C.D.F ,F(x)

Question

The random variable X has pdf given by
f(x)=(1+x),-1<x≤0 and f(x)=(1-x),0<x<1      &

 

and 

f(x)=(1+x),-1<x<0 and f(x)=(1-x),0<x<1    &&

Could you help me how can I Find C.D.F ,F(x) and are both & and && have the same C.D.F ,F(x).

 

Thanks in advance 

Answer 1

The CDF is just P(X≤x) = ∫^x_-∞ f(t) dt

 

If two PDF's differ in a countable subset of R (which includes a finite set of x's), then the CDF's are identical.

 

This is the case here so the CDFs are the same.

 

Let's compute the CDF common to both cases.

 

Since f(x) = 0 for all x≤0 in both cases, F(x)=0 in (-∞,-1].

 

Next, for (-1,0] region,

 

F(x) = 0 + ∫_-1^x (1+t) dt

 

= (t+t2/2)|_-1^x

 

= (x+x²/2) - ((-1)+(-1)²/2)

 

= x²/2 + x + 1/2

 

= (x+1)²/2

 

Next, for (0,1) region,

 

F(x) = F(0)+ ∫₀^x (1-t) dt

 

= 1/2 + (t - t²/2)|₀^x

 

= 1/2 + x - x²/2

 

= 1 - (x - 1)²/2

 

Finally, F(x) = 1 for the interval [1,∞).