quiz3 - ISEN 609 Quiz 3 Oral Quiz Oral Quiz • There are...

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Unformatted text preview: ISEN 609 Quiz 3 Oral Quiz Oral Quiz • There are 25 questions in total are 25 questions in total • At the end of the quiz, based on your answers score on 10 would be assigned answers a score on 10 would be assigned • We will randomly select the person who gets th fi the first question and keep going in a roundrobin fashion (will explain this in person). • In TRUE/FALSE questions an explanation is needed. Question 1 • If X is the sum of two i.i.d. random variables is the sum of two random variables X1 and X2, i.e. X = X1+X2, then X can also be written as written as X = X1+X1 = 2X1. True or False? 2X True or False? Question 2 • If X and Y are random variables with the and are random variables with the same mean and standard deviation, then the CDF must be the same for both True or CDF must be the same for both. True or False? Question 3 • In the recent repair-shop problem where X2 the recent repair problem where depends on X1 so that E[X2|X1] = 4X1 and Std Std.Dev.[X2|X1] = X1, is X1 independent of is of X2? In other words, is E[X1|X2]=E[X1]? Question 4 • A r.v. X has PDF f(x) = (1-|x|)/2 for -1 < x < 1 has PDF f(x) (1 for and f(x) = 0 otherwise. What is E[X]? Question 5 • • • • F(x) = 0 if if x < 0 F(x) = sin(x) if 0 <= x <= /2 F(x) = 1 if x > /2 Can the above F(x) be a CDF of a r.v.? Question 6 • If a continuous r.v. X has standard deviation continuous has standard deviation equal to the mean, then X is exponentially distributed True or False? distributed. True or False? Question 7 • All the examples of random quantities we the examples of random quantities we have seen in class are either absolutely discrete or absolutely continuous Give an discrete or absolutely continuous. Give an example of a random quantity that is partially discrete and partially continuous discrete and partially continuous. Question 8 • If X is a positive-valued continuous r.v., i.e. is positive continuous F(0) = 0, then E[ X ] [1 F ( x)]dx 0 • True or False? Question 9 • There are two paths to go from A to B. One are two paths to go from to One takes exp(1/3) time and the other takes Unif(0 time If you want to make it before Unif(0,6) time. If you want to make it before 3 time units from A to B which path would you choose? choose? Question 10 • Can you have two positive-valued continuous you have two positive continuous random variables X and Y such that the PDF of one is always greater than that of the of one is always greater than that of the other? Question 11 • Can you have two positive-valued continuous you have two positive continuous random variables X and Y such that the CDF of one is never less than that of the other? of one is never less than that of the other? Question 12 • A double-exponential r.v. has PDF for all x as double has PDF for all as f ( x) • What is F(0)? is F(0)? 2 e | x| Question 13 • For a CDF CDF x0 0 ax b 0 x 1 F ( x) cx 0.5 1 x 2 1 x2 • Compute a, b and c Question 14 • If 25 samples are drawn according to an 25 samples are drawn according to an exponential distribution with mean 1 and the 25 numbers are sorted then what would you 25 numbers are sorted, then what would you expect the 13th number to be in this sorted list? In other words what is the median of an list? In other words, what is the median of an exponential random variable? Question 15 • Let X ~ exp(), and for some constant a exp( and for some constant (such that a>0), Y = aX • True or false? Y ~ exp(/a) or false? exp( Question 16 • X1, X2, …, Xn are i.i.d. r.v. with mean and with mean standard deviation . • X = (X1+X2+…+Xn)/n (X • What are E[X] and Var[X]? Question 17 • For a positive-valued r.v. X, the coefficient of positive the coefficient of variation is defined as Var[ X ] COV ( X ) E[ X ] • If X is Erlang, then COV(X) <= 1 and if X is exponential COV(X)=1. True or False? Question 18 • True or false? Var[X] can be expressed as or false? Var[X] can be expressed as Var[ X ] E { X E[ X ]} 2 Question 19 • The mode of a continuous r.v. X is the value mode of continuous is the value of x for which the PDF f(x) attains the maximum value What is the mode of an maximum value. What is the mode of an Erlang(2,) r.v.? Question 20 • If X ~ exp(3), what is P{X>3}? exp(3) what is P{X>3}? Question 21 • If X ~ unif(a,b), what is P{X<(a+b)/2}? unif(a what is P{X<(a+b)/2}? Question 22 • If X ~ Erlang(k,), what is E[X2] ? Erlang(k what is E[X Question 23 • If X ~ Normal(0,1), and P{X>3} = a, then Normal(0 and P{X>3} then what is P{-3 < X < 3} in terms of a? Question 24 • If X and Y are iid Erlang(3,) r.v.’s what is and are iid Erlang(3 what is E[X+Y] ? Question 25 • If X ~ Unif(4,10), what is the standard Unif what is the standard deviation of X? ...
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