PE Exam Exam > PE Exam Notes > Engineering Fundamentals Revision for PE > Cheatsheet: Probability Distributions
Cheatsheet: Probability Distributions
1. Fundamental Probability Concepts
1.1 Basic Definitions
| Term | Definition |
|---|---|
| Random Variable (RV) | A variable whose value is determined by the outcome of a random phenomenon |
| Discrete RV | Takes countable values (finite or infinite) |
| Continuous RV | Takes any value within an interval or intervals |
| Probability Mass Function (PMF) | f(x) = P(X = x) for discrete RV; gives probability at specific values |
| Probability Density Function (PDF) | f(x) for continuous RV; P(a ≤ X ≤ b) = ∫ab f(x)dx |
| Cumulative Distribution Function (CDF) | F(x) = P(X ≤ x); non-decreasing function from 0 to 1 |
1.2 Key Properties
- PMF: Σf(x) = 1 for all x; 0 ≤ f(x) ≤ 1
- PDF: ∫-∞∞ f(x)dx = 1; f(x) ≥ 0
- CDF: F(x) = Σf(xi) for discrete; F(x) = ∫-∞x f(t)dt for continuous
- Relationship: f(x) = dF(x)/dx for continuous distributions
1.3 Expected Value and Variance
| Measure | Formula |
|---|---|
| Expected Value (Mean) μ | Discrete: E[X] = Σx·f(x); Continuous: E[X] = ∫x·f(x)dx |
| Variance σ² | Var(X) = E[(X - μ)²] = E[X²] - (E[X])² |
| Standard Deviation σ | σ = √Var(X) |
| E[aX + b] | aE[X] + b |
| Var(aX + b) | a²Var(X) |
2. Discrete Probability Distributions
2.1 Bernoulli Distribution
| Parameter | Value/Formula |
|---|---|
| Notation | X ~ Bernoulli(p) |
| PMF | f(x) = px(1-p)1-x for x ∈ {0,1} |
| Mean | μ = p |
| Variance | σ² = p(1-p) |
| Application | Single trial with two outcomes (success/failure) |
2.2 Binomial Distribution
| Parameter | Value/Formula |
|---|---|
| Notation | X ~ B(n,p) |
| PMF | f(x) = C(n,x)·px·(1-p)n-x for x = 0,1,2,...,n |
| Mean | μ = np |
| Variance | σ² = np(1-p) |
| Standard Deviation | σ = √(np(1-p)) |
| Conditions | n independent trials; constant probability p; two outcomes per trial |
2.3 Poisson Distribution
| Parameter | Value/Formula |
|---|---|
| Notation | X ~ Poisson(λ) |
| PMF | f(x) = (e-λ·λx)/x! for x = 0,1,2,... |
| Mean | μ = λ |
| Variance | σ² = λ |
| Standard Deviation | σ = √λ |
| Parameter λ | Average rate of occurrences per unit interval (time, space, volume) |
| Application | Count of rare events in fixed interval; arrivals, defects, accidents |
- Approximation: Binomial with large n, small p (n > 20, p < 0.05);="" use="" λ="">
2.4 Geometric Distribution
| Parameter | Value/Formula |
|---|---|
| Notation | X ~ Geometric(p) |
| PMF | f(x) = (1-p)x-1·p for x = 1,2,3,... |
| Mean | μ = 1/p |
| Variance | σ² = (1-p)/p² |
| Application | Number of trials until first success |
2.5 Hypergeometric Distribution
| Parameter | Value/Formula |
|---|---|
| Notation | X ~ Hypergeometric(N,K,n) |
| PMF | f(x) = [C(K,x)·C(N-K,n-x)]/C(N,n) |
| Mean | μ = n(K/N) |
| Variance | σ² = n(K/N)(1-K/N)[(N-n)/(N-1)] |
| Parameters | N = population size; K = successes in population; n = sample size |
| Application | Sampling without replacement from finite population |
3. Continuous Probability Distributions
3.1 Uniform Distribution
| Parameter | Value/Formula |
|---|---|
| Notation | X ~ U(a,b) |
| f(x) = 1/(b-a) for a ≤ x ≤ b; 0 otherwise | |
| CDF | F(x) = (x-a)/(b-a) for a ≤ x ≤ b |
| Mean | μ = (a+b)/2 |
| Variance | σ² = (b-a)²/12 |
| Application | Equal probability over interval; random number generation |
3.2 Normal (Gaussian) Distribution
| Parameter | Value/Formula |
|---|---|
| Notation | X ~ N(μ,σ²) |
| f(x) = (1/(σ√(2π)))·e-(x-μ)²/(2σ²) | |
| Mean | μ (location parameter) |
| Variance | σ² (scale parameter) |
| Standard Deviation | σ |
3.2.1 Standard Normal Distribution
