Probability Fundamentals
- P(A) ranges from 0 to 1; P(sample space) = 1
- Addition rule: P(A or B) = P(A) + P(B) - P(A and B)
- Multiplication rule: P(A and B) = P(A)*P(B|A)
- Independent events: P(A and B) = P(A)*P(B)
- Bayes' theorem: P(A|B) = P(B|A)*P(A)/P(B)
- Complement rule: P(A') = 1 - P(A)
Discrete Distributions
- Binomial: n independent trials, each with probability p of success
- P(X=k) = C(n,k)*p^k*(1-p)^(n-k); mean = np; variance = np(1-p)
- Poisson: models rare events over a fixed interval
- P(X=k) = lambda^k*e^(-lambda)/k!; mean = variance = lambda
- Geometric: number of trials until first success; P(X=k) = (1-p)^(k-1)*p
Continuous Distributions
- Normal distribution: symmetric, bell-shaped; defined by mu (mean) and sigma (standard deviation)
- 68-95-99.7 rule: 68% within 1 sigma, 95% within 2 sigma, 99.7% within 3 sigma
- Standard normal: Z = (X - mu)/sigma transforms any normal to N(0,1)
- Exponential: models time between events; f(t) = lambda*e^(-lambda*t); mean = 1/lambda
- Uniform: constant probability over interval [a,b]; mean = (a+b)/2; variance = (b-a)^2/12
- t-distribution: used when sigma is unknown and sample size is small; approaches normal as n grows
Statistical Inference
- Point estimate of population mean: x_bar = Sum(x_i)/n
- Confidence interval: x_bar +/- z*(sigma/sqrt(n)) for known sigma
- Use t-distribution when sigma is unknown: x_bar +/- t*(s/sqrt(n))
- Hypothesis testing: H0 (null) vs. Ha (alternative)
- Test statistic: z = (x_bar - mu_0)/(sigma/sqrt(n))
- Type I error (alpha): rejecting a true H0; Type II error (beta): failing to reject a false H0
- p-value: probability of observing the data (or more extreme) under H0
Regression Analysis
- Linear model: y = b0 + b1*x
- Least squares estimates minimize the sum of squared residuals
- R^2 (coefficient of determination): proportion of variance explained by the model
- Correlation coefficient r: measures linear association (-1 to +1)
- Residual = observed - predicted value
- Multiple regression: y = b0 + b1*x1 + b2*x2 + ...
Quality Control
- Control charts: monitor process stability over time
- X-bar chart: monitors process mean; R chart: monitors process variability
- Upper/Lower control limits: typically set at +/- 3 sigma from the center line
- Process capability: Cp = (USL - LSL)/(6*sigma)
- Cpk = min[(USL - mu)/(3*sigma), (mu - LSL)/(3*sigma)]
- Six Sigma: process with Cp >= 2.0 and defect rate of 3.4 per million