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Let $X_1 , X_2, X_3$ have the joint cumulative distribution function:

$$ F(X_1 , X_2 , X_3 ) = \left \{ \begin{array}{11} 0 \qquad X_1 < 4 \space \text{or} \space X_2 < 5 \space \text{or} \space X_3 < 6 \\ 1 \qquad X_1 \geq 4 \space \text{and} \space X_2 \geq 5 \space \text{and} \space X_3 \geq 6. \end{array} \right. $$

Find $E[X_1 X_2 X_3]$.

The examples that I have been studying consist of instances only going from a joint PDF to a expected value, so here I am unsure. I believe that I must use the marginals of each of the variables to find the expected value, but do not know how this might be done.

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Since the joint cumulative distribution jumps from 0 to 1, this means that only values occur with the specific limitations: $X_1 = 4, X_2 = 5, X_3 = 6$. As such, we have:

$$E[X_1X_2X_3] = 4 \cdot 5 \cdot 6 = 120$$

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