| Property | Value |
|---|---|
| Notation | Z ~ N(0,1) |
| Standardization | Z = (X - μ)/σ |
| φ(z) = (1/√(2π))·e-z²/2 | |
| CDF | Φ(z) = P(Z ≤ z); from standard normal table |
3.2.2 Empirical Rule (68-95-99.7)
- P(μ - σ ≤ X ≤ μ + σ) ≈ 0.68 (68.27%)
- P(μ - 2σ ≤ X ≤ μ + 2σ) ≈ 0.95 (95.45%)
- P(μ - 3σ ≤ X ≤ μ + 3σ) ≈ 0.997 (99.73%)
3.2.3 Key Properties
- Symmetric about mean μ
- Bell-shaped curve
- Inflection points at μ ± σ
- Linear combination of normal RVs is normal
- Central Limit Theorem: Sum of independent RVs approaches normal as n increases
3.3 Exponential Distribution
| Parameter | Value/Formula |
|---|---|
| Notation | X ~ Exp(λ) |
| f(x) = λe-λx for x ≥ 0; 0 otherwise | |
| CDF | F(x) = 1 - e-λx for x ≥ 0 |
| Mean | μ = 1/λ |
| Variance | σ² = 1/λ² |
| Parameter λ | Rate parameter (events per unit time) |
| Application | Time between events in Poisson process; waiting times; lifetimes |
- Memoryless property: P(X > s+t | X > s) = P(X > t)
- Relationship to Poisson: If events follow Poisson(λ), time between events is Exp(λ)
3.4 Weibull Distribution
| Parameter | Value/Formula |
|---|---|
| Notation | X ~ Weibull(α,β) |
| f(x) = (β/α)(x/α)β-1e-(x/α)^β for x ≥ 0 | |
| CDF | F(x) = 1 - e-(x/α)^β |
| Mean | μ = αΓ(1 + 1/β) |
| Parameters | α = scale parameter; β = shape parameter |
| Application | Reliability engineering; failure time modeling; wind speed distribution |
- β < 1:="" decreasing="" failure="" rate;="" β="1:" constant="" (exponential);="" β=""> 1: increasing failure rate
3.5 Lognormal Distribution
| Parameter | Value/Formula |
|---|---|
| Notation | X ~ Lognormal(μ,σ²) |
| f(x) = (1/(xσ√(2π)))·e-(ln x - μ)²/(2σ²) for x > 0 | |
| Mean | E[X] = eμ+σ²/2 |
| Variance | Var(X) = (eσ² - 1)e2μ+σ² |
| Relationship | If ln(X) ~ N(μ,σ²), then X ~ Lognormal(μ,σ²) |
| Application | Multiplicative processes; income distribution; particle size; stock prices |
4. Special Distributions
4.1 Gamma Distribution
| Parameter | Value/Formula |
|---|---|
| Notation | X ~ Gamma(α,β) |
| f(x) = (1/(βαΓ(α)))xα-1e-x/β for x > 0 | |
| Mean | μ = αβ |
| Variance | σ² = αβ² |
| Parameters | α = shape parameter; β = scale parameter |
| Application | Sum of exponential RVs; waiting time for α events in Poisson process |
- Gamma(1,β) = Exp(1/β)
- Gamma function: Γ(α) = ∫0∞ tα-1e-tdt; Γ(n) = (n-1)! for integer n
4.2 Beta Distribution
| Parameter | Value/Formula |
|---|---|
| Notation | X ~ Beta(α,β) |
| f(x) = (Γ(α+β)/(Γ(α)Γ(β)))xα-1(1-x)β-1 for 0 ≤ x ≤ 1 | |
| Mean | μ = α/(α+β) |
| Variance | σ² = αβ/[(α+β)²(α+β+1)] |
| Application | Proportions and percentages; Bayesian analysis; project completion times (PERT) |
4.3 Chi-Square Distribution
| Parameter | Value/Formula |
|---|---|
| Notation | X ~ χ²(k) |
| f(x) = (1/(2k/2Γ(k/2)))xk/2-1e-x/2 for x > 0 | |
| Mean | μ = k |
| Variance | σ² = 2k |
| Parameter k | Degrees of freedom |
| Relationship | Sum of k squared independent N(0,1) variables |
| Application | Hypothesis testing; goodness-of-fit tests; variance estimation |
4.4 Student's t-Distribution
| Parameter | Value/Formula |
|---|---|
| Notation | T ~ t(ν) |
| f(t) = (Γ((ν+1)/2)/(√(νπ)Γ(ν/2)))(1 + t²/ν)-(ν+1)/2 | |
| Mean | μ = 0 for ν > 1 |
| Variance | σ² = ν/(ν-2) for ν > 2 |
| Parameter ν | Degrees of freedom |
| Application | Small sample inference; confidence intervals when σ unknown |
- Symmetric about 0; heavier tails than normal
- Approaches N(0,1) as ν → ∞
4.5 F-Distribution
| Parameter | Value/Formula |
|---|---|
| Notation | F ~ F(d₁,d₂) |
| Mean | μ = d₂/(d₂-2) for d₂ > 2 |
| Variance | σ² = (2d₂²(d₁+d₂-2))/(d₁(d₂-2)²(d₂-4)) for d₂ > 4 |
