In a binary decision problem, Neyman-Pearson criterion is used to decide which of two signals, one with amplitude A0=−2 and another with amplitude A1=3, is present in zero-mean Gaussian noise of variance σ
2
Z
=5. The probability of false alarm, defined as deciding in favor of A1 when in fact A0 is present, is set to 25%. What is the probability of correct detection (deciding in favor of A1 when A1 is indeed present)? Express your answer in percentage and write it in the space provided below.

Question

In a binary decision problem, Neyman-Pearson criterion is used to decide which of two signals, one with amplitude A0=−2 and another with amplitude A1=3, is present in zero-mean Gaussian noise of variance σ
2
Z
=5. The probability of false alarm, defined as deciding in favor of A1 when in fact A0 is present, is set to 25%. What is the probability of correct detection (deciding in favor of A1 when A1 is indeed present)? Express your answer in percentage and write it in the space provided below.

BlackTom AI · Accepted Answer

We are dealing with a binary detection problem under Gaussian noise. The two hypotheses correspond to the presence of one of two signals: H0 with signal amplitude A0 = −2 and H1 with amplitude A1 = 3, with noise n ~ N(0, σ^2) and σ^2 = 5.

First, recall the Neyman–Pearson setup for Gaussian signals with equal variances: the decision boundary is determined by the likelihood ratio p(x|H1)/p(x|H0) compared to a threshold chosen to meet a specified false-alarm probability P_FA.

Under H0, the observed statistic X is distributed as X ~ N(A0, σ^2) = N(−2, 5).
Under H1, the observed statistic X is distributed as X ~ N(A1, σ^2) = N(3, 5).

The false-alarm probability is defined as P_FA = P(decide H1 | H0) = 0.25. If we assume a simple threshold rule of the form decide H1 if X > t, then P_FA = P0(X > t) where X ~ N(−2, 5).

To find t from P_FA = 0.25, we use the 75th percentile of the N(−2, 5) distribution. The standard deviation is σ = sqrt(5) ≈ 2.236. The 0.75 quantile of a standard normal is z ≈ 0.6745. Thus:

t = A0 + z·σ = −2 + 0.6745·2.236 ≈ −2 + 1.507 ≈ −0.493.

Now, the probability of correct detection P_D (also called probability of detection) is P_H1(X > t) with X ~ N(A1, σ^2) = N(3, 5).

Compute the standardized value under H1:
Z = (t − A1)/σ = (−0.493 − 3) / 2.236 ≈ (−3.493) / 2.236 ≈ −1.561.

Therefore, P_D = P(X > t | H1) = P(Z > −1.561) = Φ(1.561), where Φ is the standard normal CDF.

Looking up or estimating Φ(1.561) gives approximately 0.94 (more precisely around 0.9408).

Therefore, the probability of correct detection is about 94.08% (rounded as requested).

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