Show f is a probability density function

[saizen21]

Homework Statement

Let f(x) = (1 + ux)/2 for -1<= x <= 1 and 0 otherwise . where -1<= u <= 1

a) show f is a density

Homework Equations

TO show
1. f(x) >= 0

2. intergeral f (from -infinity to infinity) = 1

The Attempt at a Solution

I have done 2. and proved that it is 1 by taking the intergeral of f.

However, how do u show f >= 0.

i have found the dervavtive of f to be u / 2 for -1<= x <= 1 and 0 otherwise.

I dun know where to go?

 

[Dick]

If you've shown f'(x)=u/2 then that means the max and min of f(x) must be at x=(-1) or x=1, the boundaries of your interval. Can you take it from there?

 

[saizen21]

is this also possible?

1<=x<=1
-1<=ux<=1
0<=1+ux<=2
0<=(1+ux)/2<=1
therefore its always >= 0

 

[Dick]

is this also possible?

1<=x<=1
-1<=ux<=1
0<=1+ux<=2
0<=(1+ux)/2<=1
therefore its always >= 0

That works too.