| Parameters | d₁, d₂ = degrees of freedom (numerator, denominator) |
| Relationship | Ratio of two independent chi-square variables divided by their d.f. |
| Application | ANOVA; comparing variances; regression analysis |
5. Distribution Relationships and Approximations
5.1 Normal Approximations
| Distribution | Approximation Conditions |
|---|---|
| Binomial to Normal | np > 5 and n(1-p) > 5; use μ = np, σ² = np(1-p) with continuity correction |
| Poisson to Normal | λ > 10; use μ = λ, σ² = λ with continuity correction |
5.1.1 Continuity Correction
- P(X = k) → P(k - 0.5 < x="">< k="" +="">
- P(X ≤ k) → P(X < k="" +="">
- P(X < k)="" →="" p(x="">< k="" -="">
- P(X ≥ k) → P(X > k - 0.5)
- P(X > k) → P(X > k + 0.5)
5.2 Common Distribution Relationships
| Relationship | Description |
|---|---|
| Binomial(n,p) = Σ Bernoulli(p) | Sum of n independent Bernoulli trials |
| Geometric → Exponential | Discrete waiting time vs. continuous waiting time |
| Gamma(α,β) = Σ Exp(1/β) | Sum of α independent exponential variables |
| χ²(k) = Gamma(k/2, 2) | Special case of gamma distribution |
| Exp(λ) = Gamma(1, 1/λ) | Special case of gamma distribution |
| U(0,1) → any distribution | Inverse transform method: X = F-1(U) |
5.3 Central Limit Theorem
- If X₁, X₂, ..., Xn are i.i.d. with mean μ and variance σ², then sum Sn = ΣXi → N(nμ, nσ²) as n → ∞
- Sample mean: X̄ = Sn/n → N(μ, σ²/n) as n → ∞
- Standardized: Z = (X̄ - μ)/(σ/√n) → N(0,1)
- Rule of thumb: n ≥ 30 for reasonable approximation
6. Key Formulas and Calculations
6.1 Combination and Permutation
| Formula | Expression |
|---|---|
| Combination | C(n,k) = n!/(k!(n-k)!) = number of ways to choose k from n |
| Permutation | P(n,k) = n!/(n-k)! = number of ordered arrangements of k from n |
| Factorial | n! = n × (n-1) × ... × 2 × 1; 0! = 1 |
6.2 Moment Generating Function (MGF)
| Property | Formula |
|---|---|
| Definition | MX(t) = E[etX] |
| Discrete | MX(t) = Σetxf(x) |
| Continuous | MX(t) = ∫etxf(x)dx |
| Mean from MGF | E[X] = M'X(0) |
| Second moment | E[X²] = M''X(0) |
| Linear transformation | MaX+b(t) = ebtMX(at) |
6.3 Reliability Functions
| Function | Formula |
|---|---|
| Reliability R(t) | R(t) = P(X > t) = 1 - F(t) |
| Hazard rate h(t) | h(t) = f(t)/R(t) = f(t)/(1 - F(t)) |
| MTTF | Mean Time To Failure = E[X] = ∫0∞ R(t)dt |
6.4 Common Z-scores
| Confidence Level | Z-value |
|---|---|
| 90% | z = 1.645 |
| 95% | z = 1.96 |
| 99% | z = 2.576 |
| 99.9% | z = 3.291 |
7. Distribution Selection Guide
7.1 Discrete Distribution Selection
| Scenario | Distribution |
|---|---|
| Single trial, two outcomes | Bernoulli |
| Count successes in n independent trials | Binomial |
| Count rare events in fixed interval | Poisson |
| Number of trials until first success | Geometric |
| Sampling without replacement | Hypergeometric |
7.2 Continuous Distribution Selection
| Scenario | Distribution |
|---|---|
| Equal probability over interval | Uniform |
| Natural phenomena, measurement errors, CLT applies | Normal |
| Time between events, waiting time, memoryless | Exponential |
| Reliability, failure modeling with varying hazard rate | Weibull |
| Multiplicative process, positive skewed data | Lognormal |
| Waiting time for multiple events | Gamma |
| Proportion between 0 and 1 | Beta |
7.3 Statistical Inference Distributions
| Application | Distribution |
|---|---|
| Sample variance estimation | Chi-square |
| Small sample mean inference (σ unknown) | t-distribution |
| Comparing variances, ANOVA | F-distribution |
| Large sample mean inference | Normal (Z) |
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