Prib. of events. et P(A) Prob. of complex events. eg.P(A and B) Random Variable. Categorical X is the color of the student that enters the door next {X=Red} {X=Blue} Quantitative Generic Distribution. Special Distribution. Generic Distribution. X 0 1 2 3 P(X=i) 0.2 0.3 0.1 0.4 P(X<=1) = Area (X<=1) / Area (S) = (0.2+0.3) / 1 = 0.5 Similar to categorical variable & prob. of events. Special Distribution. discrete. 离散 Binomial distribution { Geometric distribution Continuous is normal 连续 uniform eg 1. HHH HHT HTH HTT V THH THT V TTH V TTT 3 coin toss. exactly 1 head. P(1 head)=3/8 total # of outcomes Binomial Settings (Remember!!!) 1). fixed number of trials n 2) only 2 outcomes each trial success or failure. define yourself. 3). fixed success prob. P. 4). independent trials. X represent the number of success in n trials. Binomial formula. Binomial PDF. P(X=k) = (n choose k) p^k (1-p)^(n-k) eg2: P(X=2) = (5 choose 2) 0.2^2 (1-0.2)^3 (5 choose 2) = 5! / (2!(5-2)!) (n choose k) = n! / (k!(n-k)!) Binomial CDF. P(X<=k) = P(X=0) + P(X=1) + ... + P(X=K) X ~ Binom (n, P) prob. of success for each trial. R.V 服从二项分布 max trials E(X) = Mu_x = np expected value mean of X Std(X) = Sigma_x = sqrt(np(1-p)) std. of X. 当P -> 0.5, X -> symmetric 当np >= 10, n(1-p) >= 10 时, X -> Binom -> Normal skew to the right. 0 当P -> 1 时, 且 n(1-p) < 10 时, 右裁断. Skew to the left n X~ Geometric (p). 几何分布 X 是多次实验, 第一次成功的序数。 P(X=k)=(1-p)^(k-1) P