Question

2.   The joint density function for random variables X, Y, and Z is

f (x,y,z) = {Cxyz, if 0 ≤ x ≤ 2, 0 ≤ y ≤ 2, 0 ≤ z ≤ 2

0, otherwise

a)   Find the value of C.

b) Find P(X ≤ 1,Y ≤ 1, Z ≤ 1).

Answer 1

Here, $f(x,y,z)$ is a probability density function.

So,

a) $\int_0^2\int_0^2\int_0^2 Cxyzdzdydx=1$

$\implies$ $\int_0^2\int_0^2\left\{\frac{Cxyz²}{2}\right\}_0^2dydx=1$

$\implies$ $\int_0^2\int_0^22Cxydydx=1$

$\implies$ $\int_0^2\left\{Cxy²\right\}_0^2dx=1$

$\implies$ $\int_0^24Cxdx=1$

$\implies$ $\left\{\frac{4Cx²}{2}\right\}_0^2=1$

$\implies$ $\left\{2Cx²\right\}_0^2=1$

$\implies$ $2C(2)²-2C(0)²=1$

$\implies$ $8C=1$

$\implies$ $C=\frac{1}{8}$

b) $P(X≤1,Y≤1,Z≤1)$

$=\int_0^1\int_0^1\int_0^1\frac{xyz}{8}dzdydx$

$=\int_0^1\int_0^1\left\{\frac{xyz²}{16}\right\}_0^1dydx$

$=\int_0^1\int_0^1\frac{xy}{16}dydx$

$=\int_0^1\left\{\frac{xy²}{32}\right\}_0^1dx$

$=\int_0^1\frac{x}{32}dx$

$=\left\{\frac{x²}{64}\right\}_0^1=\frac{1}{64}$