Bivariate Distributions

Characteristics of Bivariate Distributions

Versión en Español
Colección de JavaScript Estadísticos en los E.E.U.U.
Sitio Espejo para América Latina

This site is a part of the JavaScript E-labs learning objects for decision making. Other JavaScript in this series are categorized under different areas of applications in the MENU section on this page.

Professor Hossein Arsham   

In entering your data to move from cell to cell in the data-matrix use the Tab key not arrow or enter keys. Blank boxes are not included in the calculations but zeros are.

Risk Assessment Process: Clearly, different subjective probability models are plausible they can give quite different answers. These examples show how important it is to be clear about the objectives of the modeling. An important application of subjective probability models is in modeling the effect of state-of-knowledge uncertainties in consequence models. Often it turns out that dependencies between uncertain factors can be important in driving the output of the models. For example, consider two portfolios having random X and Y returns; the ratio:

Cov(X, Y) / Var (X)

is called the beta of the random variable X with respect to Y. Various methods are available to model these dependencies, in particular proportional to the Beta values methods.

Buying Gold or Foreign Currencies Investment Decision: Consider the following two investments with the given rate of returns. Given you wish to invest $12,000 over a period of one year, how do you invest for the optimal strategy?

States of Nature (Economy)
Growth Medium G No Change Low
Actions Buy Currencies (C) 5 4 3 -1
Buy Gold (G) 2 3 4 5

Using the JavaScript with equally likelihood (0.25), we obtain:

Beta (Currencies) = -0.457831,    and    Beta (Gold) = -1.9

Now, one may distribute the total capital ($12000) proportional to the Beta values:

Sum of Beta’s = -0.457831 -1.9 = -2.357831

Y1 = 12000 (-0.457831 /-2.357831) = 12000(0.194175) = $2330,    Investing on Foreign Currencies

Y2 = 12000 (-1.9/-2.357831) = 12000(0.805825) = $9670,    Investing on Gold

That is, the optimal strategic decision based upon the Beta criterion is:

Buy $2330 foreign Currencies,    and    $9670 Gold.

Notice that Beta1 and Beta2 are directly related, for example, the multiplication of the two provides the correlation square, i.e. r2. The r2 which always between [0, 1] is a number without any dimensional units, and it represent strong is the linear dependency between the rates of return of one portfolios against the other one. When any beta is negative, and the r2 is large enough, then the two portfolios are related inversely and strongly. In such a case, diversification of the total capital is recommended.


 Variable X

 Variable Y

Marginal Characteristics of Random Variable X
Expected Value: Variance:
Standard Deviation: Coefficient of Variation (%):
Marginal Characteristics of Random Variable Y
Expected Value: Variance:
Standard Deviation: Coefficient of Variation (%):
Join Characteristics of Random Variables X, and Y
Covariance: Correlation:
Beta 1: Beta 2:

For Technical Details, Back to:
Statistical Thinking for Decision Making

Kindly email your comments to:
Professor Hossein Arsham

Muestreo Estadístico de Doble Variación
Nota para los usuarios de habla hispana:
Introduzca hasta 42 pares de datos muestrales donde las mediciones son hechas sobre dos variables aleatorias (X, Y) por caso, y luego presione el botón Calculate (Calcular). Los espacios en blanco no son asumidos como ceros ni incluidos en los cálculos, pero los números cero si se incluyen. Esta matriz reconoce al punto (.) como el signo decimal en vez de la coma (,).
Mientras entre sus datos en la matriz, muévase de celda a celda usando la tecla Tab, no use la flecha o la tecla de entrada.
Los resultados que usted obtendrá de esta matriz son:
Mean (X) = Media (X)
Mean (Y) = Media(Y)
Variance (X) = Varianza (X)
Variance (Y) = Varianza (Y)
Covariate (X, Y) = Covarianza de (X, Y)
Cov(X, Y) / Var (X)= Beta Para Detalles Técnicos y Aplicaciones, Vuelta a:
Razonamiento Estadístico para la Toma de Decisiones Gerenciales


Decision Tools in Economics & Finance

Probabilistic Modeling


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Kindly e-mail me your comments, suggestions, and concerns. Thank you.

Professor Hossein Arsham   

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