17) the sequence is definitely not converging so it doesn't have a limit. but there are 2 limit points 1 and -1.
u can read the difference between limit and limit points.
26) f(Ө)= 1/pi ;-pi/2<Ө<pi/2
= 0 ; elsewhere
P(X<x) = P(sinӨ<x) =P(Ө< sin inverse x) = Integral [ x=-pi/2 to x= sin inverse x] 1/pi dӨ = <b>(sin inverse x)/pi + 1/2
27) X~N(0, σ²)
(X-0)/ σ² ~ N(0,1)
X^2/ σ² ~ chi square (1) square of standard normally distributed rv is chi square distribution with r =1
so E[X^2/ σ²]=1 mean of chi square is = r (which is 1 in this case)
E[X^2] = σ²
we know that |x|-|y|<=|x+y|<=|x|+|y|
replacing y with -y
so (i) is true
(ii) is false. Check with x=2 and y=-3
54) It was explained somewhere in this forum quite formally.
"You don't have to believe in God, but you should believe in The Book." -Paul Erdős