# Notes sur des exercices de probabilités - 2° partie, Notes de Mathématiques et dstatistiques. Université des Sciences et Technologies de Lille (Lille I)

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Notes sur les sciences mathématiques concernant des exercices de probabilités - 2° partie. Les principaux thèmes abordés sont les suivants: the curve, The temperature of a particle located, the equation.
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May 2001 - Course 1 SOA/CAS

May 2001 20 Course 1

14. The stock prices of two companies at the end of any given year are modeled with

random variables X and Y that follow a distribution with joint density function

2 for 0 1, 1 ( , )

0 otherwise. x x x y x

f x y < < < < +

=  

What is the conditional variance of Y given that X = x ?

(A) 1 12

(B) 7 6

(C) x + 1 2

(D) x2 - 1 6

(E) x2 + x + 1 3

May 2001 21 Course 1

15. Let C be the curve defined by:

2

2

2 1 and

3 1

x t t

y t t

= + −

= − +

for .t−∞ < < ∞

What is the slope of the line tangent to C at (0, 5) ?

(A) -5

(B) -1

(C) 3 5

(D) 5 3

(E) 7

May 2001 22 Course 1

16. A certain state has an income tax rate of 0% on the first 10 of income, 2% on the next 10,

and 4% on the excess over 20 . Let T(x) represent the ratio of total tax to total income x .

Which graph below best represents the graph of T for 0 ≤ x ≤ 30 ?

(A) (B)

(C) (D)

(E)

May 2001 23 Course 1

17. An auto insurance company insures an automobile worth 15,000 for one year under a

policy with a 1,000 deductible. During the policy year there is a 0.04 chance of partial

damage to the car and a 0.02 chance of a total loss of the car. If there is partial damage

to the car, the amount X of damage (in thousands) follows a distribution with density

function

/ 20.5003 for 0 15 ( )

0 otherwise.

xe x f x

− < < =  

What is the expected claim payment?

(A) 320

(B) 328

(C) 352

(D) 380

(E) 540

May 2001 24 Course 1

18. The temperature of a particle located at the point ( , ) is ( , ) uvu v f u v e= . The location is

determined by two inputs x and y such that

∂ ∂

=

∂ ∂

=

∂ ∂

=

∂ ∂

=

u x

y

u y

x

v x

x

v y

y

2

2

2

2 .

Also, (u, v) = (4, 5) when (x, y) = (2, 1) .

Calculate the rate of change of temperature as y changes, when (x, y) = (2, 1) .

(A) 6e20

(B) 12e20

(C) 20e20

(D) 28e20

(E) 54e20

May 2001 25 Course 1

19. A company manufactures a brand of light bulb with a lifetime in months that is normally

distributed with mean 3 and variance 1 . A consumer buys a number of these bulbs with

the intention of replacing them successively as they burn out. The light bulbs have

What is the smallest number of bulbs to be purchased so that the succession of light bulbs

produces light for at least 40 months with probability at least 0.9772 ?

(A) 14

(B) 16

(C) 20

(D) 40

(E) 55

May 2001 26 Course 1

20. A device that continuously measures and records seismic activity is placed in a remote

region. The time, T, to failure of this device is exponentially distributed with mean

3 years. Since the device will not be monitored during its first two years of service, the

time to discovery of its failure is X = max(T, 2) .

Determine E[X] .

(A) 612 3

e−+

(B) 2 / 3 4 / 32 2 5e e− −− +

(C) 3

(D) 2 / 32 3e−+

(E) 5

May 2001 27 Course 1

21. The rate at which a disease spreads through a town can be modeled by the differential

equation

( )dQ Q N Q dt

= −

where Q(t) is the number of residents infected at time t and N is the total number of

residents.

Which of the following is a solution for Q(t)?

(A) aet where a is a constant

(B) 1 t

t aNe

ae − where a is a constant

(C) 1 t

t aNe

ae + where a is a constant

(D) 1

Nt

Nt

aNe ae

where a is a constant

(E) 1

Nt

Nt aNe

ae+ where a is a constant

May 2001 28 Course 1

22. The waiting time for the first claim from a good driver and the waiting time for the first

claim from a bad driver are independent and follow exponential distributions with means

6 years and 3 years, respectively.

What is the probability that the first claim from a good driver will be filed within

3 years and the first claim from a bad driver will be filed within 2 years?

(A) 1 18

1 2 3 1 2 7 6− − +− − −e e e/ / /d i

(B) 1 18

7 6e− /

(C) 1 2 3 1 2 7 6− − +− − −e e e/ / /

(D) 1 2 3 1 2 1 3− − +− − −e e e/ / /

(E) 1 1 3

1 6

1 18

2 3 1 2 7 6− − +− − −e e e/ / /

May 2001 29 Course 1

23. A hospital receives 1/5 of its flu vaccine shipments from Company X and the remainder

of its shipments from other companies. Each shipment contains a very large number of

vaccine vials.

For Company X’s shipments, 10% of the vials are ineffective. For every other company,

2% of the vials are ineffective. The hospital tests 30 randomly selected vials from a

shipment and finds that one vial is ineffective.

What is the probability that this shipment came from Company X ?

(A) 0.10

(B) 0.14

(C) 0.37

(D) 0.63

(E) 0.86

May 2001 30 Course 1

24. A device contains two components. The device fails if either component fails. The

joint density function of the lifetimes of the components, measured in hours, is ( ),f s t ,

where 0 1 and 0 1 .s t< < < <

What is the probability that the device fails during the first half hour of operation?

(A) ( ) 0.5 0.5

0 0

, f s t ds dt∫ ∫

(B) ( ) 1 0.5

0 0

, f s t ds dt∫ ∫

(C) ( ) 1 1

0.5 0.5

, f s t ds dt∫ ∫

(D) ( ) ( ) 0.5 1 1 0.5

0 0 0 0

, , f s t ds dt f s t ds dt+∫ ∫ ∫ ∫

(E) ( ) ( ) 0.5 1 1 0.5

0 0.5 0 0

, , f s t ds dt f s t ds dt+∫ ∫ ∫ ∫

May 2001 31 Course 1

25. The volume, V, and the surface area, S, of a spherical balloon with radius r are:

3

2

4 3 4 .

V r

S r

π

π

=

=

The volume of the balloon increases at a rate of 60 cm3/min when the balloon’s

diameter is 6 cm.

How fast is the surface area of the balloon increasing when the balloon’s diameter

is 6 cm?

(A) 20 cm2/min

(B) 40 cm2/min

(C) 80 cm2/min

(D) 113 cm2/min

(E) 120 cm2/min

May 2001 32 Course 1

26. A company offers earthquake insurance. Annual premiums are modeled by an

exponential random variable with mean 2 . Annual claims are modeled by an

exponential random variable with mean 1 . Premiums and claims are independent.

Let X denote the ratio of claims to premiums.

What is the density function of X ?

(A) 1 2 1x +

(B) 2 2 1 2( )x +

(C) xe

(D) 22 xe

(E) xxe

May 2001 33 Course 1

27. Claim amounts for wind damage to insured homes are independent random variables

with common density function

( ) 4 3 for 1

0 otherwise

x f x x

 >=  

where x is the amount of a claim in thousands.

Suppose 3 such claims will be made.

What is the expected value of the largest of the three claims?

(A) 2025

(B) 2700

(C) 3232

(D) 3375

(E) 4500

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