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Continuous distributions Uniform, Normal, Chi-square, and Log-normal distributions

Set 3. Continuous distributions Uniform, Normal, Chi-square, and Log-normal distributions. Density function f(x). X is continuous Value of x can be any number in an interval The points in an interval are not countable f(x) is the equation of an idealized histogram

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Continuous distributions Uniform, Normal, Chi-square, and Log-normal distributions

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  1. Set 3 Continuous distributions Uniform, Normal, Chi-square, and Log-normal distributions

  2. Density function f(x) • Xis continuous • Value of x can be any number in an interval • The points in an interval are not countable • f(x)is the equation of an idealized histogram • Density is always above or on the x-axis,f(x) >0 • Total area under f(x) = 1 • The probability of the values of x in each interval a<x<b, is given by the area under f(x) • Proportion of area for any single value of x is zero

  3. An example of a density function P(a < x < b)

  4. Median,Mean, Mode • Skewed density function • Right skewed 50% 50% Mode Mean Median

  5. Cumulative distribution function • Cumulative Distribution Function (CDF): CDFy(a)= P(Y < a) • Continuous distribution (area under the density function) • 0 < CDFy(a) < 1 • CDFy(a)is a non-decreasing • P(a < Y < b) = CDFy(b) - CDFy(a)

  6. Percentiles • Cumulative distribution function CDF(a) = P(X<a) • CDF(a) = .15, a is the 15th percentile • CDF(a) = .25, a is the 25th percentile (1st quartile) • CDF(a) = .50, a is the 50th percentile (median) • CDF(a) = .75, a is the 75th percentile (3rd quartile) • CDF(a) = .95, a is the 95th percentile

  7. Uniform Distribution • Density function P( X <c) = ? f (x) x a c b

  8. Uniform distribution • Density function • Total area =1 • CDF(1.2) = P(X < 1.2) = ? • P(X > 1.2) = ? • P(X=1.2) = ? • Compute the median • Compute the 90th percentile 0.5 0 1.2 2

  9. Normal distribution • Normal density function • The bell curve • Symmetric • Unimodal (has one peak) • Two parameters determine the distribution • The center m • The standard deviation s • Our notation, N(m,s) s s m Mean = Median = Mode

  10. 68.3%, 95.5%, 99.7% Rule 68.3% 95.5% 99.7%

  11. Standard normal distribution • Standardized variable mz= 0, sz = 1 • If x is normal, so is the standardized variable z • znormal N(0,1) • Proportions of the area under f(z) are tabulated

  12. Normal distribution (general ) • Density function of N(m,s) • Transform to standard Normal • CDF: Normal table

  13. Example • Normal table gives CDF • CDF(1.28) = P(Z < 1.28) = .90 .90 Density f(z) .5 .90 CDF(z) Height = Fz(1.28) Area = CDFz(1.28) 1.28 1.28

  14. Standard normal table (z table) • The first column giveszto the first decimal place • Column labels are the second decimal place forz • Entries are area under the curve • to the left of z, P(Z < z) • Example:Area over z< 1.54 From z table .9382 0 z = 1.54

  15. Lower tail probability • Example: Lower tail area for z = -2.00 • Area for z< -2.00 From z table .0227 z = -2.00

  16. Upper tail probability • Example: Upper tail area for z = 1.54 • Area for z> 1.54 Total Area = 1 1-.9382=.0618 .9382 z = 1.54

  17. Two-tail probability • Example: Two-tail area for z = 1.77 • Area for |z|> 1.77 • P(z < -1.77) + P(z > 1.77) • = .039 + .039 =.078 Z=-1.77 Z=1.77

  18. Proportion over a finite interval • Example: Proportion for -1 < z < 1.54 P(z < 1.54) = 0.9382 P(z < -1.00) = 0.1587 P(-1< z < 1.54) = 0.9382 – 0.1587

  19. Percentiles of the standard normal • Find 90th percentile of the standard normal From z table 90% z = 1.28 z = ?

  20. Symmetric middle interval • Example: Find z values for middle 95% area • Find z values for two-tail area of .05 • Find z values for lower and upper areas of .025 z = -1.96 and z = 1.96 .025 .025 .95 Z=-1.96 Z=1.96

  21. Proportion for any normal distribution • P(a < x < b)when xis N(m,s). • Standardizeaandb • Use the z table From z Table Example:N(50,8) • P(x < 60) = ? • Compute the z value .8944 z = 1.25

  22. Percentiles of any normal distribution • Compute the 75th percentile for N(50,8) • Find the 75th percentile of the standard normal • Compute x = 50 + .675 (8) = 55.40 .7500 z =.675

  23. MINITAB calculation • (Ref. Discussion Problems DP3)

  24. Chi-square distribution • Chi-sq. distribution has one parameter • The parameter is called “degrees of freedom” (df) • Right skewed • Applications: Statistical analysis, time to occurrence of events, … • Examples of Chi-sq. densities

  25. Example of a Chi-square density • Chi-sq. Table • First column gives df for each row • Columns correspond to upper tail areas • MINITAB calculations (Ref. Discussion Problems DP3)

  26. Log-normal distribution • Y=log X • Transformation of X • Non-linear function (increasing) • Distribution of Y is N(m,s) • Distribution of X is called Log-normal LN(m,s) • m is called the location parameter (m is not the mean) • s is called the scale parameter (s is not the standard deviation) • The Log-normal distribution can have a third parameter called “threshold” (we always use threshold=0) • Right skewed • Many applications: Income, price, sales, …

  27. Example of a Log-normal density • Calculations • Use normal calculation for X and transform (increasing function) • P(X<a) = P(Log X < Log a) = P(Y< Log a), f(y) is normal • MINITAB calculations (Ref. Discussion Problems DP3)

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