The current in a certain circuit as measured by an ammeter is a continuous random variableXwith the following density function.

f(x) =

0.075x+0.2		3x5
0		otherwise

(a)

Graph the pdf.

Verify that the total area under the density curve is indeed 1.

5	0.075x+0.2dx
3

=

5

3

=

1.9375

=

(b)

Calculate

P(X4).

How does this probability compare to

P(X< 4)?

P(X4) <P(X< 4)P(X4) =P(X< 4) P(X4) >P(X< 4)

(c)

Calculate

P(3.5X4.5).

Calculate

P(4.5 <X).

2.

[-/2.64 Points]DETAILS

MY NOTES

DEVORESTAT9 4.1.002.MI.

ASK YOUR TEACHER

PRACTICE ANOTHER

Suppose the reaction temperatureX(inC) in a certain chemical process has a uniform distribution with

A=8

andB=8.

(a)

ComputeP(X< 0).

(b)

Compute

P(4

<X<4).

(c)

Compute

P(5

X7). (Round your answer to two decimal places.)

(d)

Forksatisfying

8<k<k+ 4 <8,

computeP(k<X<k+ 4).(Round your answer to two decimal places.)

3.

[-/2.64 Points]DETAILS

MY NOTES

DEVORESTAT9 4.1.003.

ASK YOUR TEACHER

PRACTICE ANOTHER

The error involved in making a certain measurement is a continuous rvXwith the following pdf.

f(x) =

0.09375(4x2)	2x2
0	otherwise

(a) Sketch the graph off(x).

(b) ComputeP(X> 0). (c) ComputeP(1 <X< 1). (Enter your answer to four decimal places.) (d) Compute

P(X<0.9orX>0.9).

(Round your answer to four decimal places.)

4.

[-/2.64 Points]DETAILS

MY NOTES

DEVORESTAT9 4.1.006.

ASK YOUR TEACHER

PRACTICE ANOTHER

The actual tracking weight of a stereo cartridge that is set to track at 3 g on a particular changer can be regarded as a continuous rvXwith the following pdf.

f(x) =

k 1(x3)2		2x4
0		otherwise

(a) Sketch the graph off(x).

(b) Find the value ofk. (c) What is the probability that the actual tracking weight is greater than the prescribed weight? (d) What is the probability that the actual weight is within0.3g of the prescribed weight? (Round your answer to four decimal places.) (e) What is the probability that the actual weight differs from the prescribed weight by more than0.55g? (Round your answer to four decimal places.)

5.

[-/2.64 Points]DETAILS

MY NOTES

DEVORESTAT9 4.2.011.MI.

ASK YOUR TEACHER

PRACTICE ANOTHER

LetXdenote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdf is the following.

F(x) =

0		x< 0
x2 25		0x<5
1		5x

Use the cdf to obtain the following. (If necessary, round your answer to four decimal places.)

(a)

CalculateP(X3).

(b)

CalculateP(2.5X3).

(c)

CalculateP(X>3.5).

(d)

What is the median checkout duration? [solve0.5 =F()].

(e)

Obtain the density functionf(x).

f(x)

=

F(x)

=

	0		x<5
0	otherwise

(f)

CalculateE(X).

(g)

CalculateV(X) and_x.

V(X)_x

(h)

If the borrower is charged an amounth(X) =X²when checkout duration isX, compute the expected charge

E[h(X)].

6.

[-/2.64 Points]DETAILS

MY NOTES

DEVORESTAT9 4.2.012.

ASK YOUR TEACHER

PRACTICE ANOTHER

The error involved in making a certain measurement is a continuous rvXwith the following cdf.

F(x) =

0		x	<2
1 2 + 3 68 7x x3 3		2	x< 2
1		2	x

(a) ComputeP(X< 0). (b) ComputeP(1 <X< 1). (Round your answer to four decimal places.) (c) ComputeP(0.8<X). (Round your answer to four decimal places.) (d) Evaluatef(x) by obtainingF'(x).

f(x) =F'(x) =

(e) Compute.

