Normal Distribution

Characterization of the Normal Distribution

The normal distribution is fully defined by two parameters, μ and σ2, through its probability density function as

where μ is the mean parameter and could take any real value and parameter σ2 is the variance of the normal distribution (equivalently, σ is standard deviation) with σ > 0. For example, for diastolic blood pressure, the parameters might be μ = 80 mmHg, σ = 10 mmHg; for birthweight, they might be μ = 120 oz, σ = 20 oz. Figure 1 shows the plot of the probability density function for a normal distribution with μ = 80 and σ = 10:

Figure 1 Graphical Illustration of Probability Density Function of Normal Distribution n (80, 100)

The density function of the normal distribution resembles a bell-shaped curve, with the mode at μ and the most frequently occurring values around μ. The curve is unimodal and symmetric around m, and for normal distribution, the mean, median, and mode all equal to μ. The curve has an inflection point on each side of μ at μ − σ and μ + σ, respectively. A point of inflection is a point where the slope of the curve changes direction. The distances from μ to points of inflection provide a good visual sense of the magnitude of the parameter σ.

To indicate that a random variable X is normally distributed with mean μ and variable σ2, we write X ∼ N(μ, σ2). The symbol ∼ indicates ‘is distributed as.’ The entire shape of the normal distribution is determined by μ and σ2. The mean μ is a measure of central tendency, while the standard deviation σ is a measure of spread of the distribution. The parameter μ is called the location parameter, and σ2 is the scale parameter. To see how these parameters affect location and scale, density functions of normal distribution with different means or variances can be compared. For instance, if two normal distributions have different means μ1 and μ2 but the same variance σ2, where μ1 > μ2, then the two density functions will have same shape but the curve with larger mean (μ1) will be shifted to the right relative to the curve with the smaller mean (μ2). Figure 2 shows the comparison of [Page 745]density curves of N(μ1 = 80, s2 = 100) and N(μ2 = 60, s2 = 100). On the other hand, if two normal distributions with the same mean μ and different variance σ21 and σ22, where σ21 < σ22 are compared, then the two density functions will have same mode but the curve with larger variance (σ22)willbemore spread out compared with the other curve with the smaller variance (σ21). Varianceσ2 determines the scale of the distribution. Figure 3 shows the comparison of density curves of N(μ = 80,σ21= 100) and N(μ = 80, σ21 =

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