Properties of the Normal Distribution

The graph of the probability density function of the normal distribution with parameters m and s is a bell-shaped curve that is symmetric about the ordinate x = m. The shape of the curve is determined by s; the larger s is, the greater is the variation among the values in the domain of the density function , and the flatter the curve is. The bell-shaped curve approaches the horizontal axis asymptotically in both directions. Because the normal distribution is a continuous probability distribution, the area bounded by its graph and the x-axis is equal to 1; also, the probability that a normally distributed variable assumes a value in the interval (a, b) is equal to the area bounded by the curve of its density function, the x-axis, and the two ordinates x = a and x = b.

The mean, mode and median of the normal distribution are all equal to each other and have the value of m. This means that the graph of the normal distribution reaches its maximum ordinate value at x = m. The above statement also asserts that an observation from a normally distributed population with mean m is less than m half the time, and greater than m half the time.

The notation X ~ N(m, s₂) denotes that the random variable X is normally distributed with parameters m and s. The mean and variance of X are m and s₂, respectively. Each of the uncountable ordered pairs (m, s) represents a normal distribution with a unique density function and bell-shaped curve.

The sum of two independent normal random variables is itself a normal random variable. Specifically, if X and Y are independent normal random variables,