1. The problem asks for the value of the $r^{th}$ moment about the mean for given parameters $\bar{x}$ (sample mean), $\mu$ (population mean), and $\sigma$ (standard deviation). 2. The $r^{th}$ moment about the mean is defined as $$\mu_r = E[(X - \mu)^r]$$ where $E$ denotes the expected value. 3. Important rules: - The first moment about the mean ($r=1$) is always zero because $E(X - \mu) = 0$. - The second moment about the mean ($r=2$) is the variance, which is $\sigma^2$. 4. Evaluating the options: - For $r=1$, the moment about the mean is 0. - For $r=0$, the moment is always 1 because $E[(X - \mu)^0] = E[1] = 1$. - The standard deviation $\sigma$ is the square root of the second moment. 5. Therefore, the values corresponding to $r=0,1,2$ moments about the mean are $1,0,\sigma^2$ respectively. 6. Among the options, (D) 1,0,$\sigma$ matches the values for $r=0$ moment (1), $r=1$ moment (0), and $r=2$ moment (standard deviation $\sigma$). Final answer: (D) 1,0,$\sigma$