Mathematical Methods in PDEs and Statistical Analysis

Posted on May 5, 2026 in Statistics

Partial Differential Equations (PDE)

A Partial Differential Equation (PDE) involves a function u(x, y, …) and its partial derivatives.

  • Homogeneous: If every term in the equation contains the dependent variable u or its derivatives. The general solution is simply the Complementary Function (C.F.).
  • Non-Homogeneous: If there is a term that is a function of the independent variables only (f(x, y)). The solution is u = C.F. + P.I. (Particular Integral). Example: ∇2u = f(x, y) (Poisson’s Equation).

The Multiplier Method (Lagrange’s Linear PDE)

To solve Pp + Qq = RS, we use the auxiliary equation: dx/P = dy/Q = dz/R.

The Method: We find multipliers (l, m, n) such that lP + mQ + nR = 0, which implies l dx + m dy + n dz = 0. By integrating this, we get a constant u(x, y, z) = C1. We repeat the process to find v(x, y, z) = C2. The general solution is Φ(u, v) = 0.

Probability Distributions

Binomial Distribution

Used for n independent trials with a constant probability of success p.

Formula:
Mean: np
Variance: npq

Poisson Distribution

A limiting case of the Binomial distribution where n approaches infinity and p approaches 0, such that np = λ (a constant).

Formula:
Mean = Variance = λ

Normal Distribution (Gaussian)

The most important continuous distribution for natural phenomena.

Standard Normal Variable (Z): Z = (X – μ) / σ, which centers the mean at 0 and variance at 1.

Moments and Generating Functions

Moments about Mean vs. Origin

  • About Origin (μ′r): The r-th moment is E[Xr]. The first moment μ′1 is the Mean.
  • About Mean (μr): Also called central moments. μr = E[(X – μ)r].
    • μ1 = 0
    • μ2 = σ2 (Variance)
    • Relationship: μ2 = μ′2 – (μ′1)2

Moment Generating Function (MGF)

The MGF “encodes” all moments of a distribution into a single function.

Derivation of Moments: To find the r-th moment about the origin, differentiate the MGF r times and evaluate at t = 0.

Skewness

Measures the lack of symmetry in a distribution.

  • Positive Skew: Tail extends to the right (Mean > Median > Mode).
  • Negative Skew: Tail extends to the left (Mode > Median > Mean).
  • Formula (Karl Pearson): Sk = (Mean – Mode) / σ

Covariance, Correlation, and Regression

Covariance: Cov(X, Y) = E[XY] – E[X]E[Y]. It shows the direction of the linear relationship.

Karl Pearson Correlation (r): Normalizes covariance to a range of -1 to +1.

Rank Correlation (Spearman’s): Used when data is ranked (qualitative).

Regression Analysis

Predicts the dependent variable based on the independent variable.

Regression Line (Y on X): y – ȳ = Byx(x – x̄)
Regression Coefficient: Byx represents the slope of the line.

Curve Fitting (Method of Least Squares)

To fit a curve y = f(x) to a set of points, we minimize the sum of the squares of the residuals.

For a straight line (y = a + bx):
1. Σy = na + bΣx
2. Σxy = aΣx + bΣx2

Inferential Statistics: Hypothesis Testing

Test of Hypothesis: A procedure to decide whether to accept or reject a statistical claim.

  • Null Hypothesis (H0): Statement of no change or no effect.
  • Alternative Hypothesis (H1): What we suspect is true.
  • Level of Significance (α): Usually 5% (0.05).
  • Critical Region: If the calculated test statistic falls here, we reject H0.

Chi-Square (χ2) Test

Used to check if observed frequencies (O) match expected frequencies (E).

Formula: χ2 = Σ [ (Oi – Ei)2 / Ei ]
Degrees of Freedom (df): (n – 1) for a 1D table, or (r – 1)(c – 1) for a contingency table.

Recursion (Statistical Context)

In probability, recursion is often used to find moments for complex distributions. For the Poisson Distribution, the central moments follow a specific recurrence relation.

Extended Reference and Review

A PDE involves a function u(x, y, …) and its partial derivatives. Homogeneous: If every term in the equation contains the dependent variable u or its derivatives. The general solution is simply the Complementary Function (C.F.).

Non-Homogeneous: If there is a term that is a function of the independent variables only (f(x, y)). The solution is u = C.F. + P.I. (Particular Integral). Example: ∇2u = f(x, y) (Poisson’s Equation).

The Multiplier Method (Lagrange’s Linear PDE)

To solve Pp + Qq = RS, we use the auxiliary equation: dx/P = dy/Q = dz/R. The Method: We find multipliers (l, m, n) such that lP + mQ + nR = 0, which implies l dx + m dy + n dz = 0. By integrating this, we get a constant u(x, y, z) = C1. We repeat the process to find v(x, y, z) = C2. The general solution is Φ(u, v) = 0.

Distribution Summary

Binomial Distribution: Used for n independent trials with a constant probability of success p. Formula: Mean: np, Variance: npq.

Poisson Distribution: A limiting case of Binomial where n goes to infinity and p goes to 0, such that np = λ (a constant). Formula: Mean = Variance = λ.

Normal Distribution (Gaussian): The most important continuous distribution for natural phenomena. Standard Normal Variable (Z): Z = (X – μ) / σ, which centers the mean at 0 and variance at 1.

Moments and Generating Functions

Moments about Mean vs. Origin:
About Origin (μ′r): The r-th moment is E[Xr]. The first moment μ′1 is the Mean.
About Mean (μr): Also called central moments. μr = E[(X – μ)r]. μ1 = 0, μ2 = σ2 (Variance). Relationship: μ2 = μ′2 – (μ′1)2.

Moment Generating Function (MGF): The MGF “encodes” all moments of a distribution into a single function. Derivation of Moments: To find the r-th moment about the origin, differentiate the MGF r times at t = 0.

Skewness: Measures the lack of symmetry in a distribution. Positive Skew: Tail extends to the right (Mean > Median > Mode). Negative Skew: Tail extends to the left (Mode > Median > Mean). Formula (Karl Pearson): Sk = (Mean – Mode) / σ.

Correlation and Regression

Covariance and Correlation: Cov(X, Y) = E[XY] – E[X]E[Y]. It shows the direction of the linear relationship. Karl Pearson Correlation (r): Normalizes covariance to a range of -1 to +1. Rank Correlation (Spearman’s): Used when data is ranked (qualitative).

Regression Analysis: Predicts the dependent variable based on the independent variable. Regression Line (Y on X): y – ȳ = Byx(x – x̄). Regression Coefficient: Byx.

Curve Fitting (Method of Least Squares): To fit a curve y = f(x) to a set of points, we minimize the sum of the squares of the residuals. For a straight line (y = a + bx): Σy = na + bΣx and Σxy = aΣx + bΣx2.

Hypothesis Testing and Chi-Square

Inferential Statistics: Test of Hypothesis is a procedure to decide whether to accept or reject a statistical claim. Null Hypothesis (H0): Statement of no change. Alternative Hypothesis (H1): What we suspect is true. Level of Significance (α): Usually 0.05. Critical Region: If the calculated test statistic falls here, we reject H0.

Chi-Square (χ2) Test: Used to check if observed frequencies (O) match expected frequencies (E). Formula: χ2 = Σ [ (Oi – Ei)2 / Ei ]. Degrees of Freedom (df): (n – 1) or (r – 1)(c – 1).

Recursion (Statistical context): In probability, recursion is often used to find moments for complex distributions. For the Poisson Distribution, the central moments follow a specific recurrence relation.