The normal distribution (Exhibit 33.17) is said to represent an elementary “truth about the general nature of reality”. It is a family of distributions, each defined by two parameters — mean (μ) and standard deviation (s).
The probability distribution function of a random variable X∼N(μ,σ) is:
$$f(x)=\frac{1}{\sqrt{2πσ^2}} e^{-\frac{(x-μ)^2}{2σ^2}}$$The normal distribution X∼N(0,1), μ = 0 and σ = 1, is called the standard normal distribution, and its distribution function is:
$$f(x)=\frac{e^{-x^2/2}}{\sqrt{2π}}$$A value from any normal distribution can be transformed into its corresponding value on a standard normal distribution using the following formula:
$$Z =\frac{X - μ}{σ}$$where Z is the value on the standard normal distribution, X is the value on the original distribution, μ is the mean of the original distribution, and σ is the standard deviation of the original distribution.
$$E(Z)= E\biggl(\frac{X - μ}{σ}\biggr)=\frac{E(X) - μ}{σ}=0$$ $$Var(Z)=Var\biggl(\frac{X - μ}{σ}\biggr)=\frac{Var(X)}{σ^2}=1$$To computing probabilities for normal distributions, convert to standard value and lookup the distribution function tables for the standard normal distribution:
$$P(X≤b)→P\biggl(Z≤\frac{b - μ}{σ}\biggr)$$Note: The probabilities may also be obtained using the NORM.DIST function in Excel. The Excel NORM.INV function returns the variable value, given the parameters and the probability.
If a random variable follows a normal distribution, we can expect that:
According to the Central Limit Theorem, the normal distribution can be used as an approximation to the binomial distribution, if the sample size is large enough.
If X is Binomial (n, p):
$$E(X) = np$$ $$Var(V) = np(1-p)$$The approximate equivalent normal distribution is N[np,√(np(1-p))], provided np≥ 5 and n(1- p)≥ 5 (thumb rule).
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