If fx and Fx are the values of the probability density and t

If f(x) and F(x) are the values of the probability density and the distribution function of X at x, then

P(a<=X<=b)=F(b)- F(a) for any real constants a and b with a<=b, and f(x)=dF(x)/dx where the derivative exists.

Using this theorem, solve the following problem:

The distribution function of the random variable X is given by

Find P(-1/2<X<1/2) and P(2<X<3).

Find the pdf of X

1) P(-1/2<X<1/2) = F(1/2) - F(-1/2) = (1/2 + 1)/2 - (-1/2 + 1)/2 = 1/2 = 0.5

P(2<X<3) = F(3) - F(2) = 1 - 1 = 0

2) pdf is

f(x) = 1/2 , -1 < x < 1

0 , otherwise