Ec305 Lecture notes

NORMAL DISTRIBUTION

Probability Density Function of the Normal Distribution

x is the random variable

m is its mean

s is its standard deviation

s² is its variance

Random variable can be "standardized" by subtracting its mean and dividing by its standard deviation. The resulting variable z has mean equal to 0 and variance equal to 1

This is a probability density function of the standard normal distribution. Obviously it holds for the standardized variable z that

m_z = 0 and s_z = 1

The following animations show how the shape of normal density function depends on m and s.

Constructing the Confidence Intervals

Probability is the area under the density curve. In this case the pdf is standard normal. The dark brown area under the blue curve and in between the red lines is the probability that the random variable will be in the interval from -1 to +1. Because for standard normal distribution the mean is equal to 0 and standard deviation is equal to 1 , the interval can be also interpreted as an interval from mean minus one standard deviation to mean plus one standard deviation.

( m - s , m + s)

With m = 0 and s = 1 the interval is (0 - 1, 0 + 1) or (-1, 1). The area under the curve can be calculated by integrating the probability density function from -1 to 1. As can be seen that integral is .683; that is 68.3 per cent is the probability of the random variable falling in the interval (-1, 1)..

The following two examples repeat the above construction for intervals of two and three standard deviation. The integrals show that randomly chosen value of the variable x will fall into the interval ( m - 2s , m + 2s) with the probability 95.4 per cent and into the interval ( m - 3s , m + 3s) with the probability 99.7 per cent.

Cumulative Distribution Function

.	The cumulative distribution function is the probability that the random variable will be less or equal to a specific value. In other words it is the integral of the probability density function from minus infinity to that value.


	The green curve F(x) represents what is in the picture of the probability density curve the area under the curve. The mutual relation of the two presentations is well seen from the above picture and the calculations bellow..