Find the constant k such that the function is a probability density function over the interval [3,8].
A) 5
B)
C)
D)
E)

Question

​Find the constant k such that the function  is a probability density function over the interval [3,8]. ​
A) ​5 
B) ​ 
C) ​ 
D) ​ 
E) ​

Quizplus · Accepted Answer

To be a probability density function over the interval [3,8], the function must satisfy two conditions:
1. The integral over the entire interval [3,8] must be equal to 1: ∫ 11ea8970_6aab_397f_861b_77e8a72e500a_TB1301_11 dx = 1
2. The function must be non-negative over the entire interval [3,8].
Using calculus, we can integrate the function and set it equal to 1 to solve for k: 
∫ kx^2 - 27x + 90 dx = 1 
(k/3)x^3 - (27/2)x^2 + 90x ∣_3^8 = 1 
Solving for k, we get k = 11ea8970_6aab_3980_861b_2de889f0c53b_TB1301_11. Therefore, the correct answer is B.

​Find the Constant K Such That the Function Is

Find the Constant K Such That the Function Is