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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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BuyFindarrow_forward

Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem

Using the Exponential Density Function In Exercises 33-36, find the required probabilities using the exponential probability density function

f ( t ) 1 λ e t / λ , [ 0 , ) .

Waiting Time The length of time t (in hours) required to unload trucks at a depot is exponentially distributed with λ   =   3 4 .

Find the probability that the trucks can be unloaded (a) in less than 1 hour, (b) in more than 1 hour but less than 2 hours, and (c) in at most 3 hours.

a)

To determine

To calculate: The probability that truck can be unloaded in less than 1 hour when the length of time needed to unload trucks at a depot is exponentially distributed with λ=34 by using the exponential probability density function given as,

f(t)=1λet/λ[0,)

Explanation

Given Information:

The length of time needed to unload trucks at a depot is exponentially distributed with λ=34.

And the provided exponential probability density function is given as,

f(t)=1λet/λ[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,

Calculation:

Consider the exponential probability density function,

f(t)=1λet/λ[0,)

In order to calculate the probability that truck can be unloaded in less than 1 hour, integrate f(t) over interval [0,1] with substitution λ=34 in f(t)

b)

To determine

To calculate: The probability that truck can be unloaded in more than 1 hour but less than 2 hour when the length of time needed to unload trucks at a depot is exponentially distributed with λ=34 by using the exponential probability density function given as,

f(t)=1λet/λ[0,)

c)

To determine

To calculate: The probability that truck can be unloaded in at most 3 hours when the length of time needed to unload trucks at a depot is exponentially distributed with λ=34 by using the exponential probability density function given as,

f(t)=1λet/λ[0,)

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