Properties of Normal Distribution

i. The curve is a bell shaped and has a single peak.ii. The mean of a normally distributed population lies at the center of its normal curve.

iii. Because of the symmetry of the normal probability distribution, the mean, median and mode are the same value.

iv. The two tails of the normal probability distribution extend indefinitely and never touch the horizontal axis. (Graphically it is not possible to show).

v. No matter what the values of μ and δ are, areas under normal curve remain in certain fixed proportions within a specified number of standard deviations on either side of μ. For example

a. μ ± δ will always contain 68.26%b. μ ± 2δ will always contain 95.44%c. μ ± 3δ will always contain 99.7%

Assumptions for OLS

i. Error term i.e. ε is a random variable.

ii. E(εi) = 0 i.e. The expected value of error term is zero. It implies that the expected value

of Y is related to X in the population by a straight line.

iii. Var (εi) = E (εi2) = δ2 for all i. i.e. the variance of error term is constant. It means that

distribution of error has same variance for all values of X (Homoscedasticity

assumption).

iv. E ( εi, εj) = 0 for all i ≠ j i.e. error terms are independent of each other (assumption of no

serial or auto correlation b/w εs)

v. E (X, εi) = 0 i.e. X and ε are independent of each other.

vi. εis are normally distributed with a mean of zero and a constant Variance δ2. This implies

that Y values are also normally distributed. The distribution of y and ε are identical

except that they have different means. This assumption is required for estimation and

testing of hypothesis on linear regression.

Properties of Ordinary Least Squares:

i. The least squares regression line always goes through the point (Ȳ, ), the means of the

data.

ii. The sum of the deviations of the observed values Yi from the least squares regression line

is always equal to zero, i.e. Ʃ (Yi – Ŷ) = 0.

iii. The sum of squares of the deviations of the observed values from the least-squares

regression line is a minimum, i.e. Ʃ (Yi – Ŷ)2 = minimum

iv. The least squares regression line obtained from a random sample is the line of best fit

because a and b are the unbiased estimates of the parameters α and β.