What is Jacobian of change of variables?

Question: What is Jacobian of change of variables?

Answer:

In multivariable calculus, the Jacobian of a change of variables is a determinant that represents the local scaling factor of a transformation from one coordinate system to another. It's essential for evaluating multiple integrals after a change of variables.

Here's a breakdown of key concepts:

Change of Variables:

    It involves switching from one set of variables (e.g., x and y) to a new set (e.g., u and v) to simplify a problem.
    This often involves functions like x = g(u, v) and y = h(u, v).

Jacobian Matrix:

    A matrix containing the partial derivatives of the new variables with respect to the original ones: 
| ∂x/∂u ∂x/∂v | 
| ∂y/∂u ∂y/∂v |

Jacobian Determinant:

    The determinant of this matrix, denoted as J(u, v) or ∂(x, y)/∂(u, v).

Role in Integrals:

    When changing variables in a double integral, the Jacobian is used to adjust the area element: 
    ∬f(x, y) dx dy = ∬f(g(u, v), h(u, v)) |J(u, v)| du dv
    It ensures the integral's value remains consistent after the transformation.

Geometric Interpretation:

    The Jacobian's absolute value represents the factor by which a small area element in the u-v plane transforms into a corresponding area element in the x-y plane.

Key Points:

    The Jacobian is a 2x2 determinant for 2D transformations, 3x3 for 3D, and so on.
    If the Jacobian is zero at a point, the transformation might distort or fold space, potentially leading to issues in integration.

Example:

    For polar coordinates (x = r cosθ, y = r sinθ), the Jacobian is r. This means a small square in the r-θ plane transforms into a corresponding sector in the x-y plane with area r times larger.

If x and y are two random variables, what is E[xy]?

Question: If x and y are two random variables, what is E[xy]?

If $x$ and $y$ are two random variables, the expected value of their product $xy$ is given by:

$E[xy] = \sum_{i}\sum_{j} x_iy_j P(X=x_i, Y=y_j)$

In continuous cases, where $x$ and $y$ are continuous random variables, this is expressed as an integral:

$E[xy] = \int\int xy f(x, y) \,dx\,dy$

where $f(x, y)$ is the joint probability density function of $x$ and $y$.

In simpler terms, the expected value of the product of two random variables is found by multiplying each pair of possible values of $x$ and $y$ by the probability of that particular combination occurring, and then summing or integrating over all possible pairs.

It's important to note that if $x$ and $y$ are independent, then $E[xy] = E[x] \cdot E[y]$. If they are not independent, the expected value of the product is influenced by the covariance between $x$ and $y$, and it can be expressed as:

$E[xy] = E[x] \cdot E[y] + \text{cov}(x, y)$

where $\text{cov}(x, y)$ is the covariance between $x$ and $y$.

Unit 11 - Confidence intervals (Statistics)

1: Expected value with empirical probabilities

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2: Conditions for a confidence interval (z interval) for a proportion

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3: Finding the critical value z for a desired confidence level

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Tags: Math,Statistics

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Derivatives of Hyperbolic Functions

Derivative of sinh(x)





Note: sinh'(x) = cosh(x)

Derivative of cosh(x)





Note: cosh'(x) = sinh(x)

Derivative of tanh(x)

An alternative to the logistic sigmoid is the hyperbolic tangent, or tanh function (Figure 1, green curves):

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