3 random variable x,y,z, the corr between x and y is 0.8 and corr between x and z is 0.8 what is the max and min of corr between y and z
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We have three random variables x, y, and z. We know the correlation between x and y is 0.8, and the correlation between x and z is also 0.8. We need to find the maximum and minimum possible correlation between y and z. This is a correlation matrix problem where we must ensure the matrix remains positive semi-definite.
To find the valid range of correlation between y and z, we need to calculate the determinant of the correlation matrix. The determinant of a 3 by 3 correlation matrix has a specific formula. Substituting our known values of 0.8 for both correlations with x, we get a quadratic expression in terms of the unknown correlation between y and z.
To find when the determinant is non-negative, we solve the quadratic inequality. Rearranging gives us a standard quadratic form. Using the quadratic formula, we calculate the discriminant and find the two roots: 0.28 and 1.00. Since the parabola opens upward, the inequality is satisfied between these roots.
Here we visualize the quadratic function as a parabola. The parabola opens upward and crosses the x-axis at our two roots: 0.28 and 1.00. The green shaded region shows where the function is negative or zero, which corresponds to our valid correlation range. This confirms that the correlation between y and z must lie between 0.28 and 1.00.
In conclusion, given that the correlation between x and y is 0.8, and the correlation between x and z is also 0.8, we have determined the valid range for the correlation between y and z. The minimum possible correlation between y and z is 0.28, and the maximum possible correlation is 1.00. This range ensures that the correlation matrix remains positive semi-definite, which is a fundamental requirement for any valid correlation matrix.