Probability density function pdf continuous distributions
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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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Probability density function pdf continuous distributions l.jpg
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 l.jpg
Cumulative Distribution Function (cdf)

  • F(x)=Pr(x<=X)probability X is <= x

    • area under pdf from -inf up to x


Frequency tabulation l.jpg
FREQUENCY TABULATION

  • 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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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)


Example computations l.jpg
Example Computations

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

  • SECOND MOMENT

  • VARIANCE


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