91Ó°ÊÓ

X ,Y,Z are the non-negative random variables such that \({\rm P}\left( {X + Y + Z < 1.3} \right) = 1\)

If X,Y,Z each random variable have the uniform distribution, then

By the property of expectation

\(E\left( {X + Y + Z} \right) = E\left( X \right) + E\left( Y \right) + E\left( Z \right)\)

\(\begin{align}E\left( X \right) &= \frac{{\left( {b + a} \right)}}{2}\\ &= \frac{{\left( {1 + 0} \right)}}{2}\end{align}\)

\(E\left( X \right) = \frac{1}{2}\)

Similarly\(E\left( Y \right)\)and\(E\left( Z \right)\)is\(\frac{1}{2}\).

therefore

\(\begin{align}E\left( X \right) + E\left( Y \right) + E\left( Z \right) &= \frac{1}{2} + \frac{1}{2} + \frac{1}{2}\\ &= \frac{3}{2}\end{align}\)

But given is\(X + Y + Z < 1.3\)

HenceX, Y ,and Z cannot possibly have a joint distribution under which each of their marginal distributions is the uniform distribution on the interval\(\left( {0,1} \right)\).