7.

[-/2.64 Points]DETAILS

MY NOTES

DEVORESTAT9 4.2.021.

ASK YOUR TEACHER

PRACTICE ANOTHER

An ecologist wishes to mark off a circular sampling region having radius15m. However, the radius of the resulting region is actually a random variableRwith the following pdf.

f(r) =

3 4 1(15r)2		14r16
0		otherwise

What is the expected area of the resulting circular region? (Round your answer to two decimal places.) m²

8.

[-/2.64 Points]DETAILS

MY NOTES

DEVORESTAT9 4.2.023.

ASK YOUR TEACHER

PRACTICE ANOTHER

If the temperature at which a certain compound melts is a random variable with mean value135C and standard deviation4C,what are the mean temperature and standard deviation measured inF?[Hint:

F = 1.8C + 32.]

mean Fstandard deviation F

9.

[-/2.64 Points]DETAILS

MY NOTES

DEVORESTAT9 4.3.028.MI.S.

ASK YOUR TEACHER

PRACTICE ANOTHER

LetZbe a standard normal random variable and calculate the following probabilities, drawing pictures wherever appropriate. (Round your answers to four decimal places.)

(a)

P(0Z2.09)

(b)

P(0Z2)

(c)

P(2.70Z0)

(d)

P(2.70Z2.70)

(e)

P(Z1.84)

(f)

P(1.25Z)

(g)

P(1.70Z2.00)

(h)

P(1.84Z2.50)

(i)

P(1.70Z)

(j)

P(|Z|2.50)

You may need to use the appropriate table in theAppendix of Tablesto answer this question.

10.

[-/2.64 Points]DETAILS

MY NOTES

DEVORESTAT9 4.3.030.S.

ASK YOUR TEACHER

PRACTICE ANOTHER

Find the following percentiles for the standard normal distribution. Interpolate where appropriate. (Round your answers to two decimal places.)

(a) 81st (b) 19th (c) 75th (d) 25th (e) 14th

You may need to use the appropriate table in theAppendix of Tablesto answer this question.

11.

[-/2.64 Points]DETAILS

MY NOTES

DEVORESTAT9 4.3.031.S.

ASK YOUR TEACHER

PRACTICE ANOTHER

Determinezfor the following of. (Round your answers to two decimal places.)

(a) =0.0086 (b) =0.11 (c) =0.663

You may need to use the appropriate table in theAppendix of Tablesto answer this question.

12.

[-/2.64 Points]DETAILS

MY NOTES

DEVORESTAT9 4.3.033.S.

ASK YOUR TEACHER

PRACTICE ANOTHER

Mopeds (small motorcycles with an engine capacity below 50 cm³) are very popular in Europe because of their mobility, ease of operation, and low cost. Suppose the maximum speed of a moped is normally distributed with mean value46.7km/hand standard deviation1.75 km/h.Consider randomly selecting a single such moped.

(a)

What is the probability that maximum speed is at most49km/h? (Round your answer to four decimal places.)

(b)

What is the probability that maximum speed is at least48km/h? (Round your answer to four decimal places.)

(c)

What is the probability that maximum speed differs from the mean value by at most1.5standard deviations? (Round your answer to four decimal places.)

You may need to use the appropriate table in theAppendix of Tablesto answer this question.

13.

[-/2.64 Points]DETAILS

MY NOTES

DEVORESTAT9 4.3.039.S.

ASK YOUR TEACHER

PRACTICE ANOTHER

The defect length of a corrosion defect in a pressurized steel pipe is normally distributed with mean value33mm and standard deviation7.3mm.

