Engineering Mathematics 2 — Probability, Statistics, Numerical Methods

Unit 1: Probability

Basic Probability

• Sample Space (S): Set of all possible outcomes
• Event (A): Subset of sample space
• Probability: P(A) = favourable outcomes / total outcomes

Axioms:

• 0 ≤ P(A) ≤ 1
• P(S) = 1
• P(A∪B) = P(A) + P(B) - P(A∩B)

Conditional Probability: P(A|B) = P(A∩B)/P(B)

Independent Events: P(A∩B) = P(A)·P(B)

Bayes' Theorem: P(A|B) = P(B|A)·P(A) / P(B)

Probability Distributions

Binomial Distribution B(n,p):

• n trials, each with probability p of success
• P(X=r) = C(n,r)·pʳ·(1-p)ⁿ⁻ʳ
• Mean = np; Variance = np(1-p)

Poisson Distribution:

• Models rare events in fixed interval
• P(X=k) = e⁻λ·λᵏ/k! (λ = average rate)
• Mean = Variance = λ

Normal Distribution N(μ,σ²):

• Bell-shaped, symmetric around mean μ
• 68-95-99.7 rule: 68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ
• Standard normal: Z = (X-μ)/σ

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Unit 2: Statistics

Measures of Central Tendency:

• Mean: Sum/count
• Median: Middle value (or avg of two middle values)
• Mode: Most frequent value

Measures of Dispersion:

• Range: Max - Min
• Variance: σ² = Σ(x-μ)²/n
• Standard Deviation: σ = √Variance
• Coefficient of Variation: CV = (σ/μ)×100%

Correlation: Measures relationship between two variables

• r = Σ[(x-x̄)(y-ȳ)] / [n·σx·σy]
• r = +1: perfect positive; r = -1: perfect negative; r = 0: no correlation

Regression Line: y = a + bx

• b = r·(σy/σx)
• a = ȳ - b·x̄

Chi-Square Test: Tests if observed frequencies match expected

• χ² = Σ(O-E)²/E

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Unit 3: Numerical Methods

Root Finding Methods

Bisection Method: If f(a)·f(b) < 0 (sign change), root exists between a and b. Repeatedly halve the interval: c = (a+b)/2

• If f(c)=0: root found
• If f(a)·f(c)<0: root in [a,c]
• Else: root in [c,b]
• Convergence: guaranteed but slow

Newton-Raphson Method: Xₙ₊₁ = Xₙ - f(Xₙ)/f'(Xₙ)

• Fast convergence (quadratic)
• Needs derivative
• May diverge if starting point is bad

Secant Method: Xₙ₊₁ = Xₙ - f(Xₙ)·(Xₙ-Xₙ₋₁)/(f(Xₙ)-f(Xₙ₋₁))

• Like Newton-Raphson but without needing derivative

Interpolation

Lagrange Interpolation: P(x) = Σ yᵢ·Lᵢ(x) where Lᵢ(x) = Π(x-xⱼ)/(xᵢ-xⱼ) for j≠i

Newton's Divided Difference: Used when data points are unequally spaced

Newton's Forward Difference: Used for equally spaced points; good near beginning

Numerical Integration

Trapezoidal Rule: ∫ₐᵇ f(x)dx ≈ h/2·[f(x₀) + 2f(x₁) + ... + 2f(xₙ₋₁) + f(xₙ)] Error: O(h²)

Simpson's 1/3 Rule: (n must be even) ∫ₐᵇ f(x)dx ≈ h/3·[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + f(xₙ)] Error: O(h⁴) — more accurate than trapezoidal

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Unit 4: Fourier Series

A periodic function can be expressed as sum of sine and cosine terms.

Fourier Series: f(x) = a₀/2 + Σ[aₙcos(nπx/L) + bₙsin(nπx/L)]

Coefficients:

• a₀ = (1/L)∫₋ₗᴸ f(x)dx
• aₙ = (1/L)∫₋ₗᴸ f(x)cos(nπx/L)dx
• bₙ = (1/L)∫₋ₗᴸ f(x)sin(nπx/L)dx

Even function: f(-x) = f(x) → only cosine terms (bₙ = 0) Odd function: f(-x) = -f(x) → only sine terms (aₙ = 0)

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Solved PYQ Questions

Q1 (2023): In a batch of 1000 items, mean=50, SD=5. Find P(45 ≤ X ≤ 55). Z₁ = (45-50)/5 = -1; Z₂ = (55-50)/5 = +1 P(-1 ≤ Z ≤ 1) = 2×P(0≤Z≤1) = 2×0.3413 = 0.6826 = 68.26%

Q2 (2023): Apply Newton-Raphson to find root of x³ - 2x - 5 = 0 starting from x₀=2. f(x) = x³-2x-5; f'(x) = 3x²-2 x₁ = 2 - f(2)/f'(2) = 2 - (8-4-5)/(12-2) = 2 - (-1/10) = 2.1 x₂ = 2.1 - f(2.1)/f'(2.1) ≈ 2.0946

Q3 (2022): Find variance of 9. Mean = (2+4+4+4+5+5+7+9)/8 = 40/8 = 5 Variance = [(2-5)²+(4-5)²+(4-5)²+(4-5)²+(5-5)²+(5-5)²+(7-5)²+(9-5)²]/8 = [9+1+1+1+0+0+4+16]/8 = 32/8 = 4 SD = 2

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Quick Revision Checklist

• Bayes' theorem formula and application
• Binomial: mean=np, variance=np(1-p)
• Poisson: mean=variance=λ
• Normal: 68-95-99.7 rule; standardize Z=(X-μ)/σ
• Bisection: guaranteed; Newton-Raphson: fast but needs derivative
• Trapezoidal error O(h²); Simpson error O(h⁴)
• Correlation: r range -1 to +1
• Fourier: even function → cosine only; odd function → sine only