Probability Statistics Notes & Formulas

1. Definition

Probability: Mathematical framework for uncertainty. Statistics: Methods for collecting, analyzing, and interpreting data.

2. Basic Probability (Very Important)

• P(A∪B) = P(A) + P(B) − P(A∩B)
• P(A|B) = P(A∩B) / P(B) [Conditional Probability]
• A and B independent iff P(A∩B) = P(A)·P(B)
• P(A') = 1 − P(A)

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

Total Probability: P(B) = Σ P(B|Aᵢ)·P(Aᵢ) for partition {Aᵢ}

3. Random Variables

• Discrete RV: Countable values. PMF: P(X=x) ≥ 0, ΣP(X=x) = 1
• Continuous RV: PDF f(x) ≥ 0, ∫f(x)dx = 1. P(a≤X≤b) = ∫ₐᵇ f(x)dx

Expectation & Variance:

• E[X] = Σ x·P(X=x) [discrete] or ∫x·f(x)dx [continuous]
• Var(X) = E[X²] − (E[X])²
• E[aX+b] = aE[X]+b
• Var(aX+b) = a²Var(X)

4. Standard Distributions (Very Important)

Discrete:
| Distribution | PMF | Mean | Variance |
|---|---|---|---|
| Bernoulli(p) | P(X=1)=p | p | p(1−p) |
| Binomial(n,p) | ⁿCₓ pˣ(1−p)ⁿ⁻ˣ | np | np(1−p) |
| Poisson(λ) | e⁻λλˣ/x! | λ | λ |
| Geometric(p) | (1−p)ˣ⁻¹p | 1/p | (1−p)/p² |

Continuous:
| Distribution | PDF | Mean | Variance |
|---|---|---|---|
| Uniform(a,b) | 1/(b−a) | (a+b)/2 | (b−a)²/12 |
| Exponential(λ) | λe⁻λx | 1/λ | 1/λ² |
| Normal(μ,σ²) | Bell curve | μ | σ² |

5. Normal Distribution

• Standard normal: Z = (X−μ)/σ ~ N(0,1)
• 68-95-99.7 rule: P(μ±σ)=68%, P(μ±2σ)=95%, P(μ±3σ)=99.7%
• Sum of independent normals: X+Y ~ N(μ₁+μ₂, σ₁²+σ₂²)

6. Statistics

Central Tendency: Mean = Σxᵢ/n, Median = middle value, Mode = most frequent
Dispersion: Variance = Σ(xᵢ−x̄)²/n, SD = √Variance

Covariance & Correlation:

• Cov(X,Y) = E[XY] − E[X]E[Y]
• ρ = Cov(X,Y) / (σₓ·σᵧ), −1 ≤ ρ ≤ 1
• Independent → Cov = 0, but Cov = 0 does NOT imply independence

7. Conditional Expectation

E[X+Y] = E[X] + E[Y] always (linearity)
E[XY] = E[X]·E[Y] only if X,Y independent

8. GATE Trick

Poisson: both mean and variance = λ (unique property). Exponential distribution is memoryless: P(X>s+t|X>s) = P(X>t). Bayes' theorem questions: carefully identify prior, likelihood, and evidence. Binomial → Poisson approximation when n large, p small, np=λ moderate.