Page 133 - Applied statistics and probability for engineers

Section 4-2/Probability Distributions and Probability Density Functions 111

Exercises FOR SECTION 4-2

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion.
4-1. Suppose that f x( ) = e − x for 0 < x. Determine the 4-10. The probability density function of the length of
.
following: a cutting blade is f x ( ) = .1 25 for 74 6. < <x 75 4 millimeters.
(a) P 1< ) (b) P 1< X < 2 5) Determine the following:
(
(
.
X
(
(
(
.
(
(
X
(c) P X = ) 3 (d) P X < 4) (e) P 3 ≤ ) (a) P X < 74 8. ) (b) P X < 74 8. or X > 75 2)
(
X
(f) x such that P x < ) = .0 10 (c) If the speciications for this process are from 74.7 to 75.3
millimeters, what proportion of blades meets speciications?
(
(g) x such that P X ≤ x) = .10 4-11. The probability density function of the length of a
0
⁄
<
.
2
x
4-2. Suppose that f x ( ) = ( 3 8 x − ) 256 for 0 < x 8 . metal rod is f x ( ) = 2 for 2 3. < x < 2 8 meters.
Determine the following: (a) If the speciications for this process are from 2.25 to 2.75
(
(
(
(a) P X < 2) (b) P X < 9) (c) P 2 < X < 4) meters, what proportion of rods fail to meet the speciications?
(
(d) P X > 6) (e) x such that P X < x) = .0 95 (b) Assume that the probability density function is f x ( ) = 2 for
(
an interval of length 0.5 meters. Over what value should the
⁄
⁄
2
4-3. Suppose that f x ( ) = .0 5 cos x for − π 2 < x < π . Deter- density be centered to achieve the greatest proportion of
mine the following: rods within speciications?
(
(
(
π )
(a) P X < 0) (b) P X < −π ) 4 (c) P − π⁄ <4 X < ⁄ 4 4-12. An article in Electric Power Systems Research [“Mod-
⁄
(
( ⁄ (e) x such that P X < eling Real-Time Balancing Power Demands in Wind Power
0
(d) P X > −π ) 4 x) = .95
Systems Using Stochastic Differential Equations” (2010, Vol.
4-4. The diameter of a particle of contamination (in 80(8), pp. 966–974)] considered a new probabilistic model to
micrometers) is modeled with the probability density function balance power demand with large amounts of wind power. In
3
x
f x ( ) = 2 ⁄ for x >1. Determine the following: this model, the power loss from shutdowns is assumed to have
(
(a) P X < 2) (b) P X > 5) (c) P 4 < X < 8) a triangular distribution with probability density function
(
(
( X > 8) (e) x such that P X < ⎧ × −4 × −6 x, ∈
(
(d) P X < 4 or x) = .95 ⎪ − . 5 56 10 + .56 10 x [100 ,500 ]
0
5
×
×
4
(
x
4-5. Suppose that f x ( ) = x for 3 < < 5. f x) = ⎨ . 4 44 10 −3 − .44 10 −6 x, , x ∈[500 ,1000 ]
8 ⎪
Determine the following probabilities: ⎩ , 0 otherwise
(a) P X < 4) (b) P X > 3 5. ) (c) P 4 < X < 5)
(
(
(
(
(d) P X < 4 5. ) (e) P X < 3 5 . or X > 4 5) Determine the following: X ≤ 200 )
(
.
(b) P(100 <
)
(a) P X( < 90
4-6. Suppose that f x ( ) = e −( x − ) 4 for 4 < x . Determine the (c) P X( > 800 ) (d) Value exceeded with probability 0.1.
following: 4-13. A test instrument needs to be calibrated periodically to
(
(a) P 1< ) (b) P 2 ≤ X < 5) (c) P 5 < ) prevent measurement errors. After some time of use without cal-
(
(
X
X
(
(
(d) P 8 < X 12) (e) x such that P X < x) = 0 90. ibration, it is known that the probability density function of the
<
x
measurement error is f x( ) = − .1 0 5 x for 0 < < 2 millimeters.
2
x
. 1
4-7. Suppose that f x ( ) = .1 5 for −1< x < Determine (a) If the measurement error within 0.5 millimeters is accept-
the following: able, what is the probability that the error is not acceptable
(a) P 0 ( < X) (b) P 0 5. ( < X) before calibration?
( ( (b) What is the value of measurement error exceeded with
0
(c) P − . ≤0 5 X ≤ . ) 5 (d) P X < − ) 2
( X > 0 5) ( . probability 0.2 before calibration?
−
.
X
(e) P X < 0 or (f) x such that P x < ) = .0 05
(c) What is the probability that the measurement error is
4-8. The probability density function of the time to fail- exactly 0.22 millimeters before calibration?
ure of an electronic component in a copier (in hours) is f x ( ) = 4-14. The distribution of X is approximated with a triangu-
e − x/1000 /1000 for x > 0. Determine the probability that lar probability density function f x( ) = .0 025 x − .0375 for
0
x
(a) A component lasts more than 3000 hours before failure. 30 < < 50 and f x) = − .025 x + .0875 for 50 < < 70.
x
0
0
(
(b) A component fails in the interval from 1000 to 2000 hours. Determine the following:
(c) A component fails before 1000 hours. (a) P X( ≤ 40 ) (b) P(40 < X ≤ 60 )
(d) The number of hours at which 10% of all components (c) Value x exceeded with probability 0.99.
have failed. 4-15. The waiting time for service at a hospital emergency depart-
4-9. The probability density function of the net weight ment (in hours) follows a distribution with probability density
in pounds of a packaged chemical herbicide is f x ( ) = .2 0 for function f x( ) = .0 5 exp(− .5 x) for 0 < x. Determine the following:
0
49 75 < < 50 25 pounds. (a) P X( < .0 5 ) (b) P X > 2 )
.
.
x
(
(a) Determine the probability that a package weighs more (c) Value x (in hours) exceeded with probability 0.05.
than 50 pounds. 4-16. If X is a continuous random variable, argue that
(
<
P
P
(b) How much chemical is contained in 90% of all packages? P x 1 ≤ ≤ ) = (x 1 < X x 2 P ≤ X x 2 )= (x 1 < < ).
X x 2
≤ )= (x 1
X x 2