ISYE 6644 EXAM 2 QUESTIONS AND
SOLUTIONS
◉ 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.
SOLUTIONS
◉ 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.
