**Mumbai University > Electronics and Telecommunication Engineering > Sem 5 > Random Signal Analysis

Marks: 10M

Year: May 2016

The statistics of P {ζ ∈S : y(t + c, ζ) ∈ A} , can be rewritten as:

P {ζ∈S : T(x(t + c, ζ), ζ) ∈ A} ,

which, for a given memoryless T, can be replaced by:

P {ζ ∈S : x(t + c, ζ) ∈ B} , (1)

where B ≜{x ∈ R: T (x, ζ) ∈ A}.

By the SSS of x(t), (1) is equal to: P {ζ∈ S : x(t, ζ) ∈ B} ,

which in turn equals

P {ζ∈S : T(x(t, ζ), ζ) ∈ A} = P {ζ∈ S : y(t, ζ) ∈ A} .

Since the above proof is valid for any c, the proof holds.