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How many of the following situations are possible? The student is encouraged to come up with graphical example(s) in cases where the situation is possible.

i. A random variable $X$ with mean=$0$, has a nonsymmetric probability density function $f(x)$ which is positive in the interval [-5,5] and zero everywhere else.

ii. A probability density function $f(x)$ is nonzero for all real numbers $x$.

iii. A probability density function $f(x)$ is nonzero in interval $[a,b]$ where $a$ and $b$ are real numbers such that $a

iv. A probability density function $f(x)$ is nonzero for all real numbers $x$ except at $x=0$, monotonically increasing in the interval $(-\infty,0)$ and monotonically decreasing in the interval $(0,\infty)$.

v. A random variable $X$ has probability density function $f(x)$ such that $f(0)=10$ if the outcome of $X$ is zero and $f(0)=5$ if the outcome of $X$ is not zero.

A

$0$

B

$1$

C

$2$

D

$3$

E

$4$

F

$5$

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