Understanding a Piecewise Probability Density Function and Deriving Its CDF

Question

I have a piecewise Probability Density Function (PDF) defined as follows:

h(x) = { 0 if x < 1

1.5 * (x - 1)^2 if 1 <= x < 2

1.5 * (3 - x)^2 if 2 <= x < 3 0 if x >= 3

}

What is the CDF

Answer 1

Let's denote the CDF as H(x).

1. For x<1: H(x) = 0

1. For 1<=x<2: We integrate 1.5∗(x−1)^2 from 1 to x: H(x) = 0.5 * (x - 1)^3 + 0
2. For 2<=x<3: We integrate 1.5∗(3−x)^2 from 2 to x: H(x) = -0.5 * (3 - x)^3 + C , where C is the constant of integration. To find C, we know that H(2)=0.5 (from the previous step). So, C=0.5+0.5∗(3−2)^3=0.5+0.5=1, H(x) = - 0.5 * (3 - x)^3
3. For x>=3x>=3: H(x) = 1
4. Putting it all together, the CDF H(x)H(x) is:
5. H(x) = { 0 if x < 1

0.5 * (x - 1)^3 if 1 <= x < 2

- 0.5 * (3 - x)^3 if 2 <= x < 3

1 if x >= 3 }

Answer 2