ISYE 6644 ASY OAN EXAM 1 SIMULATION
TESTED QUESTIONS AND COMPLETE
SOLUTIONS
◉ What is Var(aX+b)?
Answer: a^2 Var(X).
◉ If Var(X)=3, what is Var(-3X-7)?
Answer: (-3)^2 * 3 = 27.
◉ What is the correlation formula?
Answer: Corr(X,Y)=Cov(X,Y)/sqrt(Var(X)Var(Y)).
◉ If Cov(X,Y)=0, what is Corr(X,Y) when variances are positive?
Answer: 0.
◉ What is Var(aX+bY)?
Answer: a^2 Var(X)+b^2 Var(Y)+2ab Cov(X,Y).
◉ What is the special formula for Var(X-Y)?
Answer: Var(X)+Var(Y)-2Cov(X,Y).
,◉ What is Var(X-2Y) in terms of Var and Cov?
Answer: Var(X)+4Var(Y)-4Cov(X,Y).
◉ If Var(X)=Var(Y)=10 and Cov(X,Y)=6, what is Var(X-2Y)?
Answer: 10 + 4(10) - 4(6) = 26.
◉ Common mistake in Var(X-2Y): should you subtract Var(Y)
because of the minus sign?
Answer: No. The coefficient is squared for variance; the minus
sign enters through the covariance term.
◉ For X~Nor(mu, sigma^2), what is the standardization formula?
Answer: Z=(X-mu)/sigma.
◉ In Nor(2,4), is 4 the standard deviation or variance?
Answer: Variance; the standard deviation is sqrt(4)=2.
◉ If X~Nor(2,4), what is P(X>0) after standardizing?
Answer: P(Z>-1)=0.8413.
◉ What standard normal symmetry helps with P(Z>-1)?
Answer: P(Z>-1)=P(Z<1).
, ◉ For a Poisson process with rate lambda, what distribution gives
the number of arrivals in one time unit?
Answer: Poisson(lambda).
◉ For a Poisson process with rate lambda, what distribution gives
interarrival times?
Answer: Exponential(lambda).
◉ In a Poisson process, the time between the 1st and 2nd arrivals
has what distribution?
Answer: Exp(lambda), same as any consecutive interarrival time.
◉ What is the exponential CDF for X~Exp(lambda)?
Answer: P(X<=x)=1-e^{-lambda x}, x>=0.
◉ If X~Exp(lambda=1/2), what is P(X<1)?
Answer: 1-e^{-1/2}=1-1/sqrt(e).
◉ What distribution is the sum of k iid exponential(lambda)
variables?
Answer: Erlang_k(lambda).
◉ If X and Y are iid Exp(1), what is X+Y?
Answer: Erlang_2(1).
TESTED QUESTIONS AND COMPLETE
SOLUTIONS
◉ What is Var(aX+b)?
Answer: a^2 Var(X).
◉ If Var(X)=3, what is Var(-3X-7)?
Answer: (-3)^2 * 3 = 27.
◉ What is the correlation formula?
Answer: Corr(X,Y)=Cov(X,Y)/sqrt(Var(X)Var(Y)).
◉ If Cov(X,Y)=0, what is Corr(X,Y) when variances are positive?
Answer: 0.
◉ What is Var(aX+bY)?
Answer: a^2 Var(X)+b^2 Var(Y)+2ab Cov(X,Y).
◉ What is the special formula for Var(X-Y)?
Answer: Var(X)+Var(Y)-2Cov(X,Y).
,◉ What is Var(X-2Y) in terms of Var and Cov?
Answer: Var(X)+4Var(Y)-4Cov(X,Y).
◉ If Var(X)=Var(Y)=10 and Cov(X,Y)=6, what is Var(X-2Y)?
Answer: 10 + 4(10) - 4(6) = 26.
◉ Common mistake in Var(X-2Y): should you subtract Var(Y)
because of the minus sign?
Answer: No. The coefficient is squared for variance; the minus
sign enters through the covariance term.
◉ For X~Nor(mu, sigma^2), what is the standardization formula?
Answer: Z=(X-mu)/sigma.
◉ In Nor(2,4), is 4 the standard deviation or variance?
Answer: Variance; the standard deviation is sqrt(4)=2.
◉ If X~Nor(2,4), what is P(X>0) after standardizing?
Answer: P(Z>-1)=0.8413.
◉ What standard normal symmetry helps with P(Z>-1)?
Answer: P(Z>-1)=P(Z<1).
, ◉ For a Poisson process with rate lambda, what distribution gives
the number of arrivals in one time unit?
Answer: Poisson(lambda).
◉ For a Poisson process with rate lambda, what distribution gives
interarrival times?
Answer: Exponential(lambda).
◉ In a Poisson process, the time between the 1st and 2nd arrivals
has what distribution?
Answer: Exp(lambda), same as any consecutive interarrival time.
◉ What is the exponential CDF for X~Exp(lambda)?
Answer: P(X<=x)=1-e^{-lambda x}, x>=0.
◉ If X~Exp(lambda=1/2), what is P(X<1)?
Answer: 1-e^{-1/2}=1-1/sqrt(e).
◉ What distribution is the sum of k iid exponential(lambda)
variables?
Answer: Erlang_k(lambda).
◉ If X and Y are iid Exp(1), what is X+Y?
Answer: Erlang_2(1).
