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

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)  •  

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Question 04 (b)
Consider the function $F(x)={\begin{cases}a&{\text{if }}x<0,\\k\arctan x&{\text{if }}0\leq x\leq 1,\\b&{\text{if }}x\geq 1.\end{cases}}$ (b) Let X be the continuous random variable with cummulative distribution function F(x) as given in part (a). Find the probability density function of X.

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
The cumulative distribution function and the probability density function are related by a simple formula that involves an integral. Use the fundamental theorem of calculus to change this integral relationship to one using a derivative instead.

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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. The cumulative distribution is obtained by integrating the probability density function, hence the probability density function (pdf) is the derivative of the cumulative distribution function. From part (a), $F(x)={\begin{cases}0&{\text{if }}x<0,\\{\frac {4}{\pi }}\arctan x&{\text{if }}0\leq x\leq 1,\\1&{\text{if }}x\geq 1.\end{cases}}$ hence the pdf is the derivative: $f(x)=F'(x)={\begin{cases}0&{\text{if }}x<0,\\{\frac {4}{\pi }}{\frac {1}{1+x^{2}}}&{\text{if }}0<x<1,\\0&{\text{if }}x>1.\end{cases}}$ If you are really picky, you might be concerned about the endpoints. Technically the derivative is not defined at the points 0 and 1 (due to "corners"); however, for continuous random variables, the probabilities will be unaffected by isolated points where the derivative of the cumulative distribution function is undefined. How we defined the pdf $f(x)$ above is completely fine.

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