---
title: "Regression coefficients and properties"
subject: "Probability and Statistics"
chapter: 5
key: "semester_5/probability_and_statistics:regression-coefficients-properties"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/regression-coefficients-properties
site: BCT Engineering
---

# Regression coefficients and properties

Probability and Statistics · Chapter 5: Correlation and regression

- Asked in: 2076 Chaitra, 5 marks; 2072 Kartik, 5 marks (asked 2 times)
- Video: [Regression - 3 Properties of Coefficients of Regression](https://www.youtube.com/watch?v=_Al9PpkBwCE) — PUAAR Academy
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/regression-coefficients-properties
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-5

## Question

What are the two regression coefficients and what do they represent? Write the properties of regression coefficients.

## Short answer

b_yx = r·σy/σx is the slope of Y on X; b_xy = r·σx/σy is the slope of X on Y. Both have the same sign as r, their product is r², and they are independent of origin but not scale. The lines intersect at (x̄, ȳ).

## Answer

The **regression coefficients** are the slopes of the two regression lines.

- **b_yx** (regression coefficient of Y on X) is the slope of the line Y = a + bX. It is the change in Y for a unit change in X:
  b_yx = r·σy/σx = Sxy/Sxx
- **b_xy** (regression coefficient of X on Y) is the slope of the line X = a′ + b′Y. It is the change in X for a unit change in Y:
  b_xy = r·σx/σy = Sxy/Syy

#### Properties

1. **Geometric mean:** r = ±√(b_yx · b_xy), i.e. b_yx · b_xy = r² ≤ 1. The sign of r is the common sign of both coefficients.
2. **Same sign:** b_yx, b_xy and r always have the same sign (all positive or all negative).
3. **Both cannot exceed 1:** if one coefficient is greater than 1 in magnitude, the other must be less than 1 in magnitude.
4. **Arithmetic mean ≥ |r|:** |b_yx + b_xy|/2 ≥ |r|, because AM ≥ GM and both coefficients have the same sign. (For positive r this is simply (b_yx + b_xy)/2 ≥ r.)
5. **Change of origin and scale:** regression coefficients are independent of change of origin but not of scale. If u = (x − a)/h and v = (y − c)/k, then b_yx = (k/h)·b_vu.
6. **Intersection:** the two regression lines intersect at (x̄, ȳ), so both lines pass through the point of means.
7. If r = 0, then b_yx = b_xy = 0 and the lines are perpendicular; if r = ±1 the lines coincide.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The regression coefficient b_yx represents:
   A. The change in X for a unit change in Y
   B. The correlation between X and Y
   C. The change in Y for a unit change in X, the slope of Y on X
   D. The intercept of the line
   Answer: C. b_yx is the slope of the line of Y on X. (Key point: Meaning of b_yx)

2. How are r, b_yx and b_xy related?
   A. r = b_yx + b_xy
   B. r = ±√(b_yx · b_xy), with the sign common to both coefficients
   C. r = b_yx / b_xy
   D. r = (b_yx · b_xy)²
   Answer: B. r is the geometric mean of the two regression coefficients, sharing their sign. (Key point: Geometric mean property)

3. If b_yx = 1.5, what must be true of b_xy?
   A. b_xy > 1.5
   B. b_xy = 1.5
   C. b_xy is negative
   D. b_xy < 1 in magnitude, since b_yx · b_xy = r² ≤ 1
   Answer: D. The product is r² ≤ 1, so both cannot exceed 1 in magnitude. (Key point: Magnitude restriction)

4. Regression coefficients are:
   A. Independent of change of origin but not of scale
   B. Independent of both origin and scale
   C. Independent of scale but not of origin
   D. Affected by both
   Answer: A. Shifting the origin does not change the slope, but changing the scale does (b_yx = (k/h) b_vu). (Key point: Origin and scale)

## Related in Probability and Statistics

- [Regression coefficients when equal; properties](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aregression-coefficients-equal-three-properties) — Probability and Statistics, chapter 5

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- [Townsend current growth with two coefficients](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Atownsend-two-coefficients) — High Voltage Engineering, chapter 6
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