Course Notes - Department of Mathematics and Statistics

Normal Distribution Notes• The graph is symmetrical about µ (centre).• The two parameters, µ and σ, completely define the normal distribution.We sayX ∼ N(µ, σ 2 ).• Demonstration: Shape of normal distributions• Increasing µ moves the curve but does not change its shape.• Increasing σ spreads the curve more widely about X = µ but doesnot alter the centre.AREAS UNDER THE CURVE• Probabilities are equivalent to areas under the normal distributioncurve.• Total area under the curve is equal to 1 (0.5 either side of mean).• The probability P r(a < X < b) is found using the area under thecurve between X = a and X = b.• Areas under the curve can be found by integrating the equationfor the normal curve.EQUATION OF THE NORMAL CURVE• For the general normal distribution:f(X) = 1 √2πσe − 1 2( X−µσ) 2 .• The parameters µ and σ are estimated by the sample mean, xand sample standard deviation, s.• This equation simplifies nicely for the standard normal distribution(µ = 0 and σ = 1):f(Z) = 1 √2πe − 1 2 Z2 .72