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• Parth Mandhana

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Quiz 21

3. Suppose that Luu’s current wealth ω is $400 and she faces the following lottery L. She can earn $500 with probability πA = 15 , she can earn zero (stay the same) with probability πB = 35 and she can lose $300 with probability πC = 15 . In other words, PMF of L is pL (500) = pL (−300) = 51 and pL (0) = 35 . Assume Luu’s √ utility function, defined over her wealth, is u(w) = 3 w. Is Luu risk averse? (in other words, Is u concave?) yes

Set A Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section . . . . . . . . . . . . . . . . . . Email ID . . . . . . . . . . . . . . . . . .

What is the expected value of the lottery? No copying, cheating, collaboration, computers, or cell phones are allowed. Answers should be exact unless an approximation is asked for. All parts will be weighted equally within each problem. Simplifying expressions: You don’t need to simplify compli1 2 1 2 cated expressions. For example, you can leave × + × 4 3 3 5 20! 20 exactly as it is. Likewise for expressions like or 18!2! 18

E(L) =

40

Which utility level would Luu get if she received the expected value of the lottery for sure? √ u(ω + E(L)) = 3 440 What is Luu’s expected utility from taking the lottery? Eu(ω + L) =

3 ∗ 20 = 60

4. Suppose random variable X has a CDF given below. Select a valid CDF from the following and circle it :

Good luck. Ques

1

2

3

4

5

Total

Score

/4

/4

/4

/4

/4

/20

b

1. Suppose X and Y are two random variables. X can take values 1 and 3. Y can take values 0 and 1. The following is the joint probability distribution of X and Y . X↓Y → 1 3

0 0.1 0.6

1 0.2 0.1

Evaluate the following quantities pY (1) =

0.3

pX|Y (3|0) =

6 7

E[X|Y = 0] =

19 7

ρX,Y =

11 − 21

2. Let X be a uniformly distributed random variable on (0, 3) i.e. its density function is ( 1 if 0 < x < 3 fX (x) = 3 0 otherwise

Now, evaluate the following quantities Pr(X ≤ 0.3) =

0.15

Pr(|X − 0.5| > 0.1) =

Pr(X > 0.6) =

0.7

0.9

2

Define random variable Y = X . Set of values Y takes is an interval of the form (a, b). Find a, b. Range(Y ) = (a, b) =

(0, 9)

For y ∈ (a, b), find fY (y) =

1 √ 6 y

E(Y ) =

3

V(Y ) =

7.2

1 Contact:

[email protected]

5. (a) Let X and Y be independent Bernoulli random variables with Pr(X = 1) = 12 , Pr(Y = 1) = 23 The distribution of X +Y −XY is Bernoulli(q), where q=

5 6

(b) Suppose X and Y are i.i.d Bernoulli( 12 ) random variables. What is the MGF of 3(X + Y )?  3t 2 e +1 M3(X+Y ) (t) = 2

Quiz 22

3. Suppose that Luu’s current wealth ω is $400 and she faces the following lottery L. She can earn $500 with probability πA = 15 , she can earn zero (stay the same) with probability πB = 35 and she can lose $300 with probability πC = 15 . In other words, PMF of L is pL (500) = pL (−300) = 51 and pL (0) = 35 . Assume Luu’s √ utility function, defined over her wealth, is u(w) = 4 w. Is Luu risk averse? (in other words, Is u concave?) yes

Set B Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section . . . . . . . . . . . . . . . . . . Email ID . . . . . . . . . . . . . . . . . .

What is the expected value of the lottery? No copying, cheating, collaboration, computers, or cell phones are allowed. Answers should be exact unless an approximation is asked for. All parts will be weighted equally within each problem. Simplifying expressions: You don’t need to simplify compli1 2 1 2 cated expressions. For example, you can leave × + × 4 3 3 5 20! 20 exactly as it is. Likewise for expressions like or 18!2! 18

E(L) =

40

Which utility level would Luu get if she received the expected value of the lottery for sure? √ u(ω + E(L)) = 4 440 What is Luu’s expected utility from taking the lottery? Eu(ω + L) =

4 ∗ 20 = 80

4. Suppose random variable X has a CDF given below. Select a valid CDF from the following and circle it :

Good luck. Ques

1

2

3

4

5

Total

Score

/4

/4

/4

/4

/4

/20

b

1. Suppose X and Y are two random variables. X can take values 1 and 4. Y can take values 0 and 1. The following is the joint probability distribution of X and Y . X↓Y → 1 4

