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 #46336 – Math – Statistics and Probability

Question.

Obtain the mean and variance of the continuous random variable $X$ having the probability density function $f(x) = \begin{cases} 2x, & 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}$.

Solution.

The mean is $EX = \int_{-\infty}^{+\infty} x f(x) dx = \int_{0}^{1} 2x^2 dx = \frac{2}{3} x^3 \big|_{0}^{1} = \frac{2}{3}$.

$$EX^2 = \int_{-\infty}^{+\infty} x^2 f(x) dx = \int_{0}^{1} 2x^3 dx = \frac{2}{4} x^4 \big|_{0}^{1} = \frac{1}{2}.$$

The variance is $VarX = EX^2 - (EX)^2 = \frac{1}{2} - \frac{4}{9} = \frac{1}{18}$.

Answer. $EX = \frac{2}{3}$, $VarX = \frac{1}{18}$.

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