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Submitted By acumber

Words 715

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Words 715

Pages 3

Deﬁnitions:

• Sample space: A collection of all possible outcomes of a random experiment. Usually denoted by Ω.

• Probability distribution: An assignment of numbers between 0 and 1 to each possible outcomes. Sum of these numbers should be equal to 1. Usually denoted by

P.

• Random variable: A function deﬁned on a sample space. Usually denoted by X.

• Expected value of X: µx = E[X] =

P(ω)X(ω) ω∈Ω (if X is a continuous random variable, then the summation should be replaced by the integral).

• Variance of X:

2

V ar(X) = σx = E[(X − µx )2 ]

• Standard deviation of X:

E[(X − µx )2 ]

σx =

• Covariance of X and Y :

Cov(X, Y ) = E[(X − µx )(Y − µx )]

• Correlation of X and Y :

Cov(X, Y ) σx σx

Consider a sample {(x1 , y1 ), (x2 , y2 ), · · · , (xn , yn )}. Then,

• Sample mean: n 1 x= ¯ xi n i=1 ρxy =

• Sample variance: s2 x

n

1

=

n−1

(xi − x)2

¯

i=1

• Sample standard deviation: sx =

1 n−1 n

(xi − x)2

¯

i=1

• Sample covariance: sxy 1

=

n−1

n

(xi − x)(yi − x)

¯

¯ i=1 1

2

• Sample correlation: rxy =

sxy sx sy

Interpretation of correlation rxy

• −1 ≤ rxy ≤ 1 always.

• If rxy > 0, then large values of X tend to be associated with large values of Y , and small values of X tend to be associated with small values of Y . In this case, X and

Y are positively linearly related.

• If rxy < 0, then large values of X tend to be associated with small values of Y , and small values of X tend to be associated with large values of Y . In this case, X and

Y are negatively linearly related.

• Bigger |rxy | implies stronger linear relation between X and Y .

• If rxy = 0, then X and Y are not linearly related (they may have non-linear relationship).

• If rxy = 1, then all of (xi , yi ) are on a straight line with positive…...

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