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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 daily demand x for a certain product (in hundreds of pounds) is a random variable with the probability density function

f ( x ) = 6 343 x ( 7 x ) ,          [ 0 , 7 ]

(a) Find the mean and standard deviation of the demand.

(b) Find the median of the demand.

(c) Find the probability that the demand is within one standard deviation of the mean.

(a)

To determine

To calculate: The expected value(mean) and standard deviation of the probability density function f(x)=6343x(7x) on the interval [0,7] for the daily demand x as a random variable of a certain product(in hundreds of pounds).

Explanation

Given information:

The function is,

f(x)=6343x(7x)

And the interval is [0,7].

Formula used:

To find the expected value or mean of x,

μ=E(x)=abxf(x)dx

The variance of x is,

V(x)=abx2f(x)dxμ2

The standard deviation of x is,

σ=V(x)

Where, f is a probability density function with a continuous random variable x in the interval [a,b]

Calculation:

Consider the provided function is,

f(x)=6343x(7x)

Now apply the formula μ=E(x)=abxf(x)dx to find the expected value,

μ=07x[6343x(7x)]dx=634307(7x2x3)dx=6343[73x3x44]07=6343[(73(7)3744)(73(0)3044)]

Further solve,

μ=6343[(800.33600.25)]=6343[(200

(b)

To determine

To calculate: The median of the probability density function f(x)=6343x(7x) on the interval [0,7] for the daily demand x as a random variable of a certain product (in hundreds of pounds).

(c)

To determine

To calculate: The probability that the demand is within one standard deviation of the mean, if the probability density function f(x)=6343x(7x) on the interval [0,7] for the daily demand x as a random variable of a certain product (in hundreds of pounds).

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