One million random numbers are generated from a statistically stationary process with a Gaussian distribution with mean zero and standard deviation $$\sigma_0$$. The $$\sigma_0$$ is estimated by randomly drawing out 10,000 numbers of samples ($$x_n$$). The estimates $${\widehat \sigma _1}$$, $${\widehat \sigma _2}$$ are computed in the following two ways.

$$\matrix{ {\widehat \sigma _1^2 = {1 \over {100000}}\sum\nolimits\limits_{n = 1}^{10000} {x_n^2} } & {\widehat \sigma _2^2 = {1 \over {9999}}\sum\nolimits\limits_{n = 1}^{10000} {x_n^2} } \cr } $$

Which of the following statements is true?

In the following differential equation, the numerically obtained value of $$y(t)$$, at $$t=1$$ is ___________ (Round off to 2 decimal places).

$${{dy} \over {dt}} = {{{e^{ - \alpha t}}} \over {2 + \alpha t}},\alpha = 0.01$$ and $$y(0) = 0$$

Three points in the x-y plane are ($$-$$1, 0.8), (0, 2.2) and (1, 2.8). The value of the slope of the best fit straight line in the least square sense is _________ (Round off to 2 decimal places).

Consider the following equation in a 2-D real-space.

$$|{x_1}{|^p} + |{x_2}{|^p} = 1$$ for $$p > 0$$

Which of the following statement(s) is/are true.