Midterm v1 2 - this is pratice for economics PDF

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Econ 41 (Summer 2020, Session C) Department of Economics, UCLA Instructor: Kirill Ponomarev

Midterm Exam Note: use e = 2.71 for all calculations A. Let A, B and C be three events such that (i) A and C are independent, (ii) A and B are independent, (iii) B and C are independent, (iv) P (A ∩ B ∩ C) = 0. Suppose that P (A) = P (B) = P (C) = 0.3 (1) P (A ∪ B) is (a) 0.51

(b) 0.62

(c) 0.73

(d) 0.84

(c) 0.14

(d) 0.16

(2) Probability that only A happens (a) 0.10

(b) 0.12

(3) Probability that exactly one of the events happens (a) 0.30

(b) 0.32

(c) 0.36

(d) 0.38

(b) 0.37

(c) 0.41

(d) 0.44

(b) 0.45

(c) 0.52

(d) 0.67

(4) Probability that none of the events happen (a) 0.26 (5) P (A′ ∩ B ′ ∪ C) (a) 0.37

B. Draw three cards from a standard deck without replacement (6) The probability of observing two numbers and a picture is (a) 0.123

(b) 0.456

(c) 0.478

(d) 0.789

(7) The probability of observing more red cards than black cards (a)

1 2

(b)

2 5

(c)

1 3

(d)

1 4

(8) The probability of collecting a sum of 6 (no pictures allowed) is (a) 0.0012

(b) 0.0005

(c) 0.0004

1

(d) 0.0002

C. Every dog owner brings at least one dog to the Dog Beach near San Diego but some people bring more. 30% of the dog owners have German Shepherds, 40% have Retrievers among which 15% are Golden Retrievers and the rest are Labradors. Moreover, 10% have both a Shepherd and a Retriever, and within this group 5% are Golden Retrievers and 5% are Labradors. All other dog owners bring different breeds. (9) The probability that a dog owner has a Labrador is (a) 0.1

(b) 0.15

(c) 0.2

(d) 0.25

(10) The probability that a dog owner has some other breed is (a) 0.2

(b) 0.3

(c) 0.4

(d) 0.6

(11) The conditional probability that a dog owner has a Retriever, given that she has either a Retriever or a Shepherd (a)

1 4

(b)

1 3

(c)

1 2

(d)

2 3

(12) The conditional probability that a dog owner has a Labrador, given that she has a Retriever is (a)

5 7

(b)

5 8

(c)

4 7

(d)

4 9

(13) Given that a dog owner brought a Retriever and a Shepherd, the probability that she has a Golden Retriever is (a)

1 2

(b)

1 3

(c)

2 3

(d)

3 4

D. Consider a test for Coronavirus with the following properties. If a person has the virus, the test will be positive 99% of the time. If a person does not have the virus, the test will be negative 98% of the time. Suppose that in the population 1 out of 500 people has the virus. (14) Given that a randomly selected person has the virus, probability of the test being positive is (a) 1

(b) 0.99

(c) 0.98

(d) 0.90

(15) The probability that a randomly selected person has the virus and the test is positive is (a) 0.00198

(b) 0.00192

(c) 0.00194

(d) 0.099

(16) Given that a randomly selected person does not have the virus, the probability of the test being positive is 2

(a) 0.05

(b) 0.01

(c) 0.02

(d) 0

(17) The probability that a randomly selected person does not have the virus and the test is positive is (a) 0.00124

(b) 0.04302

(c) 0.05701

(d) 0.01996

(18) Given that the test is positive, the probability that the person has the virus is (a) 1

(b) 0.9

(c) 0.09

(d) 0.009

E. On the way to campus the Big Blue Bus passes three traffic lights. Each of them is either red or not independently of the others. The ҥrst light is red with probability 0.2, the second one — with probability 0.4 and the third one — with probability 0.6. The waiting time for the ҥrst light is 1 minute, for the second one — 2 minutes and for the third one — 3 minutes. (19) The probability that exactly one light is red is (a) 0.386

(b) 0.464

(c) 0.483

(d) 0.523

(20) The probability that at least one light is red is (a) 0.808

(b) 0.707

(c) 0.606

(d) 0.505

(21) Let X be a random variable that denotes the total waiting time. The support of X is (a) {1, 2, 3, 4, 5}

