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?
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?
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?
Expression, rule, or law that gives the relationship between an independent variable and dependent variable. Some important types of functions are injective function, surjective function, polynomial function, and inverse function.
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<10otherwise
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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College Algebra with Modeling & Visualization (6th Edition)
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