Show that the MLE estimate for o? is -E(vi – Bo - Bixi). Li%31
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
The question is at end of Merged document.jpg.
![# Ordinary Least Squares (OLS) Derivation
### Key Concepts:
- **Introduction to Regression Line:**
- The regression model is defined as \( y_i = \beta_0 + \beta_1 x_i + \epsilon_i \).
- **Objective:**
- Minimize the square error: \( f(\beta_0, \beta_1) = \sum_{i=1}^{n} (y_i - \beta_0 - \beta_1 x_i)^2 \).
### Mathematical Derivation:
- **Quadratic Function:**
- Minimized when derivatives are zero.
- **Solving the Derivatives:**
\[
\frac{\partial f}{\partial \beta_0} = 2 \sum_{i=1}^{n} (\beta_0 + \beta_1 x_i - y_i) = 0
\]
\[
\frac{\partial f}{\partial \beta_1} = 2 \sum_{i=1}^{n} x_i (\beta_0 + \beta_1 x_i - y_i) = 0
\]
- Simplifies to:
\[
\hat{\beta_1} = \frac{\sum_{i=1}^{n} x_i y_i - n \bar{X} \bar{Y}}{\sum_{i=1}^{n} x_i^2 - n \bar{X}^2}
\]
\[
\hat{\beta_0} = \bar{Y} - \hat{\beta_1} \bar{X}
\]
### Prediction:
- Predicted value of \( y_i \): \( \hat{y_i} = \hat{\beta_0} + \hat{\beta_1} x_i \).
# Maximum Likelihood Estimation (MLE) Derivation
### Likelihood Function:
- Definition: Probability that observations arise from a defined probability distribution.
- **Assumption:** \( \epsilon_i \) are normally distributed.
### Probability Distribution:
- Normal distribution function:
\[
f(x|\mu, \sigma) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}
\]
### Likelihood Calculation:
- Aggregated probability:
\[
p](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6bda373-d3b3-416e-bc4e-13a99b192c75%2Fd5008bf8-bfe6-4ba7-a146-305e14b1c0d3%2Fxrgazps_processed.jpeg&w=3840&q=75)
![The image describes the setup and assumptions for a linear regression model.
1. **Model Description:**
- We have a response variable \( y_i \), for \( i = 1, 2, \ldots, n \).
- One explanatory variable \( x_i \), for \( i = 1, 2, \ldots, n \).
- The linear relationship is given by:
\[
y_i = \beta_0 + \beta_1 x_i + \epsilon_i
\]
- Here, \( \epsilon_i \) are normal disturbance terms, often due to measurement error.
2. **Assumptions:**
- \( \epsilon_i \) is the only randomness of interest. The explanatory variable \( X \) can have any arbitrary distribution or be non-random.
- Parameters \( \beta_0 \) and \( \beta_1 \) need to be estimated.
- Each disturbance \( \epsilon_i \) has a mean of 0 and the same variance \(\sigma^2\).
- Disturbances \( \epsilon_i \) are independent of one another.
3. **Covariance Matrix:**
- The covariance between different \( \epsilon_i \) and \( \epsilon_j \) is:
\[
\text{cov}(\epsilon_i, \epsilon_j) =
\begin{cases}
0, & \text{if } i = j \\
\sigma^2, & \text{if } i \neq j
\end{cases}
\]
4. **Graph Description:**
- The graph shows a linear regression line \( E(Y) = \beta_0 + \beta_1 x \).
- Normal distributions are depicted around the regression line at different \( x \) values (\( x_1, x_2, x_3 \)), symbolizing the variance \(\sigma^2\) of the error terms \( \epsilon_i \).
5. **Maximum Likelihood Estimation (MLE) for Variance:**
- The goal is to show that the MLE estimate for \(\sigma^2\) is:
\[
\hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{\beta}_0 - \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6bda373-d3b3-416e-bc4e-13a99b192c75%2Fd5008bf8-bfe6-4ba7-a146-305e14b1c0d3%2Fw05ioy8_processed.jpeg&w=3840&q=75)
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