Taller variables aleatorias

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UNIVERSIDAD DE LA SALLE

Departamento de Ciencias B´

asicas

´

Area de Matem´

aticas y Estad´ıstica

Estad´ıstica 1

Variables aleatorias y distribuciones de probabilidad

DISCRETE RANDOM VARIABLES

1. Determine the range (possible values) of the random variable:

a) A batch of 500 machined parts contains 10 that do not conform to customer requirements. The random variable is the number of parts in a sample of 5 parts that do not conform to customer requirements.

b) A batch of 500 machined parts contains 10 that do not conform to customer requirements. Parts are selected successively, without replacement, until a nonconforming part is obtained. The random variable is the number of parts selected

c) Determine the range (possible values) of the random variable: The random variable is the number of computer clock cycles required to complete a selected arithmetic calculation.

d) Determine the range (possible values) of the random variable:Wood paneling can be ordered in thicknesses of 1/8, 1/4, or 3/8 inch. The random variable is the total thickness of paneling in two orders

e) An order for an automobile can select the base model or add any number of 15 options. The random variable is the number of options selected in an order.

PROBABILITY DISTRIBUTION AND PROBABILITY MASS FUNCTION OF A DISCRETE RANDOM VARIABLE

2. Verify that the following functions are probability mass functions, and determine the requested probabilities:

a)

x −2 1 0 1 2

f(x) 1/8 2/8 2/8 2/8 1/8

i. P(X ≤2)

ii. P(X >−2) iii. P(−1≤X ≤1)

iv. P(X ≤1or X= 2)

b)

f(x) =

  

2x+ 1

25 ifx= 0,1,2,3,4

0 elsewhere

i. P(X = 4) ii. P(X ≤1) iii. P(2≤X <4) iv. P(X >−10)

3. Marketing estimates that a new instrument for the analysis of soil samples will be very successful, moderately successful, or unsuccessful, with probabilities 0.3, 0.6, and 0.1, respectively. The yearly revenue associated with a very successful, moderately successful, or unsuccessful product is $10 million, $5 million, and $1 million, respectively. Let the random variable X denote the yearly revenue of the product. Determine the probability mass function ofX.

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5. An assembly consists of three mechanical components.Suppose that the probabilities that the first, second, and third components meet specifications are 0.95, 0.98, and 0.99. Assume that the components are independent. Determine the probability mass function of the number of compo-nents in the assembly that meet specifications.

CUMULATIVE DISTRIBUTION FUNCTION OF A DISCRETE RANDOM VA-RIABLE

6. Determine the cumulative distribution function for the random variable in Exercise 2a; also de-termine the following probabilities:

a) P(X ≤1,25)

b) P(X ≤2,2)

c) P(−1,1< X ≤1)

d) P(X >0)

7. Determine the cumulative distribution function for the random variable in Exercise 2b; also de-termine the following probabilities:

a) P(X <1,5)

b) P(X ≤3)

c) P(1≤X <2)

d) P(X >2)

8. Determine the cumulative distribution function for the random variable in Exercise 3.

9. Verify that the following functions are cumulative distribution functions, and determine the pro-bability mass function and the requested probabilities.

F(x) =

    

0 ifx <1 0,5 if 1≤x <3

1 ifx≥3

a) P(X ≤3)

b) P(X ≤2)

c) P(1≤X≤2)

d) P(X >2)

10. Errors in an experimental transmission channel are found when the transmission is checked by a certifier that detects missing pulses. The number of errors found in an eight-bit byte is a random variable with the following distribution:

F(x) =

        

0 ifx <1

0,7 if 1≤x <4 0,9 if 4≤x <7

1 ifx≥7

Determine each of the following probabilities:

a) P(X ≤4)

b) P(X ≥7)

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d) P(X ≥4)

MEAN AND VARIANCE OF A DISCRETE RANDOM VARIABLE

11. If the range of X is the set 0,1,2,3,4 andP(X =x) = 0,2 determine the mean and variance of the random variable.

12. Determine the mean and variance of the random variable in Exercise 2a.

13. Determine the mean and variance of the random variable in Exercise 2b.

14. Determine the mean and variance of the random variable in Exercise 3.

15. The range of the random variable X is 0,1,2,3, xwhere x is unknown. If each value is equally likely and the mean ofX is 6, determinex.

PROBABILITY DISTRIBUTION AND PROBABILITY DENSITY FUNCTION OF A CONTINUOUS RANDOM VARIABLE

16. Suppose that

f(x) =

{

e−x _if_{x >}₀

0 elsewhere

Determine the following probabilities:

a) P(1< X)

b) P(1< X <2,5)

c) P(X = 3)

d) P(X <4)

e) P(3≤X)

17. Suppose that

f(x) =

{

x/8 if 3< x <5

0 elsewhere

a) P(X <4)

b) P(X >3,5)

c) P(4< X <5)

d) P(X ≤4,5)

e) P(X <3,5 o X >4,5)

18. Suppose that

f(x) =

{

e−(x−4) _{if 4}_{< x}

0 elsewhere

a) P(1< X)

b) P(2≤X <5)

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d) P(8< X <12)

e) Determinexsuch thatP(X < x) = 0,90

19. The probability density function of the time to failure of an electronic component in a copier (in hours) is

f(x) =

{

e−x/1000

1000 ifx >0

0 elsewhere

Determine the probability that

a) A component lasts more than 3000 hours before failure.

b) A component fails in the interval from 1000 to 2000 hours.

c) A component fails before 1000 hours.

d) Determine the number of hours at which 10 % of all components have failed.

CUMULATIVE DISTRIBUTION FUNCTION OF A CONTINUOUS RANDOM VARIABLE

20. Suppose the cumulative distribution function of the random variableX is

F(x) =

    

0 ifx <−2

0,25x+ 0,5 if−2≤x <2

1 ifx≥2

Determine the following:

a) P(X <1,8)

b) P(X >−1,5)

c) P(−2> X)

d) P(−1< X <1)

21. Determine the cumulative distribution function for the distribution in Exercise 16.

24. Determine the probability density function for each of the following cumulative distribution fun-ctions.

a) F(x) = 1−e−2x_if_{x >}₀

b)

F(x) =

        

0 ifx <0 0,2x if 0≤x <4

0,04x+ 0,64 if 4≤x <9

1 ifx≥9

MEAN AND VARIANCE OF A CONTINUOUS RANDOM VARIABLE

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a)

f(x) =

{

0,25 if 0< x <4

0 elsewhere

b)

f(x) =

{

0,125x ifx >0

0 elsewhere

c)

f(x) =

{

1,5x2 _if₋₁_{< x <}₁

0 elsewhere

26. If a dealer’s profit, in units of $5000, on a new automobile can bo looked upon as a random variableX having the density function

f(x) =

{

2(1−x) if 0< x <1

0 elsewhere

find the average profit per automobile.

27. What is the dealer’s average profit per automobile if the profit on each automobile is give by

g(X) =X2_{, where}_X _{is a random variable having the density function of Exercise 26.}

Nota:Ejercicios tomados de Applied Scientists and Probability for Engineers, Third Edition. Montgomery D., Runger G. John Wiley & Sons 2003 y Probability & Statistics for Engineers & Scientists, Eighth Edition. Walpole R., Myers R., Myers S., Ye K. Pearson Education International 2007.