Question

how to find distribution function from probability density function​

Answer 1

Given

f(x)=⎧⎩⎨x,0<x<12−x,1≤x<20 everywhere else

as our P.D.F, I must find the corresponding distribution function.

I know that F(x)=P(X≤x)=∫x−∞f(t)dt is the distribution function, but I don't know how to apply it to this particular probability density function. The P.D.F has three different cases. How do I handle that? A sum of integrals with the appropriate bounds for each case of f(x)?

I'm kind of confused on how to create those bounds. How do we get rid the of the −∞?

I'm thinking:

For x>0:

F(x)=P(X≤x)=∫x−∞f(t)dt=∫10xdx+∫212−xdx

Answer2 Discern the following cases:

x≤0

F(x)=∫x−∞0dt=0

0<x≤1

F(x)=F(0)+∫x0tdt

1<x≤2

F(x)=F(1)+∫x1(2−t)dt

x>2

F(x)=F(2)=1

Answer 3

Given

f(x)=⎧⎩⎨x2−x00<x<11⩽x<2 everywhere else

Then

F(x)=⎧⎩⎨⎪⎪⎪⎪⎪⎪0∫x0xdxF(1)+∫x12−xdx1x⩽00<x<11⩽x<22⩽x

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