COMPANY LOGO Frequency and Probability Distributions COMPANY LOGO
COMPANY LOGO Frequency and Probability Distributions
COMPANY LOGO 1. Cost Estimate Example
1. Cost Estimate Example - 50 professionals estimate Project Cost. - The collected values → $26. 7 mil. ~$76. 0 mil. Ü Frequency Histogram Procedure ① Divide the range of values (5~50 intervals) ② Count the number of values occurring within each interval. ③ Draw a bar graph n One bar per interval n Bar heights → the number(or frequency) of occurrences ※ (Option) Label the y-axis with Numbers of values or Percents of the total n COMPANY LOGO
1. Cost Estimate Example COMPANY LOGO Ü Suppose that we have 500 data points More data and smaller value segment partitions provide the additional detail shown in Fig. 1. 4 n If we obtain a great many data points and use smaller intervals → Probability Density Function (p. d. f) or simply, Probability Distribution n
1. Cost Estimate Example COMPANY LOGO Ü Probability Distribution Relative likelihood of estimate values along the x-axis n The p. d. f represents a judgment about uncertainty. → Unlike a frequency distribution, the p. d. f is not representing data. Y-axis is scaled so that the area under the curve equals 1. n
COMPANY LOGO 2. Popular Central Measures
2. Popular Central Measures COMPANY LOGO Ü Two statistics annotated in Fig. 1. 5 n n Most likely value ($40 mil. ) - This peak is also called mode - Useful in describing a distribution’s shape Expected value ($45 mil. ) n Probability-weighted average n Mean value = Greek letter μ n Best single measure of value under uncertainty
COMPANY LOGO 3. Cumulative Probability Density Curve
3. Cumulative Probability Density Curve n n n COMPANY LOGO The p. d. f (Fig. 1. 5) can be converted into a cumulative (probability) density function (c. d. f) curve. Integration of p. d. f Median n centermost value, at 50% probability n another central measure n used with demographics to indicate “average” values, such as house price or salaries
3. Cumulative Probability Density Curve COMPANY LOGO n Confidence Limit n About 74% of the Project Cost estimates: below $50 mil. ☞ 74% (less-than) confidence limit(or level) is $50 mil. n Confidence Interval n About 80% of the Project Cost estimates: $33 mil. ~ $57 mil. ☞ 80% confidence interval is $33 mil. ~ $57 mil. range
COMPANY LOGO 4. Expected Value
4. Expected Value COMPANY LOGO Ü Expected Value Range $2. 2 mil. ~ $2. 7 mil. Most likely value $2. 3 mil. n Low, Most likely, and High values ▷ Triangle Distribution n The best single-point estimate is expected value(EV) n n n Performing many similar projects → average $2. 4 mil. Over the long run, estimate error will approach zero subjective objective (unbiased)
4. Expected Value COMPANY LOGO Ü Calculating Expected Value n Distribution expressed as a mathematical formula integral equation: Where n is the value of the variable and is the p. d. f of EV is the sum of the outcome values times their probabilities: Where are the outcome values, and these outcomes. are the probabilities of
4. Expected Value Ü Measurement of Dispersion n n Probability Distribution: Better for estimating Risks Standard Deviation = Outcome = Expected Value = Probabilities n n Coefficient of Variation(변동계수) n n Expected Income: 1억 Standard Deviation: 100만원 Expected Income: 100만원 Standard Deviation: 50만원 → C. O. V = COMPANY LOGO
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