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?
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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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