The phase of a sinusoid Theta is uniformly distributed over

The phase of a sinusoid, Theta , is uniformly distributed over [0, 2 pi ) so that its PDF is of the form Find Find

Solution

I will write t for theta here, please bear with the symbol.

a)

Pr(t > 3pi/4) = Integral [1/2pi d(t)] | (3pi/4, 2pi)
= (t / 2pi) | (3pi/4, 2pi)

= [2pi/2pi] - [(3pi/4)/2pi]

= 5/8 = 0.625 [answer]

*************

b)

Pr(t < pi | t > 3pi/4) = Pr(3pi/4<t<pi) / P(t>3pi/4)

= {[pi/2pi] - [(3pi/4)/2pi]} / (5/8)

= 1/5 = 0.2 [answer]

*****************

c)

Pr (cos t < 1/2) = Pr (pi/3 < t < 5pi/3)

= [(5pi/3)/2pi] - [(pi/3)/2pi]

= 2/3 = 0.66666667 [answer]