ABSTRACT
In most practical applications, when constructing a statistical model we do not know the appropriate probability distribution (or do not know it fully). If the appropriate probability distribution is fully known (e.g., if it is known that https://www.w3.org/1998/Math/MathML"> X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429114502/6d8c8b81-08fc-4b5a-8166-f9463cfdc02d/content/eq884.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> follows the normal distribution with mean 5 feet and standard deviation 6 inches), then this distribution should be used in the model. However, if a variable such as the height of females in a certain population is stated in the literature to be normal in distribution with population mean 5 feet and standard deviation 6 inches, very often these are only estimates obtained from a sample. Note that the normal distribution for height would be inappropriate formally since it gives https://www.w3.org/1998/Math/MathML"> P ( X < 0 ) > 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429114502/6d8c8b81-08fc-4b5a-8166-f9463cfdc02d/content/eq885.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , when we know that https://www.w3.org/1998/Math/MathML"> X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429114502/6d8c8b81-08fc-4b5a-8166-f9463cfdc02d/content/eq886.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> cannot be negative. This need not be a reason to reject the normal model since for this model https://www.w3.org/1998/Math/MathML"> P ( X < 0 ) = Φ ( - 10 ) = 0.7619855 × 10 - 23 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429114502/6d8c8b81-08fc-4b5a-8166-f9463cfdc02d/content/eq887.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
