Verify Property 2 of the definition of a probability density function over the given interval.

f(x)=465x3,

[2,3]

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Part 1

What is Property 2 of the definition of a probability density function?

A.

The area under the graph of f over the interval [a,b] is 1.

B.

The area under the graph of f over the interval [a,b] is b.

C.

The area under the graph of f over the interval [a,b] is a.

Part 2

Identify the formula for calculating the area under the graph of the function

y=f(x)

over the interval [a,b]. Choose the correct answer below.

A.Integral from a to b f left parenthesis x right parenthesis dx equals left bracket Upper F left parenthesis x right parenthesis right bracket Subscript a Superscript b Baseline equals Upper F left parenthesis b right parenthesis minus Upper F left parenthesis a right parenthesis

abf(x)dx=[F(x)]ba=F(b)F(a)

B.Integral from b to a f left parenthesis x right parenthesis dx equals left bracket Upper F left parenthesis x right parenthesis right bracket Subscript b Superscript a Baseline equals Upper F left parenthesis b right parenthesis minus Upper F left parenthesis a right parenthesis

baf(x)dx=[F(x)]ab=F(b)F(a)

C.Integral from b to a f left parenthesis x right parenthesis dx equals left bracket Upper F left parenthesis x right parenthesis right bracket Subscript b Superscript a Baseline equals Upper F left parenthesis a right parenthesis minus Upper F left parenthesis b right parenthesis

baf(x)dx=[F(x)]ab=F(a)F(b)

D.Integral from a to b f left parenthesis x right parenthesis dx equals left bracket Upper F left parenthesis x right parenthesis right bracket Subscript a Superscript b Baseline equals Upper F left parenthesis a right parenthesis minus Upper F left parenthesis b right parenthesis

abf(x)dx=[F(x)]ba=F(a)F(b)

Part 3

Substitute a, b, and f(x) into the left side of the formula from the previous step.

area

=

enter your response hereenter your response here465x3dx

Part 4

Next, determine F(x). First, find the antiderivative of f.

465x3dx=enter your response here

Part 5

Let

C=0

in the expression obtained above and let the resulting expression be F(x). Evaluate the result over

[2,3]

using the far right side of the formula for the area.

area

=

enter your response hereenter your response here

Part 6

area

=

enter your response here Simplify.

Part 7

Is Property 2 of the definition of a probability density function over the given interval now verified? Choose the correct answer below.

A.

Property 2 of the definition of a probability density function over the given interval has been verified since the expression in the previous step equals b.

B.

Property 2 of the definition of a probability density function over the given interval has not been verified because the expression in the previous step does not equal the expected area value.

C.

Property 2 of the definition of a probability density function over the given interval has been verified since the expression in the previous step equals 1.

D.

Property 2 of the definition of a probability density function over the given interval has been verified since the expression in the previous step equals a.