Introduction to Probability · 免费预览
矩母函数
将所有矩整合到一个函数中，从导数中读出各阶矩，并通过 MGFs 识别分布。
1. One function, every moment
One function, every moment
2. A catalog of MGFs
A catalog of MGFs
3. Existence, uniqueness, and what MGFs are for
Existence, uniqueness, and what MGFs are for
4. What the MGF stores
The r-th derivative of the MGF evaluated at t = 0 equals what?
5. Identifying by table
A random variable has MGF equal to e raised to the quantity 2 times e to the t, minus 2. Which distribution is it?
6. Bernoulli MGF moment
The Bernoulli(p) MGF is 1 - p + p e^t. Its first derivative at t = 0 is e^0 = 1 times p, plus zero. For p = 0.3, what is the mean? Enter a decimal like 0.3.
7. Exponential mean from MGF
The Expo(lambda) MGF is lambda over (lambda minus t). For lambda = 4, M'(0) = lambda squared over lambda squared, or compute directly: what is the mean? Enter a decimal like 0.25.
8. Uniform MGF check
The Uniform(0,1) MGF is (e^t minus 1) over t. Near t = 0 it limits to 1. Its series begins 1 + t/2 + t squared over 6. What is E(X)? Enter a decimal like 0.5.
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.