Find the value of the constant a that makes the given function a probability density function on the stated interval.
A)
B)
C)
D)
E) 1

Question

​Find the value of the constant a that makes the given function a probability density function on the stated interval.  ​
A) ​ 
B) ​ 
C) ​ 
D) ​ 
E) ​1

Quizplus · Accepted Answer

To be a probability density function, the function must satisfy two conditions:
1) The integral of the function over the given interval must be equal to 1.
2) The function must be non-negative on the given interval.
We have the given function f(x) = a(x^2 - 2x + 2) for 1 ≤ x ≤ 2.
1) Using integration, we get the following:
∫₁² a(x² - 2x + 2) dx = 1
=> [ax³/3 - ax² + 2ax] from 1 to 2 = 1
=> (8a/3 - 8a + 4a/3 - 2a) = 1
=> 2a/3 = 1
=> a = 3/2
2) We can observe that the term (x^2 - 2x + 2) is non-negative on the interval [1, 2]. Therefore, choosing a = 3/2 satisfies both conditions and makes the given function a probability density function.

​Find the Value of the Constant a That Makes the Given

Find the Value of the Constant a That Makes the Given