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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Demand The weekly demand x (in tons) for a certain product is described by the probability density function

f ( x ) = 1 36 x e x / 6 ,        [ 0 , ]

Find and interpret each probability.

( a ) P ( x   < 6 ) ( b ) P ( 6   < x <   12 ) ( c )       P ( 12 x 24 ) ( d )   P ( x >   12 )   =   1   - P ( x   12 )

a)

To determine

To calculate: The probability for the range P(x<6) and also interpret the result when the weekly demand x for a certain product is described by the probability density function,

f(x)=136xex/6,  [0,)

Explanation

Given Information:

The weekly demand x for a certain product is described by the probability density function,

f(x)=136xex/6,  [0,)

Formula used:

In a probability density function, the probability that x lies in interval [c,d] is given by,

P(cxd)=cdf(x) dx,

Which is shown in the figure below,

Integration by parts.

When u and v is assumed to be the differentiable functions of x then,

u dv=uvv du

Calculation:

Consider the exponential probability density function,

f(x)=136xex/6,  [0,)

In order to calculate the probability for the range P(x<6) years integrate f(x) over interval [0,6].

Thus,

P(x<6)=06136xex/6dx

In order to solve the integral 06136xex/6dx apply integration by parts formula.

Let u=x and dv=ex/6dx, then

dv=ex/6dx

Apply integral on both sides of the above equation as,

dv=ex/6dxv=6ex/6

Thus, v=6ex/6,

Then differentiate both sides of the equation u=x as,

du=dx

Substitute u=x, dv=ex/6dx, v=6ex/6 and du=dx in the formula u dv=uvv du

b)

To determine

To calculate: The probability for the range P(6<x<12) and also interpret the result when the weekly demand x for a certain product is described by the probability density function,

f(x)=136xex/6,  [0,)

c)

To determine

To calculate: The probability for the range P(12x24) and also interpret the result when the weekly demand x for a certain product is described by the probability density function,

f(x)=136xex/6,  [0,)

d)

To determine

To calculate: The probability for the range P(x>12) and also interpret the result when the weekly demand x for a certain product is described by the probability density function,

f(x)=136xex/6,  [0,)

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