A random variable X has density f(x) = 3x² for 0 ≤ x ≤ 1 and 0 elsewhere. Find E[X]. Give the answer as a fraction in lowest terms.

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Mean from a density

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A random variable $X$ has density $f (x) = 3 x^{2}$ for $0 \leq x \leq 1$ and $0$ elsewhere. Find $E [X]$ . Give the answer as a fraction in lowest terms.

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Hint

The mean is the integral of $x$ times the density.

Solution

By definition, $E [X]$ is the integral of $x \times 3 x^{2} = 3 x^{3}$ from $0$ to $1$ .
An antiderivative of $3 x^{3}$ is $\frac{3 x ^{4}}{4}$ .
So $E [X] = \frac{3}{4} - 0 = \frac{3}{4}$ .

Answer: $\frac{3}{4}$

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