Basic Probability
- P(A) = number of favorable outcomes / total number of outcomes
- P(A or B) = P(A) + P(B) - P(A and B)
- P(A and B) = P(A) * P(B|A) = P(B) * P(A|B)
- If independent: P(A and B) = P(A) * P(B)
- Complement: P(A') = 1 - P(A)
- Bayes' theorem: P(A|B) = P(B|A) * P(A) / P(B)
Combinatorics
- Permutations (order matters): P(n,r) = n! / (n-r)!
- Combinations (order does not matter): C(n,r) = n! / [r! * (n-r)!]
- Factorial: n! = n * (n-1) * (n-2) * ... * 1, where 0! = 1
Descriptive Statistics
- Mean: x_bar = Sum(x_i) / n
- Median: middle value when data is ordered
- Mode: most frequently occurring value
- Variance: s^2 = Sum((x_i - x_bar)^2) / (n-1) (sample)
- Standard deviation: s = sqrt(variance)
- Coefficient of variation: CV = s / x_bar
Common Distributions
- Binomial: P(X=k) = C(n,k) * p^k * (1-p)^(n-k); mean = np, variance = np(1-p)
- Poisson: P(X=k) = (lambda^k * e^(-lambda)) / k!; mean = lambda, variance = lambda
- Normal: bell-shaped, symmetric; defined by mean (mu) and standard deviation (sigma)
- Standard normal: Z = (X - mu) / sigma; mean = 0, std dev = 1
- Exponential: P(X > t) = e^(-lambda*t); mean = 1/lambda
- Uniform: f(x) = 1/(b-a) for a <= x <= b; mean = (a+b)/2
Confidence Intervals and Hypothesis Testing
- Confidence interval for mean: x_bar +/- z*(sigma/sqrt(n)) when sigma is known
- Use t-distribution when sigma is unknown and sample is small
- z = 1.645 for 90% CI, z = 1.96 for 95% CI, z = 2.576 for 99% CI
- Null hypothesis (H0): the default claim to be tested
- Alternative hypothesis (Ha): the claim if H0 is rejected
- p-value: probability of observing results as extreme as the test statistic under H0
- Reject H0 if p-value < significance level (alpha)
Regression
- Simple linear regression: y = b0 + b1*x
- Slope: b1 = [n*Sum(xy) - Sum(x)*Sum(y)] / [n*Sum(x^2) - (Sum(x))^2]
- Intercept: b0 = y_bar - b1*x_bar
- Coefficient of determination: R^2 = SSR/SST = 1 - SSE/SST
- R^2 ranges from 0 to 1; closer to 1 means better fit
- Correlation coefficient: r = sqrt(R^2) with sign matching slope