Science:Math Exam Resources/Courses/MATH105/April 2012/Question 04 (a)/Solution 1

Following the definition of a cdf (see the hint), we compute

• $\lim _{x\rightarrow -\infty }F(x)=a$ so $a=0$

and

• $\lim _{x\rightarrow \infty }F(x)=b$ so $b=1$.

• To impose continuity at $x=1$, we need $\lim _{x\rightarrow 1}F(x)=F(1)=b=1$. Obviously $\lim _{x\rightarrow 1^{+}}F(x)=1$. So looking at the left limit will give us k:

$\lim _{x\rightarrow 1^{-}}F(x)=\lim _{x\rightarrow 1^{-}}k\arctan x=k\arctan 1$

Therefore k = 1/(arctan 1), or simply $k={\frac {4}{\pi }}$.

• Since arctan is non-decreasing, so is F.

Hence our final answer is

$a=0,\quad b=1,\quad k={\frac {4}{\pi }}.$

As a remark, it is important to know the exact trig values for special angles. Here: $\tan {\frac {\pi }{4}}=1$ so $\arctan 1={\frac {\pi }{4}}$.