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

MATH105 April 2012

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q3 (a)  •  Q3 (b)  •  Q4 (a)  •  Q4 (b)  •  Q5 (a)  •  Q5 (b)  •  Q6 (a)  •  Q6 (b)  •  Q7 (a)  •  Q7 (b)  •  Q8 (a)  •  Q8 (b)  •  Q8 (c)  •  Q8 (d)  •  Q8 (e)  •  Q8 (f)  •  Q8 (g)  •  Q8 (h)  •  Q8 (i)  •  

   Other MATH105 Exams

  •  April 2009  •  April 2011  •  April 2010  •  April 2012  •  April 2013  •  April 2014  •  April 2015  •  April 2016  •  April 2017  •  April 2018  •  

Question 04 (a)
Consider the function $F(x)={\begin{cases}a&{\text{if }}x<0,\\k\arctan x&{\text{if }}0\leq x\leq 1,\\b&{\text{if }}x\geq 1.\end{cases}}$ (a) Find the values of a, k and b for which F is a valid cummulative distribution function of a continuous random variable. Then sketch the graph of F.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
A cumulative distribution function F of a continuous random variable must satisfy the following: $\lim _{x\to -\infty }F(x)=0$ $\lim _{x\to +\infty }F(x)=1$ F is continuous F is non-decreasing

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. 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 $\arctan 1={\frac {\pi }{4}}$.

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Probability density function, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag