- #1
JohanL
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Homework Statement
Let $$(X(n), n ∈ [1, 2])$$ be a stationary zero-mean Gaussian process with autocorrelation function
$$R_X(0) = 1; R_X(+-1) = \rho$$
for a constant ρ ∈ [−1, 1].
Show that for each x ∈ R it holds that
$$max_{n∈[1,2]} P(X(n) > x) ≤ P (max_{n∈[1,2]} X(n) > x)$$
Are there any values of ρ for which this inequality becomes an equality?
Homework Equations
The Attempt at a Solution
$$P(max_{n∈[1,2]} X(n) > x) = P ((X(1) > x) ∪ (X(2) > x)) = $$
$$ = P(X(1) > x) + P(X(2) > x) − P ((X(1) > x) ∩ (X(2) > x)) = $$
$$=2 (1 − Φ(x)) − P ((X(1) > x) ∩ (X(2) > x))$$
I can't figure out what to do with the left side of the inequality or how to use the autocorrelation function.