SOME SPECIFIC PROBABILITY DISTRIBUTIONS The next figure below shows the portion of the distribution between -4 and 0 when the mean is d o is equal to tw FIGURE 2. Normal Probability Density Function Showing P(-4<I<O Probability Between Limits is 0. 30233 0 0.18 0.16 0.14 0.04 4 1. 2. Properties of the normal random variable. (x)=u, var(x)=g b: The density is continuous and symmetric about u c: The population mean, median, and mode coinci d: The range is unbound e: There are points of inflection atμ±σ f: It is completely specified by the two parameters u and a g: The sum of two independently distributed normal random variables is normally distributed If Y= aX1 BX2 +y where X1 NN(1, 01) and X2 NN(a2, 022) and X1 and X2 are 1.3. Distribution function of a normal random variable f(s;u, o2)d Here is the probability density function and the cumulative distribution of the normal distribution with u=0 and o= 12 SOME SPECIFIC PROBABILITY DISTRIBUTIONS The next figure below shows the portion of the distribution between -4 and 0 when the mean is one and σ is equal to two. Figure 2. Normal Probability Density Function Showing P(−4 <x< 0) −8 −6 −4 −2 0 2 4 6 8 10 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Probability Between Limits is 0.30233 Density Critical Value 1.2. Properties of the normal random variable. a: E(x) = µ, Var(x) = σ2. b: The density is continuous and symmetric about µ. c: The population mean, median, and mode coincide. d: The range is unbounded. e: There are points of inflection at µ ± σ. f: It is completely specified by the two parameters µ and σ2. g: The sum of two independently distributed normal random variables is normally distributed. If Y = αX1 + βX2 + γ where X1 ∼ N(µ1,σ1 2) and X2 ∼ N(µ2,σ2 2) and X1 and X2 are independent, then Y ∼ N(αµ1 + βµ2 + γ; α2σ2 1 + β2σ2 2). 1.3. Distribution function of a normal random variable. F(x ; µ, σ2) = P r (X ≤ x) = Z x −∞ f (s ; µ, σ2 )ds (2) Here is the probability density function and the cumulative distribution of the normal distribution with µ = 0 and σ = 1