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 Chi-squared and Mahalanobis Distance (Squared standardized distance to mean) 1.       Using standardized distance to mean Ø   For univariate normal data, you determine whether an observation is likely to belong to a predefined population, according to how many standard deviations (eg. Z-score), it is away from the mean. Ø   An observation can be higher or lower than mean. So, standardized distance of an observation, from mean, can be positive or negative value. Ø   Likewise for a multinormal distribution, the distance from the mean or center of the measurements can be used to determine likelihood of membership of predefined population. Ø   For multivariate data, there is no specific positive or negative direction from center of the multivariate data. Need different metric. 2.       Using Squared standardized distance to mean Ø   For univariate data , you can calculate squared standardized dista...

Review of Formulas Thus, conditional distribution of Y given X=x is given as, Example 1 https://online.stat.psu.edu/stat414/lesson/21/21.1 Solution: Example 2. Source: https://www.sciencedirect.com/topics/mathematics/bivariate-normal-distribution Suppose that the heights of fathers and sons are r.v.'s  X  and  Y , respectively, having (approximately) Bivariate Normal distribution with parameters (expressed in inches)  μ 1  = 70,  σ 1  = 2,  μ 2  = 71,  σ 2  = 2 and  ρ  = 0.90. If for a given pair (father, son) it is observed that  X  =  x  = 69, determine: (i) The conditional distribution of the height of the son. (ii) The expected height of the son. (iii) The probability of the height of the son to be more than 72 in. Example 2. determine sigma1, sigma2, correlation coefficient that mu1=0 and mu=-1. exponent  of a bivariate normal density is,

      STANDARD NORMAL DISTRIBUTION In Univariate Context PDF for Univariate Normal Distribution contains two parameters  μ  and  σ   and PDF is given by,  by plugin  μ = 0 and  σ = 1  in the PDF of the normal distribution, the equation simplifies to Units for the standard normal distribution curve are denoted by z and are called as z-score or z-values or standard score. CDF corresponds to AUC for defined interval, usually -infinity to z value and denoted by Greek letter. where e= 2.71828 and 3.14159. In Bivariate Context (Joint Bivariate Normal Density Function) PDF for bivariate normal distribution ( with two variables, say X1 and X2) will contain five parameters, two means  μ 1  and  μ 2 ,  two  standard deviations   σ 1  and  σ 2   and the product moment correlation between the two variables,  ρ PDF for Bivariate Normal Distribution is given by, where  ∑ is variance...