Chap 7 Common continuous probability distributions

Applied Statistics by John Neter, William Wasserman, and G. A. Whitmore

Prepared by Walter Chen, Dept. of Civil Engineering, NTUTFor classroom teaching purpose

Examples of normal distributions The temperature X at noon on August 15 in a

southeastern city The weight X of a metal ingot produced in a

smelter The height X of women who are 20-29 years old

Density function

Characteristics of normal distribution Two parameters

µ (mu) σ (sigma)

Bell shaped Symmetrical

Centered at µ σ determines the spread of the distribution

Almost all of the probability in a normal distribution is located in a limited range about its mean

Mean and varianceNotation: N(μ, σ2)For example, N(100, 20)

Standard normal probability distribution Is a particular member of the family of normal

distributions Mean = 0 Standard deviation = 1 Standard normal variable Z

Z = N(0, 1)

Theorem Any linear function of a normal random variable is

also a normal random variable Therefore, any normal random variable can be

transformed into the standard normal variable The probability table for the standard normal

distribution can be used for all normal distributions

X → Z

Standard normal probability table and examples

Percentiles

Probabilities for any normal distribution

Descriptions for Figure 7.5 and 7.6

Probability limits

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Cumulative probability function for a normal distribution

Sum of independent normal RVs

Example

Central limit theorem Under very general conditions, the distribution of

the sum of a number of random variables converges to, or approaches, the normal distribution as the number of variables in the sum becomes large

The theorem does not require that the random variables entering the sum have the same distribution function or even that they be entirely independent