Question

A continuous random variable X has cdf F(x)=x² b (a) Determine the constants a and b. for a < 0, for 0 < x < 1, for x &gt; 1.​

Answer 1

Any proper CDF $F(x)$ has the properties

• $\displaystyle \lim_{x\to-\infty} F(x) = 0$

• $\displaystyle \lim_{x\to+\infty} F(x) = 1$

so we have to have a = 0 and b = 1.

This follows from the definitions of PDFs and CDFs. The PDF must satisfy

$\displaystyle \int_{-\infty}^\infty f(x) \, dx = 1$

and so

$\displaystyle \lim_{x\to-\infty} F(x) = \int_{-\infty}^{-\infty} f(t) \, dt = 0 \implies a = 0$

$\displaystyle \lim_{x\to+\infty} F(x) = \int_{-\infty}^\infty f(t) \, dt = 1 \implies b = 1$