ISYE 6644 CORE 2026 TEST ACTUAL QUESTIONS
AND ANSWERS SURE A+
✔✔If each trial succeeds with probability 0.6, what is P(first success on trial 3)? -
✔✔(0.4)^2(0.6)=0.096.
✔✔In a geometric problem, what does q usually mean? - ✔✔q=1-p, the failure
probability.
✔✔For a 10-sided die, what is the success probability of rolling a 5? - ✔✔p=1/10.
✔✔Expected tosses until rolling a 5 on a 10-sided die? - ✔✔E[X]=1/p=10.
✔✔What are the possible values of a Bernoulli random variable? - ✔✔0 and 1.
✔✔For X~Bern(p), what are P(X=1) and P(X=0)? - ✔✔P(X=1)=p and P(X=0)=1-p.
✔✔What does LOTUS stand for? - ✔✔Law of the Unconscious Statistician.
✔✔For discrete X, how do you compute E[g(X)] using LOTUS? - ✔✔Sum g(x)P(X=x)
over all possible x.
✔✔If X~Bern(0.5), what is E[e^X]? - ✔✔(1+e)/2.
✔✔For affine expectation, what is E[aX+b]? - ✔✔aE[X]+b.
✔✔If E[X]=-2, what is E[-3X-7]? - ✔✔-3(-2)-7 = -1.
✔✔In E[-3X-7], why is variance irrelevant? - ✔✔Expectation of a linear expression only
needs the mean; variance is a distractor.
, ✔✔What does 'shift happens' mean for expectation? - ✔✔Adding/subtracting a constant
stays in the expectation: E[X+c]=E[X]+c.
✔✔For variance, does shift happen? - ✔✔No. Var(X+c)=Var(X).
✔✔What is Var(aX+b)? - ✔✔a^2 Var(X).
✔✔If Var(X)=3, what is Var(-3X-7)? - ✔✔(-3)^2 * 3 = 27.
✔✔What is the correlation formula? - ✔✔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? - ✔✔0.
✔✔What is Var(aX+bY)? - ✔✔a^2 Var(X)+b^2 Var(Y)+2ab Cov(X,Y).
✔✔What is the special formula for Var(X-Y)? - ✔✔Var(X)+Var(Y)-2Cov(X,Y).
✔✔What is Var(X-2Y) in terms of Var and Cov? - ✔✔Var(X)+4Var(Y)-4Cov(X,Y).
✔✔If Var(X)=Var(Y)=10 and Cov(X,Y)=6, what is Var(X-2Y)? - ✔✔10 + 4(10) - 4(6) =
26.
✔✔Common mistake in Var(X-2Y): should you subtract Var(Y) because of the minus
sign? - ✔✔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? - ✔✔Z=(X-
mu)/sigma.
✔✔In Nor(2,4), is 4 the standard deviation or variance? - ✔✔Variance; the standard
deviation is sqrt(4)=2.
✔✔If X~Nor(2,4), what is P(X>0) after standardizing? - ✔✔P(Z>-1)=0.8413.
✔✔What standard normal symmetry helps with P(Z>-1)? - ✔✔P(Z>-1)=P(Z<1).
✔✔For a Poisson process with rate lambda, what distribution gives the number of
arrivals in one time unit? - ✔✔Poisson(lambda).
✔✔For a Poisson process with rate lambda, what distribution gives interarrival times? -
✔✔Exponential(lambda).
AND ANSWERS SURE A+
✔✔If each trial succeeds with probability 0.6, what is P(first success on trial 3)? -
✔✔(0.4)^2(0.6)=0.096.
✔✔In a geometric problem, what does q usually mean? - ✔✔q=1-p, the failure
probability.
✔✔For a 10-sided die, what is the success probability of rolling a 5? - ✔✔p=1/10.
✔✔Expected tosses until rolling a 5 on a 10-sided die? - ✔✔E[X]=1/p=10.
✔✔What are the possible values of a Bernoulli random variable? - ✔✔0 and 1.
✔✔For X~Bern(p), what are P(X=1) and P(X=0)? - ✔✔P(X=1)=p and P(X=0)=1-p.
✔✔What does LOTUS stand for? - ✔✔Law of the Unconscious Statistician.
✔✔For discrete X, how do you compute E[g(X)] using LOTUS? - ✔✔Sum g(x)P(X=x)
over all possible x.
✔✔If X~Bern(0.5), what is E[e^X]? - ✔✔(1+e)/2.
✔✔For affine expectation, what is E[aX+b]? - ✔✔aE[X]+b.
✔✔If E[X]=-2, what is E[-3X-7]? - ✔✔-3(-2)-7 = -1.
✔✔In E[-3X-7], why is variance irrelevant? - ✔✔Expectation of a linear expression only
needs the mean; variance is a distractor.
, ✔✔What does 'shift happens' mean for expectation? - ✔✔Adding/subtracting a constant
stays in the expectation: E[X+c]=E[X]+c.
✔✔For variance, does shift happen? - ✔✔No. Var(X+c)=Var(X).
✔✔What is Var(aX+b)? - ✔✔a^2 Var(X).
✔✔If Var(X)=3, what is Var(-3X-7)? - ✔✔(-3)^2 * 3 = 27.
✔✔What is the correlation formula? - ✔✔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? - ✔✔0.
✔✔What is Var(aX+bY)? - ✔✔a^2 Var(X)+b^2 Var(Y)+2ab Cov(X,Y).
✔✔What is the special formula for Var(X-Y)? - ✔✔Var(X)+Var(Y)-2Cov(X,Y).
✔✔What is Var(X-2Y) in terms of Var and Cov? - ✔✔Var(X)+4Var(Y)-4Cov(X,Y).
✔✔If Var(X)=Var(Y)=10 and Cov(X,Y)=6, what is Var(X-2Y)? - ✔✔10 + 4(10) - 4(6) =
26.
✔✔Common mistake in Var(X-2Y): should you subtract Var(Y) because of the minus
sign? - ✔✔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? - ✔✔Z=(X-
mu)/sigma.
✔✔In Nor(2,4), is 4 the standard deviation or variance? - ✔✔Variance; the standard
deviation is sqrt(4)=2.
✔✔If X~Nor(2,4), what is P(X>0) after standardizing? - ✔✔P(Z>-1)=0.8413.
✔✔What standard normal symmetry helps with P(Z>-1)? - ✔✔P(Z>-1)=P(Z<1).
✔✔For a Poisson process with rate lambda, what distribution gives the number of
arrivals in one time unit? - ✔✔Poisson(lambda).
✔✔For a Poisson process with rate lambda, what distribution gives interarrival times? -
✔✔Exponential(lambda).
