Answer:
y = 2/3x+2
Step-by-step explanation:
The answer is letter B. This is through using the formula of compound interest which is divided in quarterly. This means the number of months are only 3. SO compound interest is equal to Principal amount(1+rate/number of months)^number of months x time. This will give you the answer of <span>$3,081.54.</span>
Answer:
A = 18 units²
Step-by-step explanation:
The area (A) of a trapezium is calculated as
A =
h ( b₁ + b₂ )
where h is the perpendicular height and b₁, b₂ the parallel bases
By counting squares
h = 3, b₁ = 3, b₂ = 9 , then
A =
× 3 × (3 + 9) = 1.5 × 12 = 18 units²
Answer:
The method of moment (MOM) estimator as: ![\mathbf{\hat {\theta} =(\dfrac{\overline X}{1-\overline X})^2}](https://tex.z-dn.net/?f=%5Cmathbf%7B%5Chat%20%7B%5Ctheta%7D%20%3D%28%5Cdfrac%7B%5Coverline%20X%7D%7B1-%5Coverline%20X%7D%29%5E2%7D)
![\overline X = \dfrac{4}{9}](https://tex.z-dn.net/?f=%5Coverline%20X%20%3D%20%5Cdfrac%7B4%7D%7B9%7D)
![\mathbf{\hat {\theta} =\dfrac{16}{25} }](https://tex.z-dn.net/?f=%5Cmathbf%7B%5Chat%20%7B%5Ctheta%7D%20%3D%5Cdfrac%7B16%7D%7B25%7D%20%7D)
Step-by-step explanation:
From the question, the correct format for the probability density function is:
![fx(x ; \theta) = \left \{ {{\sqrt{\theta x}^{\sqrt{\theta}-1}}\ \ 0 \leq x \leq 1 \atop {0} \ \ \ \ \ \ \ otherwise } \right.](https://tex.z-dn.net/?f=fx%28x%20%3B%20%5Ctheta%29%20%3D%20%5Cleft%20%5C%7B%20%7B%7B%5Csqrt%7B%5Ctheta%20x%7D%5E%7B%5Csqrt%7B%5Ctheta%7D-1%7D%7D%5C%20%5C%20%200%20%5Cleq%20x%20%5Cleq%20%201%20%5Catop%20%7B0%7D%20%5C%20%20%5C%20%20%20%5C%20%5C%20%20%5C%20%5C%20%20%5C%20otherwise%20%7D%20%5Cright.)
where θ > 0 is an unknown parameter.
(a) The MOM estimator can be calculated as follows:
![E(X) = \int ^1_0x. \sqrt{\theta} \ x^{\sqrt{\theta}-1} \ dx](https://tex.z-dn.net/?f=E%28X%29%20%3D%20%5Cint%20%5E1_0x.%20%5Csqrt%7B%5Ctheta%7D%20%5C%20x%5E%7B%5Csqrt%7B%5Ctheta%7D-1%7D%20%5C%20dx)
![E(X) = \int ^1_0 \sqrt{\theta} \ x^{\sqrt{\theta}} \ dx](https://tex.z-dn.net/?f=E%28X%29%20%3D%20%5Cint%20%5E1_0%20%5Csqrt%7B%5Ctheta%7D%20%5C%20x%5E%7B%5Csqrt%7B%5Ctheta%7D%7D%20%5C%20dx)
![E(X) = \dfrac{\sqrt{\theta} }{\sqrt{\theta} +1 } ( x ^{\sqrt{\theta}+1})^1_0](https://tex.z-dn.net/?f=E%28X%29%20%3D%20%5Cdfrac%7B%5Csqrt%7B%5Ctheta%7D%20%7D%7B%5Csqrt%7B%5Ctheta%7D%20%2B1%20%7D%20%28%20x%20%5E%7B%5Csqrt%7B%5Ctheta%7D%2B1%7D%29%5E1_0)
![E(X) = \dfrac{\sqrt{\theta} }{\sqrt{\theta} +1 }](https://tex.z-dn.net/?f=E%28X%29%20%3D%20%5Cdfrac%7B%5Csqrt%7B%5Ctheta%7D%20%7D%7B%5Csqrt%7B%5Ctheta%7D%20%2B1%20%7D)
suppose E(X) = ![\overline X](https://tex.z-dn.net/?f=%5Coverline%20X)
Then;
![\overline X = \dfrac{\sqrt{\theta} }{\sqrt{\theta} +1 }](https://tex.z-dn.net/?f=%5Coverline%20X%20%3D%20%5Cdfrac%7B%5Csqrt%7B%5Ctheta%7D%20%7D%7B%5Csqrt%7B%5Ctheta%7D%20%2B1%20%7D)
![\dfrac{1}{\overline X} = \dfrac{\sqrt{\theta} +1 }{\sqrt{\theta}}](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B%5Coverline%20X%7D%20%3D%20%5Cdfrac%7B%5Csqrt%7B%5Ctheta%7D%20%2B1%20%20%7D%7B%5Csqrt%7B%5Ctheta%7D%7D)
![\dfrac{1}{\overline X} =1 + \dfrac{1}{\sqrt{\theta}}](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B%5Coverline%20X%7D%20%3D1%20%2B%20%20%5Cdfrac%7B1%7D%7B%5Csqrt%7B%5Ctheta%7D%7D)
