# The relationship between distribution and

Probability distributions have a surprising number inter-connections. A dashed line in the chart below indicates an approximate (limit) relationship between two. In spite of the variety of the probability distributions, many of them are related to each other by different kinds of relationship. Deriving the probability distribu-. The relationship between distribution and market share has increasingly become a matter of concern for consumer goods marketers. Competition for shelf space.

Type of Report Technical presentation of model; formulation of propositions; test against empirical data Objective To develop a model of the distribution-market share relationship that explicitly accounts for: Method Identifies relevant areas of the literature and conceptualizes a model wherein consumer "pull" and manufacturer "push" interact to affect market share; defines the constructs used in the conceptualization; formalizes the model and sets forward two propositions of trade behavior under specific assumptions; tests the model with empirical data; examines the implications of the postulated between distribution and market share; suggests opportunities for further research.

Audience This technical report is directed toward modelers and others with considerable quantitative proficiency.

Principal Findings The model developed in this paper relates distribution and market share. In doing so, it uses very specific definitions for consumer preference, resistance to compromise, distribution, and in-store attractiveness.

**ST102 notes: Relationship between t-distribution and the F-distribution**

These terms are further related in a conceptual "push-pull" model that treats "push" and "pull" as effects of marketing mix decisions rather than as traditional classifications of marketing strategies.

Understanding these differences in definition are essential to appreciating the findings of the research. In general terms, distribution and market share, as treated by the authors, are related in a nonlinear fashion.

As market share increases, distribution breadth increases more-than proportionately, reflecting the trade's tendency to stock those brands which sell well--and also the tendency to give those brands a larger share of shelf space and to stock more stock-keeping units. Conversely, as distribution is increased, the greater availability of the product, combined with the trade support for it, results in larger-than-proportional increases in market share.

Unavailability of a smaller brand, either because of stockouts or restricted distribution, benefits larger brands more than smaller ones.

### The Relationship Between Distribution and Market Share - MSI Web Site »

This result is implicit in the stocking behavior of retailers: Given the choice between a large brand and a small brand and given equal levels of trade supportretailers will preferentially choose the brand for which there is greater demand--and consequently the one with the larger market share.

Increases in consumers' willingness to search for a brand --and the trade support for that brand--combine synergistically to affect brand availability and market share.

Thus, consumer franchise-building activity pays off in two ways: First, such actions serve to enhance consumers' willingness to search for the brand and raise demand for the brand. The relationship is simpler in terms of failure probabilities: For more information, see Poisson approximation to binomial.

## The Relationship Between Distribution and Market Share

The sum of n Bernoulli p random variables is a binomial n, p random variable. For more information, see normal approximation to Poisson.

If X is a binomial n, p random variable and Y is a normal random variable with the same mean and variance as X, i. For more information, see normal approximation to binomial. For more information, see normal approximation to beta.

### Diagram of probability distribution relationships

For more information, see normal approximation to gamma. The square of a standard normal random variable has a chi-squared distribution with one degree of freedom. The sum of the squares of n standard normal random variables is has a chi-squared distribution with n degrees of freedom.

For more information, see normal approximation to t. A chi-squared distribution with 2 degrees of freedom is an exponential distribution with mean 2. An exponential random variable with mean 2 is a chi-squared random variable with two degrees of freedom.

More generally, sticking any random variable into its CDF yields a uniform random variable. More generally, applying the inverse CDF of any random variable X to a uniform random variable creates a variable with the same distribution as X. A random variable with a t distribution with one degree of freedom is a Cauchy 0,1 random variable.

- Relationships among probability distributions
- Diagram of distribution relationships
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