   # Finding the Mean and Median In Exercises 45–48, find the mean and median of the probability density function. f ( x ) = 5 6 e − 5 x / 6 , [ 0 , ∞ ) Special Probability Density Functions In Exercises 49–54, identify the probability density function. Then find the mean, variance, and standard deviation without integrating. ### 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 48RE
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 ) = 5 6 e − 5 x / 6 ,   [ 0 , ∞ ) Special Probability Density Functions In Exercises 49–54, identify the probability density function. Then find the mean, variance, and standard deviation without integrating.

To determine

To calculate: The mean and median of probability density function f(x)=56e5x/6 over the interval [0,).

### Explanation of Solution

Given Information:

The probability density function is defined as

f(x)=56e5x/6 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)=56e5x/6 over the interval [0,).

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

So,

μ=0x(56e5x/6) dx=560xe5x/6 dx

Use the formula udv=uvvdu for integration by parts. Assume u=x,dv=e5x/6dx,

u=x,dv=e5x/6dx

Integrate to calculate v as,

dv=e5x/6dxv=e5x/65/6v=65e5x/6

So, the mean is calculated as,

μ=56[[x(65e5x/6)]0065e5x/6dx]=56[65xe5x/6+65e5x/6(5/6)]0=[xe5x/6

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