < X < 1.
◉ In the practice problem P(X<1 | X>1/2) with support 0<X<2, what
is the denominator event? Answer: 1/2 < X < 2.
◉ Common mistake: Can you cancel x/2 across numerator and
denominator integrals? Answer: No. Compute each integral
separately; do not cancel variable terms inside integrals.
◉ If f(x)=x/2, what is the antiderivative? Answer: ∫(x/2)dx = x^2/4.
◉ For the practice conditional probability with f(x)=x/2, what is
P(X<1 | X>1/2)? Answer: 0.2.
◉ What distribution models the trial number of the first success?
Answer: Geometric distribution.
, ◉ For Geom(p) counting the successful trial itself, what is P(X=k)?
Answer: (1-p)^{k-1}p.
◉ For Geom(p) counting the successful trial itself, what is E[X]?
Answer: 1/p.
◉ If each trial succeeds with probability 0.6, what is P(first success
on trial 3)? Answer: (0.4)^2(0.6)=0.096.
◉ In a geometric problem, what does q usually mean? Answer: q=1-p,
the failure probability.
◉ For a 10-sided die, what is the success probability of rolling a 5?
Answer: p=1/10.
◉ Expected tosses until rolling a 5 on a 10-sided die? Answer:
E[X]=1/p=10.
◉ What are the possible values of a Bernoulli random variable?
Answer: 0 and 1.
◉ For X~Bern(p), what are P(X=1) and P(X=0)? Answer: P(X=1)=p
and P(X=0)=1-p.
◉ What does LOTUS stand for? Answer: Law of the Unconscious
Statistician.
