1、Jointly Distributed Random Variables联合概率分布Probability and Mathematical Statistics(概率与数理统计 )Xi ZHANG女生心目中好老公的标准高帅 健康阳光 安全感 能赚钱且上缴幽默 有房无贷 下班按时 回家 好厨艺30%80%70%50%60%10%60%20% Joint Distribution Function（联合分布函数） Independent Random Variables（独立随机变量） Sums of Independent Random Variables(独立变量之和） Identicall
2、y Distributed Uniform Random Variables（同分布均匀随机变量） Gamma Random Variables（伽马随机变量） Normal Random Variables（正态随机变量） Poisson and Binomial Random Variables（泊松，二项随机变量） Geometric Random Variables（几何随机变量） Conditional Distributions: Discrete Case（离散变量条件分布） Conditional Distributions: Continuous Case（连续变量条件分布）
3、Example There is a box with 4 cards:You draw two cards without replacement.What is the p.m.f. of the sum of the face values?1 2 3 4 = pairs of cards, equally likely outcomes = face value on first card = face value on second cardWe want the p.m.f. of + = P(X = 1, Y = 3) + P(X = 2, Y = 2) + P(X = 3, Y
4、 = 1)1/12 0 1/12P(X + Y = 4) = 1/6 In generalP(X + Y = z) = (x, y): x + y = z P(X = x, Y = y)To calculate ( + = ) we need to knowf(x, y) = P(X = x, Y = y)joint p.m.f. of and joint p.m.f. of and 0 1/12 1/12 1/121/12 0 1/12 1/121/12 1/12 0 1/121/12 1/12 1/12 01 2 3 41234XY 444332 555566677 8p.m.f. of
5、X + Y2 03 1/64 1/65 1/36 1/67 1/68 0Joint distribution functions (联合分布函数 ) For any two random variables and , the joint cumulative probability distribution function of and by , = , , 0, by|( , )( | )()XYf x yf x yfy If and are jointly continuous, then ,for any set | ( | )XYAP X A Y y f x y dx |( | ) | ( | )aX Y X YF a y P X a Y y f x y dx = (, If and are independent continuous random variables, the conditional density of X given that = is just the unconditional density of .