making
the subject of the formula, we have:
![\dfrac{1}{\sqrt{\theta}} =\dfrac{1}{\overline X} - 1](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B%5Csqrt%7B%5Ctheta%7D%7D%20%3D%5Cdfrac%7B1%7D%7B%5Coverline%20X%7D%20%20-%201)
![\dfrac{1}{\sqrt{\theta}} =\dfrac{1-\overline X}{\overline X}](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B%5Csqrt%7B%5Ctheta%7D%7D%20%3D%5Cdfrac%7B1-%5Coverline%20X%7D%7B%5Coverline%20X%7D)
![\sqrt{\theta} =\dfrac{\overline X}{1-\overline X}](https://tex.z-dn.net/?f=%5Csqrt%7B%5Ctheta%7D%20%3D%5Cdfrac%7B%5Coverline%20X%7D%7B1-%5Coverline%20X%7D)
squaring both sides, we have:
The method of moment (MOM) estimator as: ![\mathbf{\hat {\theta} =(\dfrac{\overline X}{1-\overline X})^2}](https://tex.z-dn.net/?f=%5Cmathbf%7B%5Chat%20%7B%5Ctheta%7D%20%3D%28%5Cdfrac%7B%5Coverline%20X%7D%7B1-%5Coverline%20X%7D%29%5E2%7D)
b) If the observations are ![\dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{2}](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B2%7D%2C%20%5Cdfrac%7B1%7D%7B3%7D%2C%20%5Cdfrac%7B1%7D%7B2%7D)
Then,
![\overline X = \dfrac{\dfrac{1}{2}+ \dfrac{1}{3}+\dfrac{1}{2}}{3}](https://tex.z-dn.net/?f=%5Coverline%20X%20%3D%20%5Cdfrac%7B%5Cdfrac%7B1%7D%7B2%7D%2B%20%5Cdfrac%7B1%7D%7B3%7D%2B%5Cdfrac%7B1%7D%7B2%7D%7D%7B3%7D)
![\overline X = \dfrac{\dfrac{3+2+3}{6}}{3}](https://tex.z-dn.net/?f=%5Coverline%20X%20%3D%20%5Cdfrac%7B%5Cdfrac%7B3%2B2%2B3%7D%7B6%7D%7D%7B3%7D)
![\overline X = \dfrac{\dfrac{8}{6}}{3}](https://tex.z-dn.net/?f=%5Coverline%20X%20%3D%20%5Cdfrac%7B%5Cdfrac%7B8%7D%7B6%7D%7D%7B3%7D)
![\overline X = \dfrac{8}{6} \times \dfrac{1}{3}](https://tex.z-dn.net/?f=%5Coverline%20X%20%3D%20%5Cdfrac%7B8%7D%7B6%7D%20%5Ctimes%20%5Cdfrac%7B1%7D%7B3%7D)
![\overline X = \dfrac{8}{18}](https://tex.z-dn.net/?f=%5Coverline%20X%20%3D%20%5Cdfrac%7B8%7D%7B18%7D)
![\overline X = \dfrac{4}{9}](https://tex.z-dn.net/?f=%5Coverline%20X%20%3D%20%5Cdfrac%7B4%7D%7B9%7D)
Finally, the point estimate of the estimator
is
![\mathbf{\hat {\theta} =\begin {pmatrix} \dfrac{\dfrac{4}{9}}{1-\dfrac{4}{9}} \end {pmatrix}^2}](https://tex.z-dn.net/?f=%5Cmathbf%7B%5Chat%20%7B%5Ctheta%7D%20%3D%5Cbegin%20%7Bpmatrix%7D%20%5Cdfrac%7B%5Cdfrac%7B4%7D%7B9%7D%7D%7B1-%5Cdfrac%7B4%7D%7B9%7D%7D%20%5Cend%20%7Bpmatrix%7D%5E2%7D)
![\mathbf{\hat {\theta} =\begin {pmatrix} \dfrac{\dfrac{4}{9}}{\dfrac{5}{9}} \end {pmatrix}^2}](https://tex.z-dn.net/?f=%5Cmathbf%7B%5Chat%20%7B%5Ctheta%7D%20%3D%5Cbegin%20%7Bpmatrix%7D%20%5Cdfrac%7B%5Cdfrac%7B4%7D%7B9%7D%7D%7B%5Cdfrac%7B5%7D%7B9%7D%7D%20%5Cend%20%7Bpmatrix%7D%5E2%7D)
![\mathbf{\hat {\theta} =\begin {pmatrix} \dfrac{4}{5} \end {pmatrix}^2}](https://tex.z-dn.net/?f=%5Cmathbf%7B%5Chat%20%7B%5Ctheta%7D%20%3D%5Cbegin%20%7Bpmatrix%7D%20%5Cdfrac%7B4%7D%7B5%7D%20%5Cend%20%7Bpmatrix%7D%5E2%7D)
![\mathbf{\hat {\theta} =\dfrac{16}{25} }](https://tex.z-dn.net/?f=%5Cmathbf%7B%5Chat%20%7B%5Ctheta%7D%20%3D%5Cdfrac%7B16%7D%7B25%7D%20%7D)
Answer:
6
Step-by-step explanation:
72 = 54 + 3x
where x represents the number of 3-point shots
72 - 54 = 3x
18 = 3x
x = 18/3
x = 6