# Tag Info

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Since this is a table of samples, we should use the corrected sample standard deviation whose square is divided by $n-1$ instead of $n$. This gives the square of the standard deviation to be $23.1439$ instead of $22.9125$.

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Note that $$\rho(r_i,r)= \frac{\text{cov}(r_i,r)}{\sigma(r_i)\cdot \sigma(r)}\\\implies \text{cov}(r_i,r) = \rho(r_i,r)\cdot \sigma(r_i)\cdot \sigma(r)$$

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Since $\ln{Y}$ is normal, $Y$ is a lognormal (actually, its a truncated lognormal at $\mu\pm 4\sigma$). Just generate data from the normal density between the two bounds, then exponentiate each data point to get the distribution of Y.

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I actually think standard deviation is exactly what you're looking for and you have made an error in calculating them to be the same in your example. The mean of $X$ is $25/6$ and the standard deviation is approximately $2.3$. The meany of $Y$ is $23/6$ and the standard deviation is approximately $3.1$. As you can see, as $Y$ is more spread out, it has a ...

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Perhaps calculating this variance will help \begin{align} var(\hat{\alpha} + x_i \hat{\beta})&= var(\hat{\alpha})+x_i^2 var(\hat{\beta}) + 2cov(\hat{\alpha},\hat{\beta}x_i)\\ &= var(\hat{\alpha})+x_i^2 var(\hat{\beta}) + 2x_icov(\hat{\alpha},\hat{\beta})\\ &= var(\hat{\alpha})+x_i^2 var(\hat{\beta}) - 2x_i \frac{\sigma^2\bar{x}}{S_{xx}} ...

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It looks correct and I get the same results as your calculation with numpy. import numpy as np x = [-1,0,1] np.std(x) 0.81649658092772603 So it is probably the way you calculate it.

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Note $$\operatorname{Var}[X] = \operatorname{E}[X^2] - \operatorname{E}[X]^2 = \operatorname{E}[X^2] = \Pr[X^2 = 1] = \frac{2}{3}.$$ The first equality comes from the calculation \begin{align*} \operatorname{Var}[X] &= \operatorname{E}[(X-\operatorname{E}[X])^2] \\ &= \operatorname{E}[X^2 - 2\operatorname{E}[X]X + \operatorname{E}[X]^2] \\ &= ...

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Apparently there are two types of standard deviation: sample standard deviation : and population standard deviation : which is confusing because it was never introduced in textbooks...

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