Assignment

Assignment 1

Question1:

a) Maxmin:

Choose A3 with payoff 10$

Maximum likelihood

Choose A1 with payoff 100$

Bays’ rule

A1 payoff 33$

A2 payoff 29$

A3 payoff 35$ (Choose A3)

b) If S1 occurs for certain

Choose A3 with payoff 10$

If S1 does not occur for certain

Expected payoffs

A1= (3/8)*10+= (5/8)*100=66.25

A2= (3/8)*20+= (5/8)*50=38.75

A3= (3/8)*10+= (5/8)*60=41.25

Choose A1

Expected payoff with perfect Info.=10*0.2+66.25*0.8=55

Expected value for perfect Info.=55-35=20

c) If S2 occurs for certain

Choose A2 with payoff 20$

If S3 occurs for certain

Choose A1 with payoff 100$

Expected payoff with perfect Info.=10*0.2+20*0.3+100*0.5=58

Expected value for perfect Info.=55-35=23

Question 2:

 = 0.8

P(X=0) = exp(-0.8)*0.8^0/fact(0) = 0.4493

P(X=1) = exp(-0.8)*0.8^1/fact(1) = 0.3595

P(X>=2) = 1-0.4493-0.3595 = 0.1912

Invest

Not invest

0 investor (=0)

0.4493

-15000

0

1 investor (=1)

0.3595

10000

0

2 or more investors (=2)

0.1912

20000

0

Part b)

Expect_value (invest) = -15000*0.4493+10000*0.3595+20000*0.1912 = 679.5

Expect_value(not invest) = 0

Part c)

Given:

P(S=0 | =0) = 0.5

P(S=0 | =1) = 0.4

P(S=0 | =2) = 0.2

P(S=1 | =0) = 0.4

P(S=1 | =1) = 0.5

P(S=1 | =2) = 0.4

P(S=2 | =0) = 0.1

P(S=2 | =1) = 0.1

P(S=2 | =2) = 0.4

P(S=0) = 0.5*0.4493+0.4*0.3595+0.2*0.1912 = 0.40669

P(S=1) = 0.4*0.4493+0.5*0.3595+0.4*0.1912 = 0.43595

P(S=2) = 0.1*0.4493+0.1*0.3595+0.4*0.1912 = 0.15736

Posterior: P(=0 | S=0) = 0.5*0.4493/0.40669 = 0.552386

P(=0 | S=0) = 0.552386 P(=1 | S=0) = 0.353586 P(=2 | S=0) = 0.094027

P(=0 | S=1) = 0.412249 P(=1 | S=1) = 0.412318 P(=2 | S=1) = 0.175433

P(=0 | S=2) = 0.285524 P(=1 | S=2) = 0.228457 P(=2 | S=2) = 0.486019

Part d)

If S = 0, do not invest  E(S=0) = 0

If S = 1, invest

E(S=1) = -15000*0.412249+10000*0.412318+20000*0.175433 = 1448.105

If S = 2, invest

E(S=2) = -15000*0.285524+10000*0.228457+20000*0.486019 = 7722.09

E(info) =...