Introduction to Probability · 免费预览
联合分布、边缘分布与条件分布
同时测量两个随机变量：联合概率质量函数与密度；边缘分布通过求和或积分得到；条件分布通过对切片重新归一化得到。
1. Joint distributions
Joint distributions
2. Marginals
Marginals
3. Conditional distributions
Conditional distributions
4. Marginals do not determine joints
Two joint tables have identical row sums and identical column sums. What can you conclude?
5. Reading a conditional from a table
A joint table has P(X = 1, Y = 2) = 0.15 and the column total P(Y = 2) = 0.5. What is P(X = 1 given Y = 2)?
6. Row sum
A joint PMF has P(X = 2, Y = 1) = 0.2, P(X = 2, Y = 2) = 0.1, and P(X = 2, Y = 3) = 0.25. What is P(X = 2)? Enter a decimal like 0.55.
7. Continuous marginal
A joint density on the unit square is f(x, y) = x + y for x and y in [0,1]. The marginal density of X at x is the integral from 0 to 1 of (x + y) dy, which equals x + one half. What is f_X(0.5)? Enter a decimal like 1.0.
8. Conditional slice
With f(x, y) = x + y on the unit square, the conditional density of X given Y = 0.5 is (x + 0.5) over f_Y(0.5). Since f_Y(0.5) = 1, the conditional density at x = 0.5 is what value? Enter like 1.0.
9. In your own words
In your own words, state the most important definition, theorem or method of this lesson and explain what it says.