In a threat-monitoring system, the discrete random variable X represents the number of suspicious packets observed in a single millisecond. A colleague proposes the following function as the probability mass function (PMF) of X:
Which of the following combinations of the normalizing constant C and the probability of observing at least three packets in a millisecond (P[X ≥ 3]) is correct?
C = 1/3 and P[X ≥ 3] = 8/81 (≈ 0.099)
C = 1/3 and P[X ≥ 3] = 8/27 (≈ 0.296)
C = 3 and P[X ≥ 3] = 8/3 (greater than 1)
No finite value of C can normalize the series; the proposed function cannot be a PMF
For a function to be a valid probability mass function (PMF), it must be non-negative and sum to 1 over its support. The sum of the given function, Σ_{k=0}^{∞} C(2/3)^k, is a geometric series. The sum Σ_{k=0}^{∞}(2/3)^k converges to 1 / (1 - 2/3) = 3. To normalize the PMF, we must have C * 3 = 1, so the normalizing constant is C = 1/3.
The probability of observing at least three packets, P[X ≥ 3], is the sum of the probabilities from k=3 onwards. An efficient way to calculate this is by using the complement: P[X ≥ 3] = 1 - P[X < 3]. P[X < 3] = P(X=0) + P(X=1) + P(X=2)P[X < 3] = (1/3)(2/3)^0 + (1/3)(2/3)^1 + (1/3)(2/3)^2 = 1/3 + 2/9 + 4/27 = 19/27. Therefore, P[X ≥ 3] = 1 - 19/27 = 8/27.
The correct combination is C = 1/3 and P[X ≥ 3] = 8/27 ≈ 0.296. The distractor with P[X ≥ 3] = 8/81 can result from a common calculation error.
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