Science:Math Exam Resources/Courses/MATH105/April 2013/Question 01 (m)

MATH105 April 2013

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Question 01 (m)
Short-Answer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. Compute the cumulative distribution function corresponding to the probability density function $f(x)=3x^{-4},\quad x\geq 1.$

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
How do you define the cumulative distribution function in terms of the probability density function?

Hint 2
Remember that the cumulative distribution function, $F(x)$, is defined in terms of the probability density function, $f(x)$, as follows: $F(t)=\int _{-\infty }^{t}f(x)\,dx.$

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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. Remember that the cumulative distribution function, $F(x)$, is defined in terms of the probability density function, $f(x)$, as follows: $F(t)=\int _{-\infty }^{t}f(x)\,dx.$ Hence, to determine the cumulative distribution function for the given probability density function is a matter of plugging the particular density function into the definition. (Remember that since the given density function is only defined for $x\geq 1$, we can assume that the density function evaluates to 0 for all $\displaystyle x<1$): $F(t)=\int _{-\infty }^{t}f(x)\,dx=\int _{1}^{t}3x^{-4}\,dx=-x^{-3}|_{1}^{t}=1-t^{-3}.$ Therefore, the cumulative distribution is ${\color {blue}F(t)=1-{\frac {1}{t^{3}}}}$

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