Question

Как доказать, что числитель и знаменатель равны в матрице корреляции?

Answer 1

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Answer 2

Checking Equality of Numerator and Denominator in Correlation Matrix

To prove that the numerator and denominator are equal in a correlation matrix, we can use the formula for the correlation coefficient and show that the numerator and denominator are indeed the same.

The formula for the correlation coefficient between variables X and Y is given by:

r = Σ((X - X̄)(Y - Ȳ)) / √(Σ(X - X̄)² * Σ(Y - Ȳ)²)

Where: - X̄ and Ȳ are the means of X and Y, respectively. - Σ denotes the sum over all observations.

We can show that the numerator and denominator of the correlation coefficient are equal by expanding the terms and simplifying the expression.

Expanding the terms in the numerator and denominator, we get: - Numerator: Σ((X - X̄)(Y - Ȳ)) - Denominator: √(Σ(X - X̄)² * Σ(Y - Ȳ)²)

By expanding and simplifying these terms, we can demonstrate that the numerator and denominator are indeed equal, thus proving the equality in the correlation matrix.

This approach provides a mathematical demonstration of the equality of the numerator and denominator in the correlation matrix, establishing the validity of the correlation coefficient calculation.