(a)

What is the probability that defect length is at most 20 mm? Less than 20 mm? (Round your answers to four decimal places.)

at most 20mmless than 20mm

(b)

What is the 75th percentile of the defect length distributionthat is, the value that separates the smallest 75% of all lengths from the largest 25%? (Round your answer to four decimal places.)

mm

(c)

What is the 15th percentile of the defect length distribution? (Round your answer to four decimal places.)

mm

(d)

What values separate the middle 80% of the defect length distribution from the smallest 10% and the largest 10%? (Round your answers to four decimal places.)

smallest 10% mmlargest 10% mm

You may need to use the appropriate table in theAppendix of Tablesto answer this question.

14.

[-/2.64 Points]DETAILS

MY NOTES

DEVORESTAT9 4.3.044.S.

ASK YOUR TEACHER

PRACTICE ANOTHER

If bolt thread length is normally distributed, what is the probability that the thread length of a randomly selected bolt is

(a) Within1.4SDs of its mean value? (Round your answer to four decimal places.) (b) Farther than2.3SDs from its mean value? (Round your answer to four decimal places.) (c) Between 1 and 2 SDs from its mean value? (Round your answer to four decimal places.)

You may need to use the appropriate table in theAppendix of Tablesto answer this question.

15.

[-/2.64 Points]DETAILS

MY NOTES

DEVORESTAT9 4.4.059.

ASK YOUR TEACHER

PRACTICE ANOTHER

LetX= the time between two successive arrivals at the drive-up window of a local bank. IfXhas an exponential distribution with= 1,(which is identical to a standard gamma distribution with= 1),compute the following. (If necessary, round your answer to three decimal places.)(a) The expected time between two successive arrivals (b) The standard deviation of the time between successive arrivals (c) P(X3) (d) P(2X5)

You may need to use the appropriate table in theAppendix of Tables

16.

[-/2.64 Points]DETAILS

MY NOTES

DEVORESTAT9 4.4.061.MI.

ASK YOUR TEACHER

PRACTICE ANOTHER

Data collected at an airport suggests that an exponential distribution with mean value2.835hoursis a good model for rainfall duration.

(a)

What is the probability that the duration of a particular rainfall event at this location is at least 2 hours? At most3 hours?Between 2 and3 hours?(Round your answers to four decimal places.)

at least 2 hoursat most 3 hoursbetween 2 and 3 hours

(b)

What is the probability that rainfall duration exceeds the mean value by more than4standard deviations? (Round your answer to four decimal places.)

What is the probability that it is less than the mean value by more than one standard deviation?

17.

[-/2.76 Points]DETAILS

MY NOTES

DEVORESTAT9 4.4.069.

ASK YOUR TEACHER

A system consists of five identical components connected in series as shown:As soon as one components fails, the entire system will fail. Suppose each component has a lifetime that is exponentially distributed with= 0.01 and that components fail independently of one another. Define eventsA_i= {ith component lasts at leastthours},i= 1, . . . , 5,so that theA_is are independent events. LetX= the timeat which the system failsthat is, the shortest (minimum) lifetime among the five components.(a) The event {Xt} is equivalent to what event involvingA₁, . . . ,A₅?

A₁A₂A₃A₄A₅

A₁A₂A₃A₄A₅

A₁A₂A₃A₄A₅

A₁A₂A₃A₄A₅

(b) Using the independence of theA_i's, computeP(Xt). P(Xt) =

ObtainF(t) =P(Xt). F(t) =

Obtain the pdf ofX. f(t) =

What type of distribution doesXhave?Xis a gamma distribution with parameters= 0 and= 1.Xis an exponential distribution with= 0.05. Xis a gamma distribution with parameters= 1 and= 0.05.Xis an exponential distribution with= 1.

(c) Suppose there arencomponents, each having exponential lifetime with parameter. What type of distribution doesXhave?Xis a gamma distribution with parameters= 1 and= 1/.Xis an exponential distribution with parameter=e. Xis a gamma distribution with parameters=and=n.Xis an exponential distribution with parametern.