Let s=0.25. What is the expected value of UC-UM?

Question

BlackTom AI · Accepted Answer

The prompt provides a short-answer question and an answer of 3.0, but there are no answer options to analyze. To properly coach a student, we need to restate what is given and identify what information is missing before evaluating or verifying the proposed result.

What is given:
- The variable s is specified as 0.25.
- The question asks for the expected value of UC-UM.
- The provided answer is 3.0.

What is missing and needed to analyze correctly:
- A clear definition of UC and UM. Are these random variables, constants, or functions of s, time, or other parameters? Do they depend on outcomes, states, or probability distributions?
- The probabilistic framework: what is the distribution over which the expectation E[UC-UM] is taken? For example, are there discrete states with probabilities, or is there a continuous distribution?
- How s enters the expressions for UC and UM. Is s a parameter that shifts or scales UC and/or UM, or does it determine which case of a piecewise definition applies?
- Any equations or relationships that link UC, UM, and s. Without those, we cannot compute an expected value or verify whether 3.0 is correct.

Generic approach to compute E[UC-UM] when definitions are provided:
- Step 1: Write the expression for UC-UM for each outcome or state. This yields a random variable Z = UC - UM.
- Step 2: Identify the probability distribution over the outcomes or states. Let p_i denote the probability of outcome i.
- Step 3: Compute the expected value as E[Z] = sum_i p_i * Z_i for discrete outcomes, or E[Z] = ∫ Z(x) f_X(x) dx for continuous cases, where Z_i or Z(x) is UC_i - UM_i.
- Step 4: Substitute s = 0.25 into UC and UM expressions if they depend on s, and simplify to obtain the numeric value of E[UC-UM].

Potential pitfalls or misconceptions to watch for:
- Assuming UC and UM are independent of s when they actually depend on it; failing to substitute s correctly can lead to wrong results.
- Treating E[UC-UM] as E[UC] - E[UM] unless UC and UM are independent; in general E[UC-UM] = E[UC] - E[UM] holds, but you still need the separate expectations and the underlying distributions.
- Confusing units or scales: a numeric result like 3.0 may come from unit-consistent quantities, but without definitions, the meaning remains unclear.

If you can provide the definitions of UC and UM, the distribution or sample space, and how s=0.25 influences those quantities, I can walk through the exact calculation step by step and verify whether the final result should be 3.0 or another value. Until then, the given answer cannot be validated or debunked with the information at hand.

COMM_V 190 101-104 2025W1 Final Exam - Practice Questions - Second Half

Let s=0.25. What is the expected value of UC-UM?

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