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Statistics and Probability

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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 $x^2f(x)$.

A

$0$

B

$1$

C

$2$

D

$3$

E

$4$

F

$5$