So I am studying for a final, and I am doing this problem that says: The radius of a sphere is a random number between $2$ and $4$. What is the expected value of its volume? My first thought was that since were given the probability distribution of the radius $\left(\text{which is}\frac12\right)$ we would just plug $\frac12$ into the radius value of the volume of a sphere formula. But I don't think this is right. Am I doing something wrong?
How do I find the expected volume of a sphere?
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0Kyle, if the formula was linear in $r$, you could get the expected value by setting $r$ to the average radius ($3$), but here larger radii correspond to much larger volumes, so substituting the average radius will significantly underestimate the average volume. – 2012-12-21
1 Answers
Expected values (almost) always correspond to sums or integrals, depending on whether the random variable is discrete or continuous. In this case, the random variable (the radius) is continuous--it can take on any value between 2 and 4--so it's going to be an integral. The form of the integral is $\mathbb E[g(X)] = \int_{-\infty}^{\infty} g(x) f(x) \; dx,$ where $g(X)$ is an arbitrary function of the random variable $X$ and $f(x)$ is the density function. In this case, $g(x)$ is the formula for the volume of a sphere, $g(x) = \frac43 \pi x^3,$ and $f(x) = \frac12$ between 2 and 4, as you noted, and zero everywhere else. So the expected value is going to be $\mathbb E[g(X)] = \int_2^4 \frac43 \pi x^3 \times \frac12 \; dx,$ which you should be able to solve.
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0And the answer is $40\pi$. – 2012-12-21