Question

Assume sin(θ)=19/23 where 2π<θ<5π/2. Compute sin(θ/2).

Answer #1

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a.
r=3 - 3cos(Θ), enter value for r on a table
when;
Θ=0, (π/3),(π/2),(2π/3),π,(4π/3),(3π/2),(5π/3) & 2π
b. plot points from a, sketch graph
c. use calculus to find slope at (π/2),(2π/3),(5π/3)
& 2π
d. find EXACT area inside the curve in 1st
quadrant

Assume sin(θ)=19/36 where π2<θ<π, and cos(ϕ)=−11/38 where
π<ϕ<3π/2. Use sum or difference formula to compute:
sin(θ +ϕ)=
sin(θ -ϕ)=
cos(θ +ϕ)
cos(θ -ϕ)=

If θ is in the interval [0, 2π) and cos(θ) = √ 2/2 , then θ must
be π/4 .
State true or false.

Find the arc length of r = θ 2 from θ = 2π to θ = 6π.

Find the dimension of the subspace U = span {1,sin^2(θ), cos 2θ}
of F[0, 2π]

Evaluate, in spherical coordinates, the triple integral of
f(ρ,θ,ϕ)=cosϕ, over the region 0≤θ≤2π, π/6≤ϕ≤π/2, 3≤ρ≤8.
integral =

For the following exercises, find all exact solutions on [0,
2π)
23. sec(x)sin(x) − 2sin(x) = 0
25. 2cos^2 t + cos(t) = 1
31. 8sin^2 (x) + 6sin(x) + 1 = 0
32. 2cos(π/5 θ) = √3

The random variable X is distributed with pdf fX(x,
θ) = (2/θ^2)*x*exp(-(x/θ)2), where x>0 and
θ>0. Please note the term within the exponential is
-(x/θ)^2 and the first term includes a θ^2.
a) Find the distribution of Y = (X1 + ... +
Xn)/n where X1, ..., Xn is an
i.i.d. sample from fX(x, θ). If you can’t find Y, can
you find an approximation of Y when n is large?
b) Find the best estimator, i.e. MVUE, of θ?

The random variable X is distributed with pdf
fX(x, θ) = c*x*exp(-(x/θ)2), where x>0
and θ>0. (Please note the equation includes the term
-(x/θ)2 )
a) What is the constant c?
b) We consider parameter θ is a number. What is MLE and MOM of
θ? Assume you have an i.i.d. sample. Is MOM unbiased?
c) Please calculate the Cramer-Rao Lower Bound (CRLB). Compare
the variance of MOM with Crameer-Rao Lower Bound (CRLB).

The random variable X is distributed with pdf fX(x,
θ) = c*x*exp(-(x/θ)2), where x>0 and θ>0. (Please
note the equation includes the term -(x/θ)2 )
Calculate the probability of X1 < X2,
i.e. P(X1 < X2, θ).

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