The discussion focuses on verifying the solution for the probability density function (PDF) and its corresponding cumulative distribution function (CDF) and inverse CDF calculations. The provided PDF is defined as p(x) = 0 for x < 0, 4x for 0 ≤ x < 0.5, and -4x + 4 for 0.5 ≤ x < 1. The CDF derived is CDF = 0 for x < 0, 2x² for 0 ≤ x < 0.5, -2x² + 4x - 1 for 0.5 ≤ x ≤ 1, and 1 for x > 1. The inverse CDF was noted to have an error in the root selection for the interval 0.5 ≤ x ≤ 1, which could yield values greater than 1.
PREREQUISITESStudents studying probability theory, statisticians verifying statistical models, and educators teaching concepts of PDFs, CDFs, and inverse functions.
You chose the wrong root for ##.5 \leq x \leq 1##. Can you see why?zzmanzz said:Homework Statement
I was hoping someone could just verify this solution is accurate.
p(x) =
0 , x < 0
4x, x < .5
-4x + 4 , .5 <= x < 1
Find CDF and Inverse of the CDF.
Homework Equations
The Attempt at a Solution
CDF =
0 , x < 0
2x^2 , 0 <= x < .5
-2x^2 + 4x - 1 , .5 <= x <= 1
1, x > 1
Inverse of the CDF
0 , x < 0
sqrt( x / 2) , 0 <= x < .5
1 + sqrt ( 1 - x) / sqrt ( 2 ) , .5 <= x <= 1
1, x > 1
Thanks[/B]
Ray Vickson said:You chose the wrong root for ##.5 \leq x \leq 1##. Can you see why?
zzmanzz said:Thanks for pointing that out. The value for that should also be between .5 and 1, and with that root, it can be greater than 1?
Thank you!Ray Vickson said:Yes, just look at it: you have ##1 + \text{something positive}##.