ISYE 6644 EXAM 2 COMPLETE TEST PAPER
WITH CORRECT ANSWERS
◉ 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.
,◉ For discrete X, how do you compute E[g(X)] using LOTUS?
Answer: Sum g(x)P(X=x) over all possible x.
◉ If X~Bern(0.5), what is E[e^X]?
Answer: (1+e)/2.
◉ For affine expectation, what is E[aX+b]?
Answer: aE[X]+b.
◉ If E[X]=-2, what is E[-3X-7]?
Answer: -3(-2)-7 = -1.
◉ In E[-3X-7], why is variance irrelevant?
Answer: Expectation of a linear expression only needs the mean;
variance is a distractor.
◉ What does 'shift happens' mean for expectation?
Answer: Adding/subtracting a constant stays in the expectation:
E[X+c]=E[X]+c.
◉ For variance, does shift happen?
Answer: No. Var(X+c)=Var(X).
, ◉ 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).
WITH CORRECT ANSWERS
◉ 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.
,◉ For discrete X, how do you compute E[g(X)] using LOTUS?
Answer: Sum g(x)P(X=x) over all possible x.
◉ If X~Bern(0.5), what is E[e^X]?
Answer: (1+e)/2.
◉ For affine expectation, what is E[aX+b]?
Answer: aE[X]+b.
◉ If E[X]=-2, what is E[-3X-7]?
Answer: -3(-2)-7 = -1.
◉ In E[-3X-7], why is variance irrelevant?
Answer: Expectation of a linear expression only needs the mean;
variance is a distractor.
◉ What does 'shift happens' mean for expectation?
Answer: Adding/subtracting a constant stays in the expectation:
E[X+c]=E[X]+c.
◉ For variance, does shift happen?
Answer: No. Var(X+c)=Var(X).
, ◉ 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).
