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Multivariable Calculus

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Moderate

Joint Probability Density

MVCALC-EYCGDL

Let the joint probability density function for random variables $X$, $Y$ and $Z$ be given by:

$$\begin{align*} f(x,y,z)= \begin{cases} C(x+2y+3z), &\text{ when } (x,y,z) \in [0,2]\times [0,2] \times [0,2] \\\ 0, &\text{ elsewhere} \end{cases} \end{align*}$$

Find the value of the constant $C$.

A

$-1$.

B

$0$

C

$1$

D

$\frac{1}{12}$

E

$\frac{1}{48}$