The Student Room Group

Variance of Cos(T)

hello

this is the question:

"An arrow, pivoted at its centre, is spun and the angle T, between the arrow and 'due North' is noted.

Suppose T is uniformly distributed between 0 and 360°.

Show that on average it points due South!

Evaluate Var(T) and calculate the probability that T points between SouthEast and SouthWest.

Find Var(Cos(T)) and E(e^(iT))"



i have done all the parts up to Var(Cos(T))
how do i do this part?

i need to work out E(cosT) and E(cos²T) but am confused on how to work those out.

please could u give me some pointers on how to do this question - preferable not just the answer

thank you :smile:
Reply 1
Note that:

eiT = Cos(T) + iSin(T)
Reply 2
Wrangler
Note that:

eiT = Cos(T) + iSin(T)


what is 'i'?
Reply 3
E(cos(T)) = 12π\frac{1}{2\pi}02π\int_0^{2\pi} cos(t) dt
E(cos^2(T)) = 12π\frac{1}{2\pi}02π\int_0^{2\pi} cos^2(t) dt
Reply 4
Jonny W
E(cos(T)) = 12π\frac{1}{2\pi}02π\int_0^{2\pi} cos(t) dt
E(cos^2(T)) = 12π\frac{1}{2\pi}02π\int_0^{2\pi} cos^2(t) dt


thanks - thats what i tried but i used the wrong limits: -1 and 1 :redface: