Question

Obtain the mean and variance of the discrete random variable X having the probability density function f( x ) = 2x for 0 ≤ x ≤ 1 and 0 otherwise.

Answer 1

Answer on Question #47994-Geography-Other

Obtain the mean and variance of the continuous random variable $X$ having the probability density function $f(x) = 2x$ for $0 \leq x \leq 1$ and 0 otherwise.

Solution

The mean of the continuous random variable $X$ is

$$\mu = \int_{-\infty}^{\infty} x f(x) \, dx = \int_{0}^{1} x \cdot 2x \, dx = 2 \left(\frac{x^3}{3}\right)_{0}^{1} = \frac{2}{3}.$$

The variance of the continuous random variable $X$ is

$$\sigma^{2} = \int_{-\infty}^{\infty} x^{2} f(x) \, dx - \mu^{2}.$$ $$\int_{-\infty}^{\infty} x^{2} f(x) \, dx = \int_{0}^{1} x^{2} \cdot 2x \, dx = 2 \left(\frac{x^4}{4}\right)_{0}^{1} = \frac{1}{2}.$$ $$\sigma^{2} = \frac{1}{2} - \left(\frac{2}{3}\right)^{2} = \frac{1}{18}.$$

Answer: $\frac{2}{3}$ and $\frac{1}{18}$.

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