Math, Statistics & Numerical Methods
DSSSB TGT CS — Section B P2 (Rank ~12). Focus: matrix/det basics, probability, averages, errors, interpolation keywords.
1. Matrices & Determinants (basics)
| Term | Meaning |
|---|---|
| Matrix | Rectangular array (m \times n) |
| Square | (n \times n) |
| Row / column matrix | Single row / column |
| Diagonal / identity | Nonzeros on diagonal; I has 1s on diagonal |
| Transpose (A^T) | Rows ↔ columns |
Operations: add (same order), scalar multiply, multiply (A_{m\times n} \times B_{n\times p}).
Determinant (square only):
- (2\times2): (ad-bc) for (\begin{bmatrix}a&b\c&d\end)
- det = 0 → singular (no inverse)
- det(AB) = det(A)det(B); det(Aᵀ)=det(A)
Inverse: (A^{-1}) exists iff det ≠ 0; (AA^{-1}=I).
2. Probability (classical)
[ P(E) = \frac{\text{favourable outcomes}}{\text{total equally likely outcomes}} ]
| Rule | Formula |
|---|---|
| Range | (0 \le P \le 1) |
| Complement | (P(A') = 1 - P(A)) |
| Addition (mutually exclusive) | (P(A\cup B)=P(A)+P(B)) |
| Addition (general) | (P(A\cup B)=P(A)+P(B)-P(A\cap B)) |
| Independent | (P(A\cap B)=P(A)P(B)) |
- Trap: Classical definition needs equally likely outcomes.
3. Mean, Median, Mode
| Measure | Idea |
|---|---|
| Mean (\bar) | (\sum x_i / n) (arithmetic average) |
| Median | Middle value when sorted (avg of two middles if even n) |
| Mode | Most frequent value |
- Skew: mean pulled toward tail; mode at peak.
- For exam data sets: compute all three quickly.
4. Variance & Standard Deviation
| Population | Sample (exam often uses /n) | |
|---|---|---|
| Variance | (\sigma^2 = \frac{\sum(x_i-\mu)^2}) | (s^2 = \frac{\sum(x_i-\bar{x})^2}) or (n-1) |
| SD | (\sigma = \sqrt{\sigma^2}) | (s=\sqrt) |
- SD = spread around mean; same units as data.
- Variance = square units.
- Trap: Low SD → values close to mean.
5. Correlation vs Regression
| Correlation | Regression | |
|---|---|---|
| Question | How strongly related? | How to predict Y from X? |
| Output | Coefficient r (e.g. −1 to +1) | Equation (line) |
| Causation | Does not prove cause | Models relationship for prediction |
- (r = +1) perfect positive; (r = -1) perfect negative; (r = 0) no linear correlation.
- Regression line: (Y = a + bX) (simple linear).
6. Numerical Methods
Newton–Raphson (idea)
Root finding: (x_{n+1} = x_n - \frac{f(x_n)})
- Needs derivative; quadratic convergence when it works.
- May fail if (f'=0) or bad start.
Interpolation
Estimate unknown (f(x)) between known tabulated points.
| Method | Idea |
|---|---|
| Lagrange | Weighted sum of y-values with Lagrange basis polynomials |
| Newton forward | Uses forward differences; good for equal spacing, start of table |
- Extrapolation = outside range (less reliable).
7. Errors
| Type | Definition |
|---|---|
| Absolute error | |true − approx| |
| Relative error | absolute / |true| |
| Percentage error | relative × 100% |
Also: truncation error (series cutoff), round-off error (finite precision).
8. Boolean / Number System (light overlap)
- Binary, octal, decimal, hex conversions (see Digital notes for depth).
- Boolean: AND, OR, NOT; identities; De Morgan.
- Link: digital logic questions may appear near numerical — keep conversions fluent.
| Dec | Bin | Hex |
|---|---|---|
| 10 | 1010 | A |
| 15 | 1111 | F |
Quick Revision Traps
- det = 0 → no inverse.
- Median uses sorted data; mean does not need sort.
- Correlation ≠ causation.
- Relative error is dimensionless ratio.
- Newton–Raphson uses (f/f').
- Lagrange vs Newton forward — both interpolate; Newton forward uses difference table.
- SD is √variance.