Question

Find the covariance s_(xy) using bivariate data.

Bivariate numerical data
ANSWERED
asked 2021-01-04
Find the covariance \(\displaystyle{s}_{{{x}{y}}}\) using bivariate data.

Answers (1)

2021-01-05
The given bivariate data is(1,6),(3,2),and(2,4).
Calculation:
First determine \(\displaystyle\sum{x}_{{i}}{y}_{{i}},\sum{x}_{{i}}{\quad\text{and}\quad}\sum{y}_{{i}}\).
\(\displaystyle{x}_{{i}}\rightarrow\) first coordinates of the ordered pairs.
\(\displaystyle{y}_{{i}}\rightarrow\) second coordinates of the ordered pairs.
\(\displaystyle\sum{x}_{{i}}{y}_{{i}}={1}\cdot{6}+{3}\cdot{2}+{2}\cdot{4}={20}\)
\(\displaystyle\sum{x}_{{i}}={1}+{3}+{2}={6}\)
\(\displaystyle\sum{y}_{{i}}={6}+{2}+{4}={12}\)
For covariance \(\displaystyle{s}_{{{x}{y}}}{u}{\sin{{g}}}{f}{\quad\text{or}\quad}\mu{l}{a}{s}_{{{x}{y}}}=\frac{{\sum{x}_{{i}}{y}_{{i}}-\frac{{\sum{x}_{{i}}-\sum{y}_{{i}}}}{{{n}}}}}{{{n}-{1}}}\).
Where n = number of ordered pairs.
n=3
\(\displaystyle{s}_{{{x}{y}}}=\frac{{{20}-\frac{{{6}\cdot{12}}}{{{3}}}}}{{{3}-{1}}}\)
\(\displaystyle=\frac{{{20}-\frac{{{72}}}{{{3}}}}}{{{3}-{1}}}\)
\(\displaystyle=\frac{{{2}-{24}}}{{2}}\)
\(\displaystyle=-\frac{{4}}{{2}}=-{2}\)
Hence \(\displaystyle{s}_{{{x}{y}}}=-{2}\).
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