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 > 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`