## Karl Pearson's Correlation Coefficient

### For Population:

\[{\rho _{XY}} = \frac{{cov(X,Y)}}{{{\sigma _X} \cdot {\sigma _Y}}}{\rm{ }}\]

### For Sample:

\[\mathop r\nolimits_{XY} = \frac{{{s_{YX}}}}{{{s_X} \cdot {s_Y}}}\] \[\left\{ \begin{array}{l}Where,\\{s_{YX}} = \frac{{\sum {\left( {X - \overline X } \right)\left( {X - \overline X } \right)} }}{n}\\{s_X} = {\rm{ }}\sqrt {\frac{{\sum {{{\left( {X - \overline X } \right)}^2}} }}{n}} \\{s_Y} = \sqrt {\frac{{\sum {{{\left( {Y - \overline Y } \right)}^2}} }}{n}} \end{array} \right.\]

Substituting the formula of Covariance and Standard Deviation: \[\mathop r\nolimits_{XY} = \frac{{\frac{{\sum {\left( {X - \overline X } \right)\left( {X - \overline X } \right)} }}{n}}}{{\sqrt {\frac{{\sum {{{\left( {X - \overline X } \right)}^2}} }}{n}} \cdot \sqrt {\frac{{\sum {{{\left( {Y - \overline Y } \right)}^2}} }}{n}} }}\] Cancelling n: \[\mathop r\nolimits_{XY} = \frac{{\sum {\left( {{X_i} - \overline X } \right)\left( {{Y_i} - \overline Y } \right)} }}{{\sqrt {\sum {{{\left( {{X_i} - \overline X } \right)}^2} \cdot \sum {{{\left( {{Y_i} - \overline Y } \right)}^2}} } } }}\] Deviation Form: \[\mathop r\nolimits_{XY} = \frac{{\sum {\mathop x\nolimits_i \mathop y\nolimits_i } }}{{\sqrt {\sum {\mathop x\nolimits_i^2 } } \cdot \sqrt {\sum {\mathop y\nolimits_i^2 } } }}\] Formula without Mean and Deviation: Formula without Mean and Deviation: \[\mathop r\nolimits_{XY} = \frac{{n\sum {\left( {{X_i}{Y_i}} \right) - \left( {\sum {{X_i}} } \right)\left( {\sum {{Y_i}} } \right)} }}{{\sqrt {n\sum {X_i^2 - {{\left( {\sum {{X_i}} } \right)}^2}} } \cdot \sqrt {n\sum {Y_i^2 - {{\left( {\sum {{Y_i}} } \right)}^2}} } }}\]

## Spearman's Rank Correlation Coefficient:

\[r' = 1 - \frac{{6\sum {{D^2}} }}{{n\left( {{n^2} - 1} \right)}}\]
Where,

D = Difference between the ranks of corresponding pairs of X and Y

## Simple Linear Regession Model:

### For Population

\[\begin{array}{l}{Y_i} = {b_0} + {b_1}{X_i}\\{Y_i} = {b_0} + {b_1}{X_i} + u\end{array}\]

### For Sample

\[{{\hat Y}_i} = {{\hat b}_0} + {{\hat b}_1}{X_i}\]### OLS Estimation

\[\begin{array}{l}{Y_i} - {{\hat Y}_i} = {e_i}\\Minimize \to {e_i} = {Y_i} - {{\hat Y}_i}\\Min\sum {\mathop e\nolimits_i^2 } = \sum {\mathop {\left( {{Y_i} - {{\hat Y}_i}} \right)}\nolimits^2 } \\Min\sum {\mathop e\nolimits_i^2 } = \sum {\mathop {\left( {{Y_i} - {{\hat b}_0} + {{\hat b}_1}{X_i}} \right)}\nolimits^2 } \\Slope:\\\mathop {\hat b}\nolimits_1 = \frac{{n\sum {\mathop X\nolimits_i \mathop Y\nolimits_i - \sum {\mathop X\nolimits_i \sum {\mathop Y\nolimits_i } } } }}{{n\sum {\mathop X\nolimits_i^2 - \mathop {\left( {\sum {\mathop X\nolimits_i } } \right)}\nolimits^2 } }}\\\mathop {\hat b}\nolimits_1 = \frac{{\sum {\mathop x\nolimits_i \mathop y\nolimits_i } }}{{\sum {\mathop x\nolimits_i^2 } }} = \frac{{{\mathop{\rm cov}} (X,Y)}}{{\mathop \sigma \nolimits_X^2 }}\\Intercept:\\\mathop {\hat b}\nolimits_0 = \overline Y - {{\hat b}_1}\overline X \end{array}\]

### Hypothesis Testing of estimators:

Null Hypothesis = Estimators are not statistically more than zero.

\[\begin{array}{l}\mathop t\nolimits_0 = \frac{{{{\hat b}_0} - {b_0}}}{{{s_{{{\widehat b}_0}}}}}\\\mathop t\nolimits_1 = \frac{{{{\hat b}_1} - {b_1}}}{{{s_{{{\widehat b}_1}}}}}\end{array}\]

#### Variance of estimators when population variance is given:

\[\begin{array}{l}Var{\rm{ }}{\widehat b_0} = \sigma _u^2\frac{{\sum {X_i^2} }}{{n\sum {x_i^2} }}\\Var{\rm{ }}{\widehat b_1} = \sigma _u^2\frac{1}{{\sum {x_i^2} }}\end{array}\]

#### Estimator of Population Variance (\(\sigma _u^2\)):

\[{s^2} = \hat \sigma _u^2 = \frac{{\sum {e_i^2} }}{{n - k}}\]

#### Variance of estimators when population variance is not given:

\[\begin{array}{l}s_{{{\widehat b}_0}}^2 = \frac{{\sum {e_i^2} }}{{n - k}} \cdot \frac{{\sum {X_i^2} }}{{n\sum {x_i^2} }}\\s_{{{\widehat b}_1}}^2 = \frac{{\sum {e_i^2} }}{{n - k}} \cdot \frac{1}{{\sum {x_i^2} }}\end{array}\]

### Coefficient of Determination:

\[\mathop R\nolimits^2 = \frac{{\sum {\hat y_i^2} }}{{\sum {y_i^2} }} = 1 - \frac{{\sum {e_i^2} }}{{\sum {y_i^2} }} = \mathop {\hat b}\nolimits_1 \frac{{\sum {{x_i}{y_i}} }}{{\sum {y_i^2} }}\]

#### Relationship between Correlation Coefficient & Coefficient of Determination:

\[r = \sqrt {{R^2}} \]

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