Find the value of the constant k so that the function is a probability density function in the indicated interval.
B)
C)
D)
E)

Question

Find the value of the constant k so that the function is a probability density function in the indicated interval. ​  ​
B) ​ 
C) ​ 
D) ​ 
E) ​

Quizplus · Accepted Answer

To be a probability density function, the function must satisfy two conditions: 
1) It must be non-negative in the given interval. 
2) The integral of the function over the given interval must be equal to 1. 
We're given that the function is k(x^2-1) for -1 ≤ x ≤ 1. 
Condition 1: This function will be non-negative if k is positive. 
Condition 2: We need to find the value of k such that the integral of the function over the interval [-1,1] is equal to 1. 
∫[-1,1] k(x^2-1) dx = 1 
= k[(x^3)/3 - x] [-1,1] = 1 
= k[(-1/3)-(-1/3)] = 1 
= 2k/3 = 1 
= k = 3/2 
Therefore, the best choice is B, with k = 3/2 making the function a probability density function.

Find the Value of the Constant K So That the Function