Let X be a random variable with probability density function f ( x ) = { c ( 1 − x 2 ) − 1 < x < 0 0 otherwise a. What is the value of c? b. What is the cumulative distribution function of X?

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Answer 1

Textbook Question

Chapter 5, Problem 5.1P

Let X be a random variable with probability density function f ( x ) = { c ( 1 − x 2 ) − 1 < x < 0 0 otherwise

a. What is the value of c?

b. What is the cumulative distribution function of X?

Expert Solution

To determine

(a) To compute: To find the value of c

Answer to Problem 5.1P

The value of c is 0.75.

Explanation of Solution

Given information:

The probability density function of X

f (x)= {c (1−x2) −1<x<10 otherwise

Calculation:

As the probability density function integrates to 1

∫−11c (1−x2) dx = 1

= [c × (x−x33)]−11 = 1

= c × (1−13−(−1−−13)) = 1

= c × (2−23)=1

= c × (43)=1

c=34=0.75

(b)

Expert Solution

To determine

To find: Cumulative distribution of X

Answer to Problem 5.1P

The cumulative distribution of X is 0.75 × (x−x33+23) .

Explanation of Solution

Formula used:

F (x) = ∫−1xf(t)dt

Calculation:

The cumulative distribution of X is

= ∫−1x0.75 × (1−t2) dt

= [0.75 × (t−t33)]−1x

=0.75 × [x−x33− (−1−−13)]

=0.75 × (x−x33+23)

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03:06

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Answer 2

Textbook Question

Chapter 5, Problem 5.1P

Let X be a random variable with probability density function f ( x ) = { c ( 1 − x 2 ) − 1 < x < 0 0 otherwise

a. What is the value of c?

b. What is the cumulative distribution function of X?

Expert Solution

To determine

(a) To compute: To find the value of c

Answer to Problem 5.1P

The value of c is 0.75.

Explanation of Solution

Given information:

The probability density function of X

f (x)= {c (1−x2) −1<x<10 otherwise

Calculation:

As the probability density function integrates to 1

∫−11c (1−x2) dx = 1

= [c × (x−x33)]−11 = 1

= c × (1−13−(−1−−13)) = 1

= c × (2−23)=1

= c × (43)=1

c=34=0.75

(b)

Expert Solution

To determine

To find: Cumulative distribution of X

Answer to Problem 5.1P

The cumulative distribution of X is 0.75 × (x−x33+23) .

Explanation of Solution

Formula used:

F (x) = ∫−1xf(t)dt

Calculation:

The cumulative distribution of X is

= ∫−1x0.75 × (1−t2) dt

= [0.75 × (t−t33)]−1x

=0.75 × [x−x33− (−1−−13)]

=0.75 × (x−x33+23)

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03:06

Answer 3

Textbook Question

Chapter 5, Problem 5.1P

Let X be a random variable with probability density function f ( x ) = { c ( 1 − x 2 ) − 1 < x < 0 0 otherwise

a. What is the value of c?

b. What is the cumulative distribution function of X?

Expert Solution

To determine

(a) To compute: To find the value of c

Answer to Problem 5.1P

The value of c is 0.75.

Explanation of Solution

Given information:

The probability density function of X

f (x)= {c (1−x2) −1<x<10 otherwise

Calculation:

As the probability density function integrates to 1

∫−11c (1−x2) dx = 1

= [c × (x−x33)]−11 = 1

= c × (1−13−(−1−−13)) = 1

= c × (2−23)=1

= c × (43)=1

c=34=0.75

(b)

Expert Solution

To determine

To find: Cumulative distribution of X

Answer to Problem 5.1P

The cumulative distribution of X is 0.75 × (x−x33+23) .

Explanation of Solution

Formula used:

F (x) = ∫−1xf(t)dt

Calculation:

The cumulative distribution of X is

= ∫−1x0.75 × (1−t2) dt

= [0.75 × (t−t33)]−1x

=0.75 × [x−x33− (−1−−13)]

=0.75 × (x−x33+23)

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03:06

Answer 4

Textbook Question

Chapter 5, Problem 5.1P

Let X be a random variable with probability density function f ( x ) = { c ( 1 − x 2 ) − 1 < x < 0 0 otherwise

a. What is the value of c?

b. What is the cumulative distribution function of X?

Expert Solution

To determine

(a) To compute: To find the value of c

Answer to Problem 5.1P

The value of c is 0.75.

Explanation of Solution

Given information:

The probability density function of X

f (x)= {c (1−x2) −1<x<10 otherwise

Calculation:

As the probability density function integrates to 1

∫−11c (1−x2) dx = 1

= [c × (x−x33)]−11 = 1

= c × (1−13−(−1−−13)) = 1

= c × (2−23)=1

= c × (43)=1

c=34=0.75

(b)

Expert Solution

To determine

To find: Cumulative distribution of X

Answer to Problem 5.1P

The cumulative distribution of X is 0.75 × (x−x33+23) .

