   # Finding the Mean and Median In Exercises 45–48, find the mean and median of the probability density function. f ( x ) = 0.25 e − x / 4 , [ 0 , ∞ ) ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
Publisher: Cengage Learning
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
Publisher: Cengage Learning
ISBN: 9781305860919
Chapter 9, Problem 47RE
Textbook Problem
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## Finding the Mean and Median In Exercises 45–48, find the mean and median of the probability density function. f ( x ) = 0.25 e − x / 4 ,   [ 0 ,   ∞ )

To determine

To calculate: The mean and median of probability density function f(x)=0.25ex/4 over the interval [0,).

### Explanation of Solution

Given Information:

The probability density function is defined as

f(x)=0.25ex/4 over the interval [0,).

Formula used:

Mean:

If f is the probability density function f for a continuous random variable x for the interval [a,b], where a and b are real numbers, then the mean of x is defined by the formula,

μ=abxf(x) dx

Median:

If f is the probability density function f for a continuous random variable x for the interval [a,b], where a and b are real numbers, then the median of x is defined by the formula,

amf(x) dx=0.5

Calculation:

Consider the provided probability density function is,

f(x)=0.25ex/4 over the interval [0,).

Use the formula μ=abxf(x) dx for the provided probability density function to calculate the mean.

So,

μ=0x(0.25ex/4) dx=0.250xex/4 dx

Use the formula udv=uvvdu for integration by parts. Assume u=x,dv=ex/4dx.

dv=ex/4dx

Integrate to calculate v as,

dv=ex/4dxv=ex/41/4v=4ex/4

So, the mean is calculated as,

μ=0.25[x(4ex/4)]004ex/4dx=0.25[xex/4+4ex/41/4]0=0.25[xex/416ex/4]0

Use the Fundamental theorem abf(x) dx=F(b)F(a) and apply the limits as,

0

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