Verify Property 2 of the definition of a probability density function over the given interval. f(x) = x*, [-3,3] %3D
Verify Property 2 of the definition of a probability density function over the given interval. f(x) = x*, [-3,3] %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Verify Property 2 of the definition of a probability density function over the given interval.
f(x) =
[-3,3]
486
What is Property 2 of the definition of a probability density function?
O A. The area under the graph of f over the interval [a,b] is 1.
O B. The area under the graph of f over the interval [a,b] is a.
O C. The area under the graph of f over the interval [a,b] is b.
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.
O A. b
O B. a
[rox) dx = [F(x); = F(b)-F(a)
F(x) dx = [F(x)1; = F(a) – F(b)
%3D
a
ОС. а
O D. b
f(x) dx = [F(x)], = F(b) – F(a)
f(x) dx = [F(x)l% = F(a) - F(b)
b
a
Substitute a, b, and f(x) into the left side of the formula from the previous step.
5
x* dx
486
area =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee2e25dc-949a-49f9-893c-6a598612cdf0%2Fcf8c8cb3-3a58-4345-b5dc-7d66412c3370%2Fb7squx_processed.png&w=3840&q=75)
Transcribed Image Text:Verify Property 2 of the definition of a probability density function over the given interval.
f(x) =
[-3,3]
486
What is Property 2 of the definition of a probability density function?
O A. The area under the graph of f over the interval [a,b] is 1.
O B. The area under the graph of f over the interval [a,b] is a.
O C. The area under the graph of f over the interval [a,b] is b.
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.
O A. b
O B. a
[rox) dx = [F(x); = F(b)-F(a)
F(x) dx = [F(x)1; = F(a) – F(b)
%3D
a
ОС. а
O D. b
f(x) dx = [F(x)], = F(b) – F(a)
f(x) dx = [F(x)l% = F(a) - F(b)
b
a
Substitute a, b, and f(x) into the left side of the formula from the previous step.
5
x* dx
486
area =
![Next, determine F(x). First, find the antiderivative of f.
.4
X dx =
486
Let C= 0 in the expression obtained above and let the resulting expression be F(x). Evaluate the result over [-3,3] using the far right side of the formula for the area.
O-D
area =
Simplify
%3D
Is Property 2 of the definition of a probability density function over the given interval now verified? Choose the correct answer below.
O 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 1.
O B. 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.
O C. 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 valu
O 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee2e25dc-949a-49f9-893c-6a598612cdf0%2Fcf8c8cb3-3a58-4345-b5dc-7d66412c3370%2F4f826m_processed.png&w=3840&q=75)
Transcribed Image Text:Next, determine F(x). First, find the antiderivative of f.
.4
X dx =
486
Let C= 0 in the expression obtained above and let the resulting expression be F(x). Evaluate the result over [-3,3] using the far right side of the formula for the area.
O-D
area =
Simplify
%3D
Is Property 2 of the definition of a probability density function over the given interval now verified? Choose the correct answer below.
O 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 1.
O B. 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.
O C. 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 valu
O 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.
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