Marginal Probability Distributions

• Marginal probability distributions are obtained by summing up the joint probability distribution over all values of one of the variables, say x.

• The resulting marginal distribution is the probability of the other variable, in this case y, without regard to x.

• The marginal distribution of X is:

  P (xi) = Σj=1m P(xi, yj)

• Likewise, the marginal distribution of X is:

  P (yj) = Σi=1n P(xi, yj)

• The probability of A being 1 regardless of the value of B (marginal probability) is 1/6.

• Likewise, the probability of B being 4 regardless of the value of A (marginal probability) is 1/6.

• Note that the joint probabilities (1/36) of six possible outcomes have been summed to obtain the marginal probability.

• Marginal cumulative probability distributions are obtained by combining the concepts of marginal and cumulative distributions.

• The marginal cumulative probability distribution of X is:

  F (xk) = Σi=1k Σ j=1m P(xi, yj)

• Likewise, the marginal cumulative probability distribution of Y is:

  F (yl) = Σi=1n Σ j=1l P(xi, yj)

• The probability of A being less than or equal to 2, regardless of the value of B, is 2/6 = 1/3.

• Likewise, the probability of B being less than or equal to 5, regardless of the value of A, is 5/6.

Conditional Probability

• Conditional probability is useful in regression analysis.

• The conditional probability is the ratio of joint and marginal distributions.

• The conditional probability of x, given y, is:

  P(x|y) = P(x, y) / P(y)

• The conditional probability of y, given x, is:

  P(y|x) = P(x, y) / P(x)

• Joint probability is the product of conditional and marginal probabilities.

• Joint probability distributions can be expressed as continuous functions.

• In this case, they are called joint density functions.

• As with univariate distributions, joint density functions have moments.

• The joint moment of order r and s about the origin (indicated with ') is defined as follows:

  μ'r,s = ∫ ∫ xr ys f(x,y) dy dx

• With r= 1 and s= 0, it reduces to the mean of x:

  μ'1,0 = ∫ x [ ∫ f(x,y) dy ] dx

• The expression within brackets is the marginal PDF of x, or f(x):

  μ'1,0 = μx = ∫ x f(x) dx

• With r= 0 and s= 1, it reduces to the mean of y:

  μ'0,1 = ∫ y [ ∫ f(x,y) dx ] dy

• The expression within brackets is the marginal PDF of y, or f(y):

  μ'0,1 = μy = ∫ y f(y) dy

• The second moments are usually written about the mean:

  μr,s = ∫ ∫ (x - μx)r (y - μy)s f(x,y) dy dx

• For r= 2 and s= 0, the second moment reduces to the variance of x.

  σ2x = ∫ (x - μx)2 f(x) dx

• Likewise, for r= 0 and s= 2, the second moment reduce to the variance of y.

  σ2y = ∫ (y - μy)2 f(y) dy

• For r= 1 and s= 1, the second moment reduces to the covariance:

  σx,y = ∫ ∫ (x - μx)(y - μy) f(x,y) dy dx

• The correlation coefficient relates the covariance σ_x,y and standard deviations σ_x and σ_y. The population correlation coefficient is:

  ρx,y = σx,y / (σxσy)

• The sample correlation coefficient is:

  rx,y = sx,y / (sxsy)

• The correlation coefficient is a measure of the linear dependence between x and y.

• It varies in the range -1 and +1.

• A value of ρ close to 1 indicates a strong linear dependence between the variables.

• A values of ρ close to or equal to -1 indicates a correlation such that large values of x are associated with small values of y, and vice versa.

• A value of ρ = 0, i.e., a zero covariance, indicates a lack of a linear dependence between x and y.

• Example 7-1.

• Example 7-1 Solution.

• Bivariate Normal Distribution.

7.2  REGRESSION ANALYSIS

• A fundamental tool of regional analysis is the equation relating two or more hydrologic variables.

• The variable for which values are given is called the predictor variable.

• The variable for which values must be estimated is called the criterion variable.

• Correlation provides a measure of the goodness of fit of the regression.

• Regression provides the parameters; correlation describes its quality.

• The principle of least squares is used to obtain the best estimates of the parameters of the prediction equation.

• It is based on the minimization of the sum of the square of the differences between observed and predicted values.