Unit Test 5 - Exploring bivariate numerical data (From Khan Academy)

Q1: Correlation coefficient

Q2: Least squares regression line

Sol 2:

Q3: Scatter plot and line of best fit

Q4: Residual

Q5: Interpretation of slope

Q6: Non linear association

Q7: Scatterplot construction

Q8: Residuals from line of regression

Q9: Coefficient of determination (sq(R))

Q10: Interpretation of scatter plot for equation of line of regression

Sol 10:

Q11: Scatter plot building

Q12: Identifying the association from scatter plot

Derive Jacobian when we are changing from 2D cartesian coordinates to polar coordinates

Here's the derivation of the Jacobian when changing from 2D Cartesian coordinates to polar coordinates:

1. Transformation Equations:

• Cartesian coordinates (x, y) are related to polar coordinates (r, θ) by:
  • x = r cos(θ)
  • y = r sin(θ)

2. Jacobian Matrix:

• The Jacobian matrix represents the partial derivatives of the new coordinates with respect to the old coordinates:

  J = | ∂x/∂r ∂x/∂θ | | ∂y/∂r ∂y/∂θ |

3. Calculating Partial Derivatives:

• ∂x/∂r = cos(θ)
• ∂x/∂θ = -r sin(θ)
• ∂y/∂r = sin(θ)
• ∂y/∂θ = r cos(θ)

4. Filling the Jacobian Matrix:

J = | cos(θ) -r sin(θ) | | sin(θ) r cos(θ) |

5. Evaluating the Determinant:

• The Jacobian, denoted as J, is the determinant of this matrix:

  J = det(J) = cos(θ) * r cos(θ) - (-r sin(θ)) * sin(θ) = r cos^2(θ) + r sin^2(θ) = r (cos^2(θ) + sin^2(θ))

6. Simplifying Using Trigonometric Identity:

• Since cos^2(θ) + sin^2(θ) = 1, we get:

  J = r

Therefore, the Jacobian for the change of variables from Cartesian coordinates to polar coordinates is r.

Significance in Integration:

• When changing variables in a double integral from Cartesian to polar coordinates, the area element dx dy transforms to r dr dθ. This is a crucial step in evaluating integrals over regions that are more naturally expressed in polar coordinates, such as circles and sectors.

Using ChatGPT

Let's go through the steps to calculate the Jacobian for the transformation from Cartesian to polar coordinates.

The transformation from Cartesian coordinates $x$ and $y$ to polar coordinates $r$ and $\theta$ is given by:

$x = r \cos(\theta)$ $y = r \sin(\theta)$

The Jacobian of this transformation is the determinant of the matrix of partial derivatives:

$J = \begin{vmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \end{vmatrix}$

1. Partial Derivatives: Calculate the partial derivatives:

   $\frac{\partial x}{\partial r} = \cos(\theta)$ $\frac{\partial x}{\partial \theta} = -r \sin(\theta)$ $\frac{\partial y}{\partial r} = \sin(\theta)$ $\frac{\partial y}{\partial \theta} = r \cos(\theta)$

2. Jacobian Determinant: Compute the determinant:

   $J = \cos(\theta) \cdot r \cos(\theta) - (-r \sin(\theta)) \cdot \sin(\theta)$ $J = r \cos^2(\theta) + r \sin^2(\theta)$ $J = r$

So, the Jacobian determinant is $r$ for the given transformation. When changing variables in an integral, you need to multiply by the absolute value of the Jacobian determinant. In polar coordinates, this means you multiply by $r$. The $dx\,dy$ becomes $r\,dr\,d\theta$ in polar coordinates.

Solving 2D Gaussian Integral w.r.t x and y and also w.r.t polar coordinates

w.r.t x and y

To find the double integral of $e^{-\frac{x^2}{2\sigma^2} - \frac{y^2}{2\sigma^2}}$ with respect to $x$ and $y$, you need to specify the limits of integration. Since the function involves the Gaussian distribution, it's common to integrate over the entire real line. The result is related to the normalization constant of the 2D Gaussian distribution.

The double integral is given by:

$\iint e^{-\frac{x^2}{2\sigma^2} - \frac{y^2}{2\sigma^2}} \,dx \,dy$

Assuming integration over the entire real line, this double integral can be evaluated, and the result is:

$\iint e^{-\frac{x^2}{2\sigma^2} - \frac{y^2}{2\sigma^2}} \,dx \,dy = 2\pi\sigma^2$

This result is obtained by expressing the double integral as the product of two Gaussian integrals (one for $x$ and one for $y$), each contributing a factor of $\sqrt{2\pi\sigma^2}$, and then multiplying those factors together.

Keep in mind that the specific limits of integration and the context in which this double integral is applied may vary, so make sure to adjust accordingly based on the problem at hand.

w.r.t. polar coordinates

When changing variables in a double integral from Cartesian to polar coordinates, the area element dx dy transforms to r dr dθ. This is a crucial step in evaluating integrals over regions that are more naturally expressed in polar coordinates, such as circles and sectors.

The 2D Gaussian integral involves integrating a Gaussian function over two variables, typically $x$ and $y$. The general form of the 2D Gaussian integral is:

$\iint e^{-\frac{x^2 + y^2}{2\sigma^2}} \,dx \,dy$

Here, $\sigma$ is a parameter representing the standard deviation of the Gaussian distribution. The limits of integration would depend on the specific context of the problem, but commonly, the integration is performed over the entire real line for both $x$ and $y$.

The 2D Gaussian integral can be evaluated by expressing it in polar coordinates. The transformation from Cartesian to polar coordinates is given by $x = r\cos(\theta)$ and $y = r\sin(\theta)$, where $r$ is the radius and $\theta$ is the angle.

The integral then becomes:

$\int_0^{2\pi} \int_0^\infty e^{-\frac{r^2}{2\sigma^2}} \,r \,dr \,d\theta$

This integral simplifies to:

$2\pi \sigma^2$

This result is obtained by recognizing that the radial part of the integral is a standard Gaussian integral, and the angular part yields a factor of $2\pi$.

So, the 2D Gaussian integral evaluates to $2\pi\sigma^2$, which is the area under the 2D Gaussian distribution.

Derive generalized Gaussian Integral from the Special Case of Gaussian Integral for Mu = 0 and Sigma = 1-by-sqrt(2)

Pre-read

What is the integral 'I' with respect to Gaussian distribution?

The integral with respect to a Gaussian distribution, often referred to as the Gaussian integral, is a classical mathematical result. The Gaussian integral is defined as follows:

$\int_{-\infty}^{\infty} e^{-x^2} \,dx = \sqrt{\pi}$

This integral is frequently encountered in probability theory and statistics, especially when dealing with normal (Gaussian) distributions. The integral represents the area under the bell-shaped curve of the standard normal distribution.

It's important to note that the Gaussian integral has a well-known result, but proving it involves more advanced mathematical techniques such as completing the square and complex analysis. The result is widely used in various branches of mathematics, physics, and engineering. If you encounter a Gaussian distribution with mean $\mu$ and variance $\sigma^2$, the integral can be generalized as:

$\int_{-\infty}^{\infty} e^{-(x-\mu)^2/(2\sigma^2)} \,dx = \sqrt{2\pi\sigma^2}$

This generalization involves a change of variable and scaling to account for the mean ($\mu$) and variance ($\sigma^2$).

Fig 1:



Fig 2:



Showing how Generalized Gaussian Integral and Special Case of Gaussian Integral are related:
Fig 3: