Suppose q=0.3. Then, the expected value of X is 2. What is the standard deviation of X?

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BlackTom AI · Accepted Answer

The question states: 'Suppose q=0.3. Then, the expected value of X is 2. What is the standard deviation of X?' However, it does not specify the distribution of X or any further parameters beyond E[X] = 2 and q = 0.3. Without knowing the underlying distribution (e.g., Poisson, Binomial, Geometric, Negative Binomial, etc.) and its parameters, the standard deviation cannot be uniquely determined.

Consider a few common scenarios to illustrate why the answer is not uniquely fixed:
- If X ~ Poisson(λ) with E[X] = λ, then E[X] = 2 implies λ = 2, and the standard deviation would be sqrt(λ) = sqrt(2) ≈ 1.414.
- If X ~ Binomial(n, p) with E[X] = np = 2 and p = 0.3 (as implied by q = 0.3 if q denotes 1 − p), then n = 2 / 0.3 ≈ 6.667, which is not an integer. In a valid Binomial model, n must be a nonnegative integer, so this combination is not feasible as stated. If we attempt to approximate with a feasible n close to 6 or 7, the resulting standard deviation would be sqrt(np(1 − p)) ≈ sqrt(2 × 0.7) ≈ sqrt(1.4) ≈ 1.183, which differs from 1.095.
- If X ~ Geometric or Negative Binomial, the relationship between mean and standard deviation follows different formulas, and without exact parameters that also relate to q = 0.3, we cannot pin down σ.

Because only E[X] = 2 and q = 0.3 are given, there are infinitely many distributions and parameterizations that could yield the same mean but different standard deviations. Therefore, the question as stated does not provide enough information to determine the standard deviation uniquely.

If you had additional details, such as:
- X ~ Poisson with λ = 2, then σ = sqrt(2) ≈ 1.414.
- X ~ Binomial(n, p) with specific integer n and p = 0.3 that satisfy np = 2 exactly (which would require a feasible integer n), then σ = sqrt(np(1 − p)) would be computed accordingly.
- X ~ another distribution with a known relationship between mean and variance, you could compute σ directly from that distribution’s parameters.

In short, with the information provided, there is no unique standard deviation to report. The result 1.0954 would require a specific distribution and parameter set not supplied in the prompt.

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

Suppose q=0.3. Then, the expected value of X is 2. What is the standard deviation of X?

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