0 0.1 0.6

1 0.2 0.1

Evaluate the following quantities pY (1) =

0.3

pX|Y (4|1) =

1 3

E[X|Y = 1] =

2

ρX,Y =

11 − 21

2. Let X be a uniformly distributed random variable on (0, 4) i.e. its density function is ( 1 if 0 < x < 4 fX (x) = 4 0 otherwise

Now, evaluate the following quantities Pr(X ≤ 0.4) =

0.2

Pr(|X − 0.5| > 0.2) =

Pr(X > 0.8) =

0.6

0.8

2

Define random variable Y = X . Set of values Y takes is an interval of the form (a, b). Find a, b. Range(Y ) = (a, b) =

(0, 16)

For y ∈ (a, b), find fY (y) =

1 √ 8 y

E(Y ) =

16 3

V(Y ) = 2 Contact:

256 5

−

[email protected]


16 2 3

5. (a) Let X and Y be independent Bernoulli random variables with Pr(X = 1) = 12 , Pr(Y = 1) = 34 The distribution of X +Y −XY is Bernoulli(q), where q=

=

1024 45

7 8

(b) Suppose X and Y are i.i.d Bernoulli( 12 ) random variables. What is the MGF of 4(X + Y )?  4t 2 e +1 M4(X+Y ) (t) = 2

Quiz 23

3. Suppose that Luu’s current wealth ω is $400 and she faces the following lottery L. She can earn $500 with probability πA = 15 , she can earn zero (stay the same) with probability πB = 35 and she can lose $300 with probability πC = 15 . In other words, PMF of L is pL (500) = pL (−300) = 51 and pL (0) = 35 . Assume Luu’s √ utility function, defined over her wealth, is u(w) = 5 w. Is Luu risk averse? (in other words, Is u concave?) yes

Set C Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section . . . . . . . . . . . . . . . . . . Email ID . . . . . . . . . . . . . . . . . .

What is the expected value of the lottery? No copying, cheating, collaboration, computers, or cell phones are allowed. Answers should be exact unless an approximation is asked for. All parts will be weighted equally within each problem. Simplifying expressions: You don’t need to simplify compli1 2 1 2 cated expressions. For example, you can leave × + × 4 3 3 5 20! 20 exactly as it is. Likewise for expressions like or 18!2! 18

E(L) =

40

Which utility level would Luu get if she received the expected value of the lottery for sure? √ u(ω + E(L)) = 5 440 What is Luu’s expected utility from taking the lottery? Eu(ω + L) =

5 ∗ 20 = 100

4. Suppose random variable X has a CDF given below. Select a valid CDF from the following and circle it :

Good luck. Ques

1

2

3

4

5

Total

Score

/4

/4

/4

/4

/4

/20

b

1. Suppose X and Y are two random variables. X can take values 1 and 5. Y can take values 0 and 1. The following is the joint probability distribution of X and Y . X↓Y → 1 5

0 0.1 0.6

1 0.2 0.1

Evaluate the following quantities pY (0) =

0.7

pX|Y (5|0) =

6 7

E[X|Y = 0] =

31 7

ρX,Y =

11 − 21

2. Let X be a uniformly distributed random variable on (0, 5) i.e. its density function is ( 1 if 0 < x < 5 fX (x) = 5 0 otherwise

Now, evaluate the following quantities Pr(X ≤ 0.5) =

Pr(X > 1) =

0.25

Pr(|X − 0.5| > 0.3) =

0.5

0.7

2

Define random variable Y = X . Set of values Y takes is an interval of the form (a, b). Find a, b. Range(Y ) = (a, b) =

(0, 25)

For y ∈ (a, b), find fY (y) =

1√ 10 y

E(Y ) =

25 3

V(Y ) = 3 Contact:

125 −


25 2 3

[email protected]

=

500 9

5. (a) Let X and Y be independent Bernoulli random variables with Pr(X = 1) = 12 , Pr(Y = 1) = 45 The distribution of X +Y −XY is Bernoulli(q), where q=

= 55.56

9 10

= 0.9

(b) Suppose X and Y are i.i.d Bernoulli( 12 ) random variables. What is the MGF of 5(X + Y )?  5t 2 e +1 M5(X+Y ) (t) = 2