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If you have two random variables $X,Y$, the correlation coefficient is $$\rho=\frac{\mu_{XY}-\mu_X \mu_Y}{\sigma_X \sigma_Y}$$ where $\mu$ denotes the mean of the variable and $\sigma$ denotes the standard deviation of the variable. Suppose now you have two Bernoulli random variables $X,Y$ (so they take on only the values $0$ and $1$) and both have total ...

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$E[X|Y] = \sum_{y} E[X|Y=y] P(Y=y) = \sum_{y} 0 P(Y=y) = 0$. Now, note $E[XY] = E[E[XY|Y]] = E[Y E[X|Y]] = E[Y (0) ] = E[0] =0$.

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Let $X_i \sim N(0,1)$ for $i \in \{1,\dots,100\}$ denote the random variables described in the question. Next, define $$Y \equiv \sum_{i=1}^{98} X_i \;\;\;\;\; \text{and} \;\;\;\;\; Z \equiv \sum_{i=1}^{100} X_i$$ The correlation between $Y$ and $Z$ is given by $$\frac{\mathbb E [(Y-\mathbb E [Y])(Z-\mathbb E [Z])]}{ \sqrt{\mathbb E [(Y-\mathbb E ... 1 This equation is the autocorrelation, see wikipedia . (or autocovariance , depending of usages :-) . Quite the same by a \sigma ). Not correlated in time is like each sample being independently distributed in time. So the correlation between x_{t0} and x_{t1} is 1 if t0=t1 otherwise 0. 1 What does the correlation coefficient of two events mean? Is it the correlation coefficient of their Bernoulli indicator random variables? Then$$\newcommand{\Chi}{{\raise{0.5ex}{\chi}}} \begin{align} \rho_{\!\lower{0.5ex}{A,B}} & = {\sf Corr}(\Chi_A, \Chi_B) \\[1ex] & = \dfrac{{\sf Cov}(\Chi_A, \Chi_B)}{\sqrt{\,{\sf Var}(\Chi_A){\sf ...

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Here is the proof my teacher gaved me today, shorter than the very good one from Daniel. \begin{align*} R^2&=\frac{ESS}{TSS}=\frac{\sum(\hat y -\bar y)^2}{\sum (y_i-\bar y)^2}\\ &=\frac{\sum (\hat\beta_1+\hat \beta_2x_i-\bar y)^2}{\sum(y_i-\bar y)^2}\\ &=\frac{\sum (\bar y- \hat \beta_2\bar x+\hat \beta_2x_i-\bar y)^2}{\sum(y_i-\bar y)^2}\\ ...

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For (mostly my own) future reference, here's my intuitive explanation of the answer. If CEO strength made no difference, then for the stronger CEO, for 100 firms, 50 would be 'more successful' and 50 would be 'less successful'. If a stronger CEO made a difference (.30 correlation), then out of 100 firms, 30 would be 'more successful' for the stronger CEO ...

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