Company LOGO 위험도 관리 및 의사결정론 한양대학교 건설관리학과

COMPANY LOGO Frequencyand Probability Distributions

COMPANY LOGO 1. Cost Estimate Example

COMPANY LOGO 1. Cost Estimate Example - 50 professionals estimate Project Cost. - The collected values → $26.7 mil. ~$76.0 mil.  Frequency Histogram Procedure 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 One bar per interval One bar per interval Bar heights → the number(or frequency) of occurrences Bar heights → the number(or frequency) of occurrences ※ (Option) Label the y-axis with Numbers of values or or Percents of the total

COMPANY LOGO  Suppose that we have 500 data points More data and smaller value segment partitions provide the additional detail shown in Fig. 1.4 More data and smaller value segment partitions provide the additional detail shown in Fig. 1.4 If we obtain a great many data points and use smaller intervals If we obtain a great many data points and use smaller intervals → Probability Density Function (p.d.f) or or simply, Probability Distribution 1. Cost Estimate Example

COMPANY LOGO 1. Cost Estimate Example  Probability Distribution Relative likelihood of estimate values along the x-axis Relative likelihood of estimate values along the x-axis The p.d.f represents a judgment about uncertainty. 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.

COMPANY LOGO 2. Popular Central Measures

COMPANY LOGO  Two statistics annotated in Fig. 1.5 Most likely value ($40 mil.) Most likely value ($40 mil.) - This peak is also called mode - Useful in describing a distribution’s shape Expected value ($45 mil.) Expected value ($45 mil.) Probability-weighted average Probability-weighted average Mean value = Greek letter μ Mean value = Greek letter μ Best single measure of value under uncertainty Best single measure of value under uncertainty 2. Popular Central Measures

COMPANY LOGO 3. Cumulative Probability Density Curve

COMPANY LOGO The p.d.f (Fig.1.5) can be converted into a cumulative (probability) density function (c.d.f) curve. 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 Integration of p.d.f Median Median centermost value, at 50% probability centermost value, at 50% probability another central measure another central measure used with demographics to indicate “average” values, such as house price or salaries used with demographics to indicate “average” values, such as house price or salaries 3. Cumulative Probability Density Curve

COMPANY LOGO Confidence Limit Confidence Limit About 74% of the Project Cost estimates: below $50 mil. About 74% of the Project Cost estimates: below $50 mil. ☞ 74% (less-than) confidence limit(or level) is $50 mil. Confidence Interval Confidence Interval About 80% of the Project Cost estimates: $33 mil. ~ $57 mil. About 80% of the Project Cost estimates: $33 mil. ~ $57 mil. ☞ 80% confidence interval is $33 mil. ~ $57 mil. range 3. Cumulative Probability Density Curve

COMPANY LOGO 4. Expected Value

COMPANY LOGO  Expected Value Range $2.2 mil. ~ $2.7 mil. Range $2.2 mil. ~ $2.7 mil. Most likely value $2.3 mil. Low, Most likely, and High values Low, Most likely, and High values ▷ Triangle Distribution The best single-point estimate is expected value(EV) The best single-point estimate is expected value(EV) Performing many similar projects → average $2.4 mil. Performing many similar projects → average $2.4 mil. Over the long run, estimate error will approach zero Over the long run, estimate error will approach zero 4. Expected Value subjective objective (unbiased)

COMPANY LOGO  Calculating Expected Value Distribution expressed as a mathematical formula integral equation: Distribution expressed as a mathematical formula integral equation: Where is the value of the variable and is the p.d.f of EV is the sum of the outcome values times their probabilities: EV is the sum of the outcome values times their probabilities: Where are the outcome values, and are the probabilities of these outcomes. 4. Expected Value

COMPANY LOGO  Measurement of Dispersion Probability Distribution: Better for estimating Risks Probability Distribution: Better for estimating Risks Standard Deviation Standard Deviation = Outcome = Outcome = Expected Value = Expected Value = Probabilities = Probabilities Coefficient of Variation( 변동계수 ) Coefficient of Variation( 변동계수 ) Expected Income: 1 억 Expected Income: 1 억 Standard Deviation: 100 만원 Expected Income: 100 만원 Expected Income: 100 만원 Standard Deviation: 50 만원 → C.O.V = 4. Expected Value

Company LOGO 위험도 관리 및 의사결정론