(b) {1, 2, 3, 4, 5, 6}

(c) {0,1,2,3,4,5,6} (d) {1, 3, 5, 7}

(22) If fX denotes the PMF of X, the value of fX (3) is (a) 0.32

(b) 0.42

(c) 0.52

(d) 0.62

(23) If FX denotes the CDF of X, the value of FX (4) is (a) 0.64

(b) 0.68

(c) 0.72

(d) 0.76

F. Let X be a random variable with PDF fX (x) = c · (|x| + 1) for x = −2, −1, 0, 1, 2 where c is some constant (24) c is equal to (a) 11

(b)

1 11

(c) 10

3

(d)

1 10

(25) The value of the CDF of X at 0 is (a)

5 11

(b)

5 10

(c)

6 11

(d)

6 10

(26) The mean of X is (a) -1

(b) 0

(c) 0.5

(d) 1

(b) 1.635

(c) 2.225

(d) 2.545

(c) 1.82

(d) 1.08

(27) The variance of X is (a) 1.480 (28) Expectation of

√

X 2 + 1 is

(a) 0.53 (29) Deҥne Y = (a)

(b) 0.78 √

X 2 + 1. Let FY (y) denote the CDF of Y. The value of FY (2) is

4 10

(b)

5 11

(c)

1 2

(d)

6 10

G. A trader wants to build a portfolio of two stocks with independent returns. The return of stock A is either -5 or +10 with probabilities 1/2. The return of stock B is either -10 or +20 with probabilities 1/2. (30) If α is the share of stock A in the portfolio, the total expected return is (a) 10 − 10α

(b) 10 − 5α

(c) 5 + 10α

(d) 5 − 5α

(c) 5 · (1 − α)2 +

(d) 10 · (1 − α)2 +

(31) The variance of such portfolio is (a) 37.5 · α2 + 150 · (1 − α)2

(b) 62.5 · α2 +

250 · (1 − α)2

10 · α2

5 · α2

(32) If the trader wants to minimize variance, he should choose α* equal to (a)

1 5

(b)

2 5

(c)

3 5

(d)

4 5

(33) If the trader wants to minimize variance but have the expected return of at least 5, he should choose α equal to (a)

1 5

(b)

2 5

(c)

3 5

(d)

4 5

(34) If the trader wants to minimize variance but have the expected return of at least 8, he should choose α equal to (a)

1 5

(b)

2 5

(c)

3 5

(d)

4 5

H. Alex liked 10 people on Tinder and he thinks that each of them will like him back (forming a match) with probability 0.4 independently of the others. Let X denote the total number of matches. 4

(35) The distribution of X is (a) Bin(10, 0.4)

(b) Bin(10, 0.6)

(c) Pois(10)

(d) Pois(0.4)

(c) 4

(d) 5

(b) 6.4

(c) 12.4

(d) 18.4

(b) 0.998

(c) 0.976

(d) 1

(36) The expected number of matches is (a) 2

(b) 3

(37) The expectation of X 2 is (a) 2.4 (38) The value of FX (8) is (a) 0.945

(39) Given that seven people did not like Alex back, the probability that he will have exactly two matches is (a) 0.288

(b) 0.256

(c) 0.212

(d) 0.192

I. There are only two cash desks in the grocery store. Let X1 and X2 denote the number of people passing through the ҥrst and the second cashiers correspondingly during a 10 minute interval. On average, the ҥrst cashier serves 4 people every 10 minutes and the second one — 2 people. They work independently of each other. (40) The distribution of X1 is 4 (a) Bin(10, 10 )

(b) Bin(4, 101 )

(c) Pois(2)

(d) Pois(4)

(c) 0.818

(d) 0.241

(c) 0.43

(d) 0.48

(41) The value of P (X1 > 2) is (a) 0.759

(b) 0.621

(42) The value of FX2 (1.5) is (a) 0.34

(b) 0.41

(43) The expected number of people that both cashiers serve in 10 minutes is (a) 3

(b) 5

(c) 6

(d) 7

(44) The probability that both cashiers together serve 5 people in 10 minutes is (a) 0.085

(b) 0.164

(c) 0.182

(d) 0.212

(45) The expected number of people that both cashiers serve in 15 minutes is (a) 7

(b) 8

(c) 9

5

(d) 10...