MATHEMATICS - III
∞ n=0 Pn(cos θ) n + 1 = log{1 + cosec(θ/2)}, 0 < θ ≤ π
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MATHEMATICS - III
$$
\begin{array}{l}
\sum_{r=0}^{n} T_{2 r}(x)=\frac{1}{2}\left(1+\frac{1}{\left(1-x^{2}\right)^{1 / 2}} U_{2 n+1}(x)\right) \\
\text { where, } U_{n}(x)=\sin \left(n \cos ^{-1} x\right) \text { and } T_{n}(x)=\cos \left(n \cos ^{-1} x\right)
\end{array}
$$
numbers 55 and 57 please 55-60 Find the slope of the tangent line to the given...
numbers 55 and 57 please
55-60 Find the slope of the tangent line to the given polar curve at the point specified by the value of θ 55, r 2 cos θ, θ = π/3 57, r-1/0, θ π 59. r= cos 2θ, θ= π/4 60. r= 1 + 2 cos θ, θ= π/3 58. r= cos(93), θ= π
Let f(x; θ) = 1 θ x 1−θ θ for 0 < x < 1, 0...
Let f(x; θ) = 1 θ x 1−θ θ for 0 < x < 1, 0 < θ < ∞.
(1) Show that ˆθ = − 1 n Pn i=1 log(Xi) is the MLE of θ. (2) Show
that this MLE is unbiased.
Exactly 6.4-8. Let f(x:0)-缸붕 for 0 < x < 1,0 < θ < oo 1 1-0 (1) Show that θ Σ-1 log(X) is the MLE of θ (2) Show that this MLE is unbiased.
MATHEMATICS - III
Pn
−
1
2
= P0
−
1
2
P2n
1
2
+ P1
−
1
2
P2n−1
1
2
+ · · · + P2n
−
1
2
P0
1
2
If sin(π/4)=cos(θ) and 0 < θ < π/2, then θ=
If sin(π/4)=cos(θ) and 0 < θ < π/2, then θ=
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Identify each of the following curves using the capital letters A, B, C, D, E, F...
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numbers 55 and 57 please
55-60 Find the slope of the tangent line to the given polar curve at the point specified by the value of θ 55, r 2 cos θ, θ = π/3 57, r-1/0, θ π 59. r= cos 2θ, θ= π/4 60. r= 1 + 2 cos θ, θ= π/3 58. r= cos(93), θ= π
Let f(x; θ) = 1 θ x 1−θ θ for 0 < x < 1, 0 < θ < ∞.
(1) Show that ˆθ = − 1 n Pn i=1 log(Xi) is the MLE of θ. (2) Show
that this MLE is unbiased.
Exactly 6.4-8. Let f(x:0)-缸붕 for 0 < x < 1,0 < θ < oo 1 1-0 (1) Show that θ Σ-1 log(X) is the MLE of θ (2) Show that this MLE is unbiased.
Problem 5. Consider least squares polynomial approximation to f(x) = cos (nx) on x E [-1,1] using the inner product 1. In finding coefficients you will need to compute the integral By symmetry, an 0 for odd n, so we need only consider even n. (a) Make a change of variables and use appropriate identities to transform the integral for a to cos (Bcos 8)cos (ne) de (b) The Bessel function of even order, (x), can be defined by the...
5. Consider Legendre equation for a function y(x) defined in the interval -1. Changing the variable y(cos θ) x cos θ in equation (1) derive the trigonometric form of Legendre equation for a function T (0) where 0 θ π: sin θ Then the general solution to (3) is T (0) y(cos θ) AP, (cos0) + BQ, (cos0).
5. Consider Legendre equation for a function y(x) defined in the interval -1. Changing the variable y(cos θ) x cos θ in...
Problem 3 (12 points) The curve with parametric equations (1 + 2 sin(9) cos(9), y-(1 + 2 sin(θ)) sin(0) is called a limacon and is shown in the figure below. -1 1. Find the point (x,y 2. Find the slope of the line that is tangent to the graph at θ-π/2. 3. Find the slope of the line that is tangent to the graph at (,y)-(1,0) ) that corresponds to θ-π/2.
Problem 3 (12 points) The curve with parametric equations...
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