Question

Three numbers \(X\), \(Y\), \(Z\) are randomly drawn from the set \(\{1, 2, 3, 4\}\). \(E(XYZ)\) is equal to:

• A. \(15\dfrac{5}{8}\)
• B. \(12\dfrac{1}{2}\)
• C. 12
• D. \(13\dfrac{1}{8}\)

Hint:

Treat \(X\), \(Y\), \(Z\) as independent uniform draws on \(\{1,2,3,4\}\) and use \(E(XYZ) = E(X)E(Y)E(Z)\).

Answer 1

The Correct Option is A

Step 1: Since \(X\), \(Y\), \(Z\) are each randomly drawn from the same set \(\{1,2,3,4\}\) independently of one another, each is a discrete uniform random variable on \(\{1,2,3,4\}\), and being independent draws, \(X\), \(Y\), \(Z\) are mutually independent random variables.

Step 2: For independent random variables, the expectation of the product equals the product of the expectations: \[E(XYZ) = E(X)\,E(Y)\,E(Z)\]

Step 3: Find \(E(X)\) for a uniform distribution on \(\{1,2,3,4\}\): \[E(X) = \frac{1+2+3+4}{4} = \frac{10}{4} = 2.5\] Similarly \(E(Y) = E(Z) = 2.5\).

Step 4: Multiply: \[E(XYZ) = (2.5)(2.5)(2.5) = (2.5)^3 = 15.625\]

Step 5: Convert to a mixed fraction: \[15.625 = 15\frac{5}{8}\]

So \(E(XYZ) = 15\dfrac{5}{8}\), which is option (A).