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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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Find the mean and median of the probability density function

f ( x ) = 2 e 2 x ,  0   x  <  .

To determine

To calculate: The mean and median of the probability density function f(x)=2e2x on the interval 0x<.

Explanation

Given information:

The function is,

f(x)=2e2x

And the interval is 0x<.

Formula used:

To find the expected value or mean of x,

μ=E(x)=abxf(x)dx

The median m of x is,

amf(x)dx=0.5

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)=2e2x

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

μ=abx[2e2x]dx

To find the integration substitute u=2x,

μ=20euu4du=2140ueudu=120ueudu

Now, apply By Parts formula u(x)v(x)dx=uvdxu(vdx)dx,

μ=[12

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