Introduction to Probability · 免费预览
变量替换
用于变换连续密度函数的雅可比方法、单调与分段单调情形，以及以对数变换为实例进行讲解。
1. The change-of-variables formula
The change-of-variables formula
2. Piecewise monotone maps
Piecewise monotone maps
3. The log transform, both directions
The log transform, both directions
4. The Jacobian's job
Under Y = 2X, the density of Y at y equals what?
5. A two-to-one split
Y = X squared, with X taking both signs and having density f symmetric about 0. Which is f_Y(y)?
6. Scaling a Uniform
X is Uniform(0, 2) with density one half. Y = 2X is Uniform(0, 4). What is the density value of Y? Enter a decimal like 0.25.
7. Inverting an exponential map
Y has density e^minus y on y over 0 (Exponential 1). X = e^Y has density e^minus ln x times 1/x. Simplify for x over 1: what is the density of X? Enter like 1/x^2.
8. Chi-square 1 mean
Z squared for standard Normal Z gives Chi-square 1. By LOTUS, E(Z squared) is the second moment of Z. What is its value? Enter an integer.
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.