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Probability Density Function (pdf) continuous distributions. f(x) >= 0 all x f(x) is the “likelihood” of x integral under f(x) is exactly 1.0. Cumulative Distribution Function (cdf). F(x)=Pr(x<=X) probability X is <= x area under pdf from -inf up to x. FREQUENCY TABULATION .

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• f(x) >= 0 all x

• f(x) is the “likelihood” of x

• integral under f(x) is exactly 1.0

• F(x)=Pr(x<=X)probability X is <= x

• area under pdf from -inf up to x

• Interval Freq Rel Cum

• Freq Prob

• 40-44 22 .11 .11

• 44-48 46 .23 .34

• 48-52 74 .37 .71

• 52-56 52 .26 .97

• 56-60 6 .03 1.00

INTERPOLATION FORMULA for FRACTILE given cumulative probability

xk = left end-point of interval

xk+1 = right end-point of interval

pk = cum prob at left end-point of interval

pk+1 = cum prob at right end-point of interval

p = given cum prob between pk and pk+1

INTERPOLATION FORMULA for cumulative probability given x

xk = left end-point of interval

xk+1 = right end-point of interval

pk = cum prob at left end-point of interval

pk+1 = cum prob at right end-point of interval

x = given variable value between xk and xk+1

MEAN VALUE FORMULA for HISTOGRAM DISTRIBUTION

• Interval Mid-points

• mk = (xk + xk+1)/2

• Interval Probabilities

• pk = Pk+1 - Pk

• Mean Value

• µ=Sum of pk * mk =

VARIANCE FORMULA for HISTOGRAM DISTRIBUTION

• AVERAGE of x2 for each interval

• mi2 = (xi2 + xixi-1 + xi-12)/3

• SECOND MOMENT

• Sum of pi mi2

• VARIANCE

• σ2=Second Moment - (mean)^2

• STANDARD DEVIATION

• sqrt(Variance)

• MEAN = .11*42 + .23*46 + .37*50 +.26*54 + .03*58 = 49.48

• SECOND MOMENT

• VARIANCE