Science:Math Exam Resources/Courses/MATH103/April 2016/Question 09 (b)

MATH103 April 2016

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Question 09 (b)
Let $x$ be a continuous random variable taking values in $[0,100]$ with probability density function $p(x)$, mean value $\mu$, variance $\sigma ^{2}$, median value $x_{\text{med}}$, and cumulative function $F(x)$. Write down the integral which expresses $\sigma ^{2}$ in terms of $p(x)$ and $\mu$.

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
Recall the definition of variance for a continuous random variable

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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. By the definition of variance for a continuous random variable, $\sigma ^{2}=\int _{-\infty }^{\infty }(x-\mu )^{2}p(x)\,dx.$ Now since $x\in [0,100],$ $\sigma ^{2}=\color {blue}\int _{0}^{100}(x-\mu )^{2}p(x)\,dx.$