Page 95 - Applied statistics and probability for engineers

Section 3-3/Cumulative Distribution Functions 73

F(x)
F(x)
1.000
1.0 0.997
0.886
0.7
0.2

–2 0 2 x 0 1 2 x
FIGURE 3-3 Cumulative distribution function FIGURE 3-4 Cumulative distribution function
for Example 3-7. for Example 3-8.

Exercises FOR SECTION 3-3

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion
3-38. Determine the cumulative distribution function of missing pulses. The number of errors found in an eight-bit byte
the random variable in Exercise 3-16. is a random variable with the following distribution:
3-39. Determine the cumulative distribution function for ⎧0 x 1<
⎪
the random variable in Exercise 3-17; also determine the fol- ⎪0 7. 1 ≤ x < 4
lowing probabilities: F x ( ) = ⎨ ⎪ 0 9 4 ≤ x < 7
.
(a) P X( ≤1 25. ) (b) P X( ≤ 2 2. ) ⎪ 1 7 ≤ x
⎩
(c) P(−1 1. < X ≤1 ) (d) P X( > 0 )
Determine each of the following probabilities:
(
(
(
3-40. Determine the cumulative distribution function for (a) P X ≤ ) 4 (b) P X > ) 7 (c) P X ≤ ) 5
the random variable in Exercise 3-18; also determine the fol- ( (
lowing probabilities: (d) P X > ) 4 (e) P X ≤ ) 2
(a) P X( < 1 5. ) (b) P X( ≤ 3 ) ⎧0 x < −10
(c) P X( > 2 ) (d) P(1< X ≤ ) 2 ⎪ −10 Ð x< 30
⎪0 25.
F x) = ⎨
3-41. Determine the cumulative distribution function for the 3-51. (
.
random variable in Exercise 3-19. ⎪ 0 75 30 ≤ x < 50
3-42. Determine the cumulative distribution function for the ⎩ 50 ≤ x
⎪ 1
(
(
random variable in Exercise 3-20. (a) P X ≤ ) (b) P X ≤ )
40
50
3-43. Determine the cumulative distribution function for the
(
(
random variable in Exercise 3-21. (c) P 40 ≤ X ≤ 60) (d) P X < ) 0
(
<
3-44. Determine the cumulative distribution function for the (e) P 0 ≤ X 10) (f) P − ( 10 < X < )
10
random variable in Exercise 3-22.
3-45. Determine the cumulative distribution function for 3-52. The thickness of wood paneling (in inches) that a
the random variable in Exercise 3-25. customer orders is a random variable with the following cumu-
3-46. Determine the cumulative distribution function for lative distribution function:
the random variable in Exercise 3-26. ⎧0 x < 1 8/
⎪
/
3-47. Determine the cumulative distribution function for ⎪0 2. 1 8 ≤ x < 1 4/
the random variable in Exercise 3-27. F x ( ) = ⎨ 0 9 1 4 ≤ x < 3 8/
/
⎪
.
3-48. Determine the cumulative distribution function for
/
⎪ 1 3 8 ≤ x
⎩
the variable in Exercise 3-28. Determine the following probabilities:
Verify that the following functions are cumulative distribution ( ) ( ) ( )
/
/
functions, and determine the probability mass function and the (a) P X ≤1 18/ (b) P X ≤1 4 (c) P X ≤ 5 16
(
(
/
requested probabilities. (d) P X >1 4/ ) (e) P X ≤1 2 )
⎧0 x < 1 3-53. Determine the cumulative distribution function for the
⎪
(
3-49. F x) = ⎨ 0 5. 1 ≤ x < 3 random variable in Exercise 3-32.
⎪ ≤ 3-54. Determine the cumulative distribution function for
⎩ 1 3 x the random variable in Exercise 3-33.
(a) P X( ≤ 3 ) (b) P X( ≤ 2 ) 3-55. Determine the cumulative distribution function for the
(c) P(1≤ X ≤ ) 2 (d) P X( > 2 ) random variable in Exercise 3-34.
3-50. Errors in an experimental transmission channel are 3-56. Determine the cumulative distribution function for the
found when the transmission is checked by a certiier that detects random variable in Exercise 3-35.