Question

Which of the following statement(s) is/are TRUE for the least squares adjustment of observations?

• A. Observations have a Chi-square distribution
• B. Random errors in the observations are assumed to have a symmetrical distribution
• C. The positive and negative random observation errors are equally likely
• D. The adjusted parameters are independent of a priori reference variance

Hint:

In LS adjustment: errors $\sim \mathcal{N}(0,\sigma^2)$ (symmetric), the normalized residual sum $\to\chi^2$, and scaling all weights by the same constant leaves $\hat{x}$ unchanged.

Answer 1

The Correct Option is B

(A) False. In least squares, the random errors (residuals) are assumed zero-mean and (usually) normally distributed. The sum of the weighted squared residuals (when divided by the reference variance) follows a $\chi^2$ distribution—not the observations themselves.
(B) True. The basic assumption is a symmetric error law about zero (normally Gaussian), so the PDF is symmetric.
(C) True. With a symmetric zero-mean distribution, $+e$ and $-e$ have equal probability.
(D) True. If the weight matrix is $W=\frac{1}{\sigma_0^2}C^{-1}$, the LS estimate $\hat{x}=(A^{\mathsf T}WA)^{-1}A^{\mathsf T}W\ell$ is unchanged by the scalar factor $\sigma_0^{-2}$; hence adjusted parameters are independent of the (scalar) a priori reference variance. Only relative weights matter.