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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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Use a symbolic integration utility to find the percent of the distribution in Example 3 that lies within 1.5 standard deviations of the mean.

Find the standard deviation of the probability density function

f ( x ) = 2 π ( x 2 2 x + 2 ) ,  0 x 2 .

What percent of the distribution lies within one standard deviation of the mean?

To determine

To calculate: The percent of the distribution function f(x)=2π(x22x+2) on the interval 0x2 that lies within 1.5 as a standard deviations of the mean by using symbolic integration utility.

Explanation

Given information:

The function is,

f(x)=2π(x22x+2)

And the interval is 0x2.

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)=2π(x22x+2)

Now apply the formula V(x)=ab(xμ)2f(x)dx to find the expected value,

μ=E(x)=02x[2π(x22x+2)]dx

To find the mean value use symbolic integration utility, by ti-83 calculator,

Step1: Press ON key.

Step2: Press “Y=” key.

Step3: Enter the equation “Y=2X/(π(X22X+2))” before Y1.

Step4: Set the window as,

Xmax=3,   Xmin=0,   Xscl=1Ymax=2,   Ymin=1,Yscl=1

Step5: Press 2nd key and then press TRACE key.

Step6: Press 7th key.

Step7: Now, enter lower limit 0 and press ENTER key.

Step8: Now, enter upper limit 2 and press ENTER key.

The obtained graph is,

Thus,

From the above diagram, it is showed that, 02f(x)dx=1, so the expected value is 1.

Hence, the expected value of the function f(x)=2π(x22x+2) is 1.

Now put μ=1, f(x)=2π(x22x+2) and [a,b]=[0,2] in the formula V(x)=abx2f(x)dxμ2,

V(x)=02x2[2π(x22x+2)]dx(1)2=02x2[2π(x22x+2)]dx1 …… (1)

To find the variance use symbolic integration utility, by ti-83 calculator,

Step1: Press ON key

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