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Chapter 4: Q143E (page 277)

Lead in metal shredder residue. On the basis of data collectedfrom metal shredders across the nation, the amount xof extractable lead in metal shredder residue has an approximateexponential distribution with mean胃= 2.5 milligramsper liter (Florida Shredder鈥檚 Association).

a. Find the probability that xis greater than 2 milligramsper liter.

b. Find the probability that xis less than 5 milligrams perliter.

Short Answer

Expert verified

a.Px>2=.0068

b.Px<5=0.9999

Step by step solution

01

Given information

X is an exponential random variable and胃=2.5 .

02

Define the probability density function

The p.d.f of X is

fx=胃e-胃x;x鈮�0=2.5e-2.5x

03

Calculate P(x>2)

a

Px>2=1-Px鈮�2=1-鈭�022.5-2.5xdx=1-2.5鈭�02e-2.5xdx=1-525-2e-552=1-e-5e-5-1=1-0.9932=0.0068

Hence, the probability of x that x is greater than 2 milligrams per liter is 0.0068.

04

Calculate P(x<5)

b

Px<5=鈭�052.5e-2.5xdx=2.5鈭�05e-2.5xdx=525-2e-25252=e-252e-252-1=0.9999

Hence, the probability of x that x is less than 5 milligrams per liter is 0.9999.

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