$X$ is a discrete random variable with possible values $\{-3,-1,0,1,5\}$. The expected value of $X$ is?

Question

BlackTom AI · Accepted Answer

Consider the question: an X that is discrete with possible values {-3, -1, 0, 1, 5}. The prompt asks for the expected value E(X).

Option: 'a. We cannot compute E(X) because we do not know the probability of each outcome.'

First, recall the definition: the expected value E(X) = sum over all x of x * P(X = x). This requires knowing the probability distribution (the probabilities assigned to -3, -1, 0, 1, and 5).

If the probabilities P(X = -3), P(X = -1), P(X = 0), P(X = 1), and P(X = 5) are not provided, then indeed we cannot compute a numeric value for E(X). Without those probabilities, the sum is undefined.

However, consider potential special cases to understand the nuance:
- If the problem had specified a particular distribution (e.g., uniform distribution over the five values), we could compute E(X) by plugging in those probabilities into the formula. For example, under a uniform distribution with P(X = x) = 1/5 for each x, E(X) would be (-3 - 1 + 0 + 1 + 5)/5 = 2/5 = 0.4.
- If the problem provided some probabilities (even if not all), we could compute a partial contribution to E(X) for the known values and note that the unknowns prevent a full calculation.

But given only the set of possible values and no probabilities, the statement in option a correctly identifies the obstacle: the expected value cannot be computed numerically without the probability distribution.

In terms of common misunderstandings: some students might think you can compute E(X) from the values alone, forgetting that the weights (probabilities) are essential. Others might assume a default like equal likelihood for all outcomes; that assumption is not stated, so it would be speculative.

Overall, option a is accurate for the information provided, because E(X) depends on the probabilities assigned to each outcome, which are not given here.

\(X\) is a discrete random variable with possible values \(\{-3,-1,0,1,5\}\). The expected value of \(X\) is?单项选择题

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