   # In Exercises 7–9, use a graphing utility to graph the function. Then determine whether the function f represents a probability density function over the given interval. If f is not a probability density function, identify the condition(s) that is (are) not satisfied. f ( x ) = 3 4 e − 3 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 9TYS
Textbook Problem
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## In Exercises 7–9, use a graphing utility to graph the function. Then determine whether the function f represents a probability density function over the given interval. If f is not a probability density function, identify the condition(s) that is (are) not satisfied. f ( x ) = 3 4 e − 3 x / 4 ,   [ 0 , ∞ )

To determine

To graph: Whether the function f(x)=34e3x/4 represents a probability density function over the interval [0,) and graph the function using a graphing utility.

### Explanation of Solution

Graph:

Use the ti-83 graphing calculator to construct the graph of the function.

Step 1: Open the ti-83 graphing calculator.

Step 2: Press Y= and enter the function Y1=(3/4)e^(3X/4).

Step 3: Press the WINDOW key and adjust the scale.

Xmin=0Xmax=8Xscl=2

And,

Ymin=0Ymax=0.8Yscl=0.2

Step 4: Press the GRAPH key.

The graph of the function is obtained as,

Consider the provided function f(x)=34e3x/4 over the interval [0,).

A function f of a continuous random variable x in the interval [a,b] is a probability density function if it is nonnegative and continuous on the interval [a,b] and

abf(x) dx=1

Consider any value x=5 in the interval [0,).

Evaluate the function at x=5.

So,

f(5)=34e3(5)/4=34(e3.75)=0.070554=0.0176

So, the function f is continuous and nonnegative for all values on the interval [0,). Now, show that abf(x) dx=1.

Evaluate the integral 034e3x/4 dx

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