Asian School of BusinessPG Programme in Management (2005-06)

Course: Quantitative Methods in Management IInstructor: Chandan Mukherjee

Session 10: Useful Theoretical Distributions (contd.)

Normal Distribution

2

2

1exp

2

1)(

x

xf

- ∞ < x < ∞ ; - ∞ < μ < ∞ ; σ > 0

X ~ N (μ, σ2)

Go to the Excel File ‘normal-distn.xls’ and fit the Normal Distribution.

Do you feel that the model describes the distribution satisfactorily?

Exercise with Normal Distribution

What is the probability that the measurements are within + or – 2mm?

Normal Distribution Properties

If X ~ N (μ, σ2) then

(X- μ)/ σ ~ N(0,1) ==> Standard Normal Distribtuion

Standard Deviate

Normal Distribution Properties

If X ~ N (μ, σ2) then

P (μ - 2σ < X <= 2σ) = ?

P (μ - 3σ < X <= 3σ) = ?

P (μ - 6σ < X <= 6σ) = ?

If your target measurement is 100 and the tolerence limit is +/– 2% then what is the tolerance limit for σ in the system?

Six Sigma Quality: “Commonly defined as 3.4 defects per million opportunities ….”

If you are interested in learning about six sigma quality standard, you may like to refer to the following site:

http://www.isixsigma.com/library/content/six-sigma-newbie.asp

Lognormal Distribution

X has a Lognormal Distribution with parameters μ and σ2, if loge(X) i.e. ln(X) has a Normal Distribution with parameters μ and σ2

E(X) =

V(X) =

Example: Fitting a Lognormal Distribution

Annual Sales (in Rs. Lakhs) of a product by a sample of 50 firms

Go to the Excel File ‘lognormal-distn.xls’ and fit the Lognormal Distribution.

Do you feel that the model describes the distribution satisfactorily?

Shares in Lognormal Distribution

If X ~ LN (μ, σ2) then -

Share of units below say, K

= P [Y <= (lnK- μ)/ σ)- σ] where Y ~ N(0,1)

•What is the share in total sales of the firms with annual sales above 5 Lakhs?

•What is the share of the top 5% of the firms?

Measuring Concentration or Inequality in a Distribution: Lorenz Curve & Lorenz Ratio (Gini Co-efficient)

•Lorenz Curve: Plot of Cummulative Shares against Cummulative Relative Frequency

•Lorenz Ratio (Gini Co-efficient) =

Twice the area between the Egalitarian Line and the Lorenz Curve. It lies between 0 and 1, no inequality and extreme equality, respectively.

•In Lognormal Distribution:

Lorenz Ratio = 2.Prob(X ≤ σ/√2) -1 where X ~ N(0,1)

Measuring Concentration or Inequality in a Distribution: Herfindahl Index

N = Number of firms in an industry or a segment of an industry

Si = Share of the ith firm (in total sales, say)

Herfindahl-Hirschman Index =

Below 1000: Competitive

1000-1800: Moderate Concentration

Above 1800: Towards Monopolistic situation

N

iiS

1

2

Measures of concentration Two measures are commonly used by economists to judge the relative amount of concentration in any industry. They are the Concentration Ratio and the Herfindahl Index.

Concentration ratio The concentration ratio is expressed in the terms CRx, which stands for the percentage of the market sector controlled by the biggest x firms. For example, CR3 = 70% would indicate that the top three firms control 70% of a market. CR4 is the most typical concentration ratio for judging what kind of an oligopoly it is. A CR4 of over 50% is generally considered a tight oligopoly; CR4 between 25 and 50 is generally considered a loose oligopoly. A CR4 of under 25 is no oligopoly at all. We would add that a CR3 of over 90% or a CR2 of over 80% should be considered a super-tight oligopoly. The problem with this measure is that CR4 does not indicate what the relative size of the four largest companies is. It may be that a CR4 of 80 means that one company controls 50% of the market, while the others have 10% apiece. That's a very different market structure than one where every firm has a 20% share.

An Additional Note on Market Concentration