ISYE 6644 EVALUATION QUESTIONS AND ANSWERS
SURE A+
✔✔Common CLT trap: why is Bin(100,0.9) not the answer for Xbar? - ✔✔Bin(100,0.9)
describes the sum of successes, not the sample mean.
✔✔For U~Unif(0,1), what does 10U produce before applying ceiling? - ✔✔A continuous
random number between 0 and 10.
✔✔For U~Unif(0,1), what does ceil(10U) simulate? - ✔✔One fair 10-sided die roll with
values 1 through 10.
✔✔What does ceil(10U)+ceil(10V) simulate for iid U,V~Unif(0,1)? - ✔✔The sum of two
independent 10-sided dice rolls.
✔✔If U~Unif(0,1), what is the distribution of 1-U? - ✔✔Unif(0,1).
✔✔What is the key log rule for products inside ln? - ✔✔ln(ABC)=ln(A)+ln(B)+ln(C).
✔✔If U~Unif(0,1), what is -ln(U)? - ✔✔Exp(1).
✔✔If U~Unif(0,1), what is -(1/lambda)ln(U)? - ✔✔Exp(lambda).
✔✔If U~Unif(0,1), what is -3ln(U)? - ✔✔Exp(1/3).
✔✔What is the distribution of -3ln(U1(1-U2)U3) for iid uniforms? - ✔✔Erlang_3(1/3).
✔✔Why does -3ln(U1(1-U2)U3) become Erlang_3(1/3)? - ✔✔It is the sum of three iid
Exp(1/3) variables.
✔✔What is the inverse transform theorem? - ✔✔If U~Unif(0,1), then F^{-1}(U) has CDF
F.
, ✔✔If Phi is the standard normal CDF, what is Phi^{-1}(U)? - ✔✔Nor(0,1).
✔✔What is the probability integral transform? - ✔✔If X has continuous CDF F, then
F(X)~Unif(0,1).
✔✔If X is continuous with CDF F, what is E[F(X)]? - ✔✔1/2.
✔✔Using LOTUS, why is E[F(X)] = 1/2? - ✔✔E[F(X)]=∫F(x)f(x)dx = [F(x)^2/2] from -∞ to
∞ = 1/2.
✔✔What is a linear congruential generator? - ✔✔A deterministic recurrence like
X_{i+1}=(aX_i+c) mod m used to generate pseudorandom integers.
✔✔What does 'mod 8' mean in a linear congruential generator? - ✔✔Take the
remainder after division by 8.
✔✔For X_{i+1}=(5X_i+1) mod 8 with X0=0, what is the cycle? - ✔✔0,1,6,7,4,5,2,3,0.
✔✔For X_{i+1}=(5X_i+1) mod 8 with X0=0, what is X99? - ✔✔X99=X3=7 because the
cycle length is 8 and 99 mod 8 = 3.
✔✔When an LCG rescales Xi by dividing by m, what does Ui=Xi/m represent? - ✔✔A
pseudorandom number intended to look Uniform(0,1).
✔✔What does the Park-Miller style PRN algorithm with multiplier 16807 do? -
✔✔Generates a deterministic integer cycle and rescales it into pseudorandom numbers
Ui that appear uniform and independent.
✔✔Do pseudorandom integer sequences eventually cycle? - ✔✔Yes.
✔✔What is a PRN? - ✔✔A pseudorandom number.
✔✔In discrete-event simulation, what is the Future Events List (FEL)? - ✔✔The internal
calendar/list of scheduled future events.
✔✔True or False: Most discrete-event simulations have a Future Events List deep
down. - ✔✔True.
✔✔Can an arrival event insert, delete, or reorder events in the FEL? - ✔✔Yes; an event
can schedule new events, cancel future events, and affect ordering.
✔✔How does next-event time advance work in discrete-event simulation? - ✔✔The
simulation clock jumps from one scheduled event time to the next.
SURE A+
✔✔Common CLT trap: why is Bin(100,0.9) not the answer for Xbar? - ✔✔Bin(100,0.9)
describes the sum of successes, not the sample mean.
✔✔For U~Unif(0,1), what does 10U produce before applying ceiling? - ✔✔A continuous
random number between 0 and 10.
✔✔For U~Unif(0,1), what does ceil(10U) simulate? - ✔✔One fair 10-sided die roll with
values 1 through 10.
✔✔What does ceil(10U)+ceil(10V) simulate for iid U,V~Unif(0,1)? - ✔✔The sum of two
independent 10-sided dice rolls.
✔✔If U~Unif(0,1), what is the distribution of 1-U? - ✔✔Unif(0,1).
✔✔What is the key log rule for products inside ln? - ✔✔ln(ABC)=ln(A)+ln(B)+ln(C).
✔✔If U~Unif(0,1), what is -ln(U)? - ✔✔Exp(1).
✔✔If U~Unif(0,1), what is -(1/lambda)ln(U)? - ✔✔Exp(lambda).
✔✔If U~Unif(0,1), what is -3ln(U)? - ✔✔Exp(1/3).
✔✔What is the distribution of -3ln(U1(1-U2)U3) for iid uniforms? - ✔✔Erlang_3(1/3).
✔✔Why does -3ln(U1(1-U2)U3) become Erlang_3(1/3)? - ✔✔It is the sum of three iid
Exp(1/3) variables.
✔✔What is the inverse transform theorem? - ✔✔If U~Unif(0,1), then F^{-1}(U) has CDF
F.
, ✔✔If Phi is the standard normal CDF, what is Phi^{-1}(U)? - ✔✔Nor(0,1).
✔✔What is the probability integral transform? - ✔✔If X has continuous CDF F, then
F(X)~Unif(0,1).
✔✔If X is continuous with CDF F, what is E[F(X)]? - ✔✔1/2.
✔✔Using LOTUS, why is E[F(X)] = 1/2? - ✔✔E[F(X)]=∫F(x)f(x)dx = [F(x)^2/2] from -∞ to
∞ = 1/2.
✔✔What is a linear congruential generator? - ✔✔A deterministic recurrence like
X_{i+1}=(aX_i+c) mod m used to generate pseudorandom integers.
✔✔What does 'mod 8' mean in a linear congruential generator? - ✔✔Take the
remainder after division by 8.
✔✔For X_{i+1}=(5X_i+1) mod 8 with X0=0, what is the cycle? - ✔✔0,1,6,7,4,5,2,3,0.
✔✔For X_{i+1}=(5X_i+1) mod 8 with X0=0, what is X99? - ✔✔X99=X3=7 because the
cycle length is 8 and 99 mod 8 = 3.
✔✔When an LCG rescales Xi by dividing by m, what does Ui=Xi/m represent? - ✔✔A
pseudorandom number intended to look Uniform(0,1).
✔✔What does the Park-Miller style PRN algorithm with multiplier 16807 do? -
✔✔Generates a deterministic integer cycle and rescales it into pseudorandom numbers
Ui that appear uniform and independent.
✔✔Do pseudorandom integer sequences eventually cycle? - ✔✔Yes.
✔✔What is a PRN? - ✔✔A pseudorandom number.
✔✔In discrete-event simulation, what is the Future Events List (FEL)? - ✔✔The internal
calendar/list of scheduled future events.
✔✔True or False: Most discrete-event simulations have a Future Events List deep
down. - ✔✔True.
✔✔Can an arrival event insert, delete, or reorder events in the FEL? - ✔✔Yes; an event
can schedule new events, cancel future events, and affect ordering.
✔✔How does next-event time advance work in discrete-event simulation? - ✔✔The
simulation clock jumps from one scheduled event time to the next.
