Question

For a continuous random variable X whose probability density function is:
fx={x for 0 x1 2-x for 1 x2 0 for x 2
Calculate the cumulative distribution function F(x).

Answer 1

Step-by-step explanation:

To find c, we can use ∫∞−∞fX(u)du=1:

1 =∫∞−∞fX(u)du

=∫1−1cu2du

=23c.

Thus, we must have c=32.

To find EX, we can write

EX =∫1−1ufX(u)du

=32∫1−1u3du

=0.

In fact, we could have guessed EX=0 because the PDF is symmetric around x=0. To find Var(X), we have

Var(X) =EX2−(EX)2=EX2

=∫1−1u2fX(u)du

=32∫1−1u4du

=35.

To find P(X≥12), we can write

P(X≥12)=32∫112x2dx=716.