a) Solve for PIX > 2) = f- (2) + fB) -1ft4) = 60

likewise ,= I - fll) - flo) = 0.60

b) EIX] = X' FIX) for all possible X .

= 010.23+510.23+210.23+310.3] -1410.17=1.9 times

c) 0×2 =VarEX]=E[ (X- EE 1=-2 (x -EADY . fix) forall possible x.

= f- 1.912.1021+(-0.95/0.2) + (0.15/0.2)+(1.1/40.3) + (2. 1140.1)var XI -- 1.69

Ox - Fatt 1.3 beach trips

d) passumingx ,

- beach trips in week one & Xz-

- beach trips in week 2 :

IX. + Xz S2 ] - PIXEONXEO] t PEX ,in =L ] + PEX ,

- I r't2--01=[0.2-0.2]+ [0.2+0.2]+[0.2-0.2] = 0.12

- PfXz3nXZ2] PIXZ 37e) Solve -for PEX 's IX > 21 = p-tx.zy-pfxzzy-f.IT -

- Zzz 0.67

Assume X is the number of flips until getting one tails , then n-

- X and :

PIX -- x] = Hz)"

for x-- I , 2 . . . .

and ENT Xl.

!k )"

E-EX] = I (E)+ 2ft)'

t 32413 t 4ft)"t

. ..

EE IX] = HEY +2 (E)3t 31E)4 t .

. .

Efx] - IE IX] = TEEN = I (E) + HEY + I (E)3+ IfE)4 t . .

.

EE# E. IET =# = I

.

.

.EIX] = EIN) = $2

where N is the value of moving a guest will makeAlternatively :Consider the geometric series : gcx)

-

- Efxn = Fx for - l exc 1

Which has a derivative of : g'

(x) = nxn- '=#p for - I ex - I

multiply both sides byX : g'

(X) -¥,

nxn =¥xp for - I exe 1

Since n-- X and E- EX] -€721142)x ,EEN7=9442 ) = YET. = $2 ,

where N is the value of money a guest will make

a) Epc . 31g, = I → c 3¥ - 3 - 1) = I → des - 4) = I

÷ c =Ies - 4

E-HI -- Ey . If = c Ty?".

=3e 37=34%34 - Yy-

- 2

=3ele'- t )

=3-

Lee: ] a 3.5595b) PLY > 37 = I - PLY = 27 - PLY=3] = I - c (E, t 37. )

= I - 9C = I -e93.4 a 0.4405

C) ELYH - ID= ugly-D . cyst. = CETI, = 5c.EE?Ty=9e3c

↳ = ELY2

] - ELY ] : .. ELY7 = 9pct 3C(e3- t ) = 344es- I )

so : var [YI = ELYY- ELYIP = 344e

'- t ) - 9eye

'- 1)

2

I 2.

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