Fisica Para Poetas

UNIVERSIDAD DE LA SALLE Departamento de Ciencias B´sicas a ´ Area de Matem´ticas y Estad´ a 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 partsin 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 thenumber 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 isthe 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 f(x) 1/8 2/8 2/8 2/8 i. P (X ≤ 2) ii. P (X > −2) iii. P (−1 ≤ X ≤ 1) iv. P (X ≤ 1 or X = 2) 2 1/8 b)   2x + 1 f (x) = 25 0 i. ii. iii.iv. P (X = 4) P (X ≤ 1) P (2 ≤ X < 4) P (X > −10) if x = 0, 1, 2, 3, 4 elsewhere

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, $5million, and $1 million, respectively. Let the random variable X denote the yearly revenue of the product. Determine the probability mass function of X. 4. An assembly consists of two mechanical components.Suppose that the probabilities that the first and second components meet specifications are 0.95 and 0.98. Assume that the components are independent. Determine the probability mass function of thenumber of components in the assembly that meet specifications. 1

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 components in the assembly that meet specifications.CUMULATIVE DISTRIBUTION FUNCTION OF A DISCRETE RANDOM VARIABLE 6. Determine the cumulative distribution function for the random variable in Exercise 2a; also determine 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 determine 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 probability mass function and the requested probabilities.  0 if x < 1  F (x) = 0,5 if 1 ≤ x < 3   1 if x ≥ 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:  if x < 1 0   0,7 if 1 ≤ x < 4 F (x) = 0,9 if 4 ≤ x < 7    1 if x ≥ 7 Determine each of the following probabilities: a) P (X ≤ 4) b) P (X ≥...