Explanation of Solution

Formula used:

F (x) = ∫−1xf(t)dt

Calculation:

The cumulative distribution of X is

= ∫−1x0.75 × (1−t2) dt

= [0.75 × (t−t33)]−1x

=0.75 × [x−x33− (−1−−13)]

=0.75 × (x−x33+23)

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03:06

Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

Expert Solution

To determine

(a) To compute: To find the value of c

Answer to Problem 5.1P

The value of c is 0.75.

Explanation of Solution

Given information:

The probability density function of X

f (x)= {c (1−x2) −1<x<10 otherwise

Calculation:

As the probability density function integrates to 1

∫−11c (1−x2) dx = 1

= [c × (x−x33)]−11 = 1

= c × (1−13−(−1−−13)) = 1

= c × (2−23)=1

= c × (43)=1

c=34=0.75

(b)

Expert Solution

To determine

To find: Cumulative distribution of X

Answer to Problem 5.1P

The cumulative distribution of X is 0.75 × (x−x33+23) .

Explanation of Solution

Formula used:

F (x) = ∫−1xf(t)dt

Calculation:

The cumulative distribution of X is

= ∫−1x0.75 × (1−t2) dt

= [0.75 × (t−t33)]−1x

=0.75 × [x−x33− (−1−−13)]

=0.75 × (x−x33+23)

Answer 8

Expert Solution

To determine

(a) To compute: To find the value of c

Answer to Problem 5.1P

The value of c is 0.75.

Explanation of Solution

Given information:

The probability density function of X

f (x)= {c (1−x2) −1<x<10 otherwise

Calculation:

As the probability density function integrates to 1

∫−11c (1−x2) dx = 1

= [c × (x−x33)]−11 = 1

= c × (1−13−(−1−−13)) = 1

= c × (2−23)=1

= c × (43)=1

c=34=0.75

Answer 9

Expert Solution

Answer 10

Expert Solution

Answer 11

Expert Solution

Answer 12

To determine

(a) To compute: To find the value of c

Answer 13

Answer to Problem 5.1P

The value of c is 0.75.

Answer 14

Explanation of Solution

Given information:

The probability density function of X

f (x)= {c (1−x2) −1<x<10 otherwise

Calculation:

As the probability density function integrates to 1

∫−11c (1−x2) dx = 1

= [c × (x−x33)]−11 = 1

= c × (1−13−(−1−−13)) = 1

= c × (2−23)=1

= c × (43)=1

c=34=0.75

Answer 15

(b)

Expert Solution

To determine

To find: Cumulative distribution of X

Answer to Problem 5.1P

The cumulative distribution of X is 0.75 × (x−x33+23) .

Explanation of Solution

Formula used:

F (x) = ∫−1xf(t)dt

Calculation:

The cumulative distribution of X is

= ∫−1x0.75 × (1−t2) dt

= [0.75 × (t−t33)]−1x

=0.75 × [x−x33− (−1−−13)]

=0.75 × (x−x33+23)

Answer 16

(b)

Expert Solution

Answer 17

(b)

Expert Solution

Answer 18

Expert Solution

Answer 19

To determine

To find: Cumulative distribution of X

Answer 20

Answer to Problem 5.1P

The cumulative distribution of X is 0.75 × (x−x33+23) .

Answer 21

Explanation of Solution

Formula used:

F (x) = ∫−1xf(t)dt

Calculation:

The cumulative distribution of X is

= ∫−1x0.75 × (1−t2) dt

= [0.75 × (t−t33)]−1x

=0.75 × [x−x33− (−1−−13)]

=0.75 × (x−x33+23)

Answer 22

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Let X be a random variable with probability density function f ( x ) = { c ( 1 − x 2 ) − 1 < x < 0 0 otherwise a. What is the value of c? b. What is the cumulative distribution function of X?

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Answer to Problem 5.1P

Explanation of Solution

Answer to Problem 5.1P

Explanation of Solution

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Chapter 5 Solutions

Additional Math Textbook Solutions

Let X be a random variable with probability density function f ( x ) = { c ( 1 − x 2 ) − 1 &lt; x &lt; 0 0 otherwise a. What is the value of c? b. What is the cumulative distribution function of X?

Concept explainers

Videos

Answer to Problem 5.1P

Explanation of Solution

Answer to Problem 5.1P

Explanation of Solution

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Chapter 5 Solutions

Additional Math Textbook Solutions

Let X be a random variable with probability density function f ( x ) = { c ( 1 − x 2 ) − 1 < x < 0 0 otherwise a. What is the value of c? b. What is the cumulative distribution function of X?