Moderate# Mean and Variance based on Graph of Probability Density Function

STATS-EJRF4X

Suppose and experiment $X$ has probability density function $f(x)$ given by:

How many of the following statements are **TRUE**?

i.The mean of $X$ is greater than zero.

ii.The output of $x^3f(x)$ can be negative for some $x$ value.

iii.In order to obtain an approximate value of $\int_{-\infty}^{\infty}x^2f(x)dx$ we can carry out experiment $X$ 100 million times, record the outcomes and obtain the mean of this dataset.

iv.If drawn, the curve $xf(x)$ will never be below the $x$-axis.

v.The variance of $X$ is found by computing the area under the curve $x^2f(x)$.