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Ryan 2023-06-17 17:23:20 +08:00
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pages/总复习2023t1.md
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@ -2690,28 +2690,22 @@
- ((648c27ed-e09d-4f1a-8100-7ae2dfff1c8e))
- ((648c27f9-f944-4bf8-9e44-cb3e9048d286))
- DONE Ch2 Random Variable
collapsed:: true
:LOGBOOK:
CLOCK: [2023-06-16 Fri 17:15:12]--[2023-06-16 Fri 21:34:48] => 04:19:36
:END:
- The Definition of a Random Variable ![section 2.1.pdf](../assets/section_2.1_1686906920851_0.pdf)
collapsed:: true
- ((648c2860-5454-45f1-80f8-1c85bce84fd1))
- ((648c290a-3532-4e56-8536-258daf7b00a7))
- Distribution Function
- ((648c2970-9091-4f46-8745-b0c10cdeb832))
collapsed:: true
- ((648c2994-36eb-44bf-8038-e49d2390a5b0))
- ((648c2a73-3961-4723-90e8-5f160bb18e0d)) ![section 2.2.pdf](../assets/section_2.2_1686907456219_0.pdf)
collapsed:: true
- ((648c2a88-3d0f-47d2-be44-21df002a1def))
- ((648c2abf-5c5e-4af9-9e13-bb566f3206e8))
- ((648c2b26-e92d-43f7-b8e1-60b51a2b5268))
- Discrete type
collapsed:: true
- Definition: ((648c2be0-787e-4e1f-8d0c-b859f0d08383))
- Continuous type
collapsed:: true
- ((648c2c54-b536-40ba-8191-ef1aeb2be0b9))
- ((648c2c9a-fcca-43e3-94a6-466f52131b48))
- ((648c2d05-82d4-472f-901c-b362852e2a3a))
@ -2723,134 +2717,99 @@
- ((648c48ac-76be-4848-af0d-1532467c8d68))
- Mathematical Expectation and Variance ![section 2.3.pdf](../assets/section_2.3_1686916498071_0.pdf)
- Expectation
collapsed:: true
- Definitions :-> ((648c4e9e-5515-4da3-bc6b-39d6969f5f14))
id:: 648c4e9f-7b40-4df5-ab81-e6eb49536831
- Examples:
collapsed:: true
- ((648c5298-fb84-4e9c-b816-7df1a7ad9a66)) #flashcard
id:: 648c52b8-5174-4dfc-8954-d58b27893865
collapsed:: true
- ((648c52d9-ea03-4835-8080-52ba5e6c3122))
- Expectation of compound **functions**: (Expectation of Functions of a Random Variable) (Replace $$x$$ with $$g(x)$$) #flashcard
id:: 648c52dc-1cfe-4c05-888a-d8ddff10c24e
collapsed:: true
- ((648c5381-49cd-4701-ba48-31c0e21474a7))
- Properties:
collapsed:: true
- ((648c568e-c617-4699-b217-fd4e8ea205bc))
- Linerarity:
((648c569c-e3d0-445d-acf6-caf0e271f7be))
- Variance
collapsed:: true
- Definition:
collapsed:: true
- ((648c57b6-225a-4926-9ce1-8d0e7aae1391))
- Alternative calculation:
collapsed:: true
- ((648c580f-8ccb-4fd3-bc0d-24ab109f240d))
- ((648c57d7-8f76-4520-8bc0-b54953bbe4cf))
- Standard variance: ((648c57e8-e79a-48b6-98cf-f68dc0e5277e))
- Properties
collapsed:: true
- ((648c5982-4b4f-4935-ad4f-de93462bc564))
- ((648c5a58-62da-4c15-b22d-00e839b8f909))
- Examples
collapsed:: true
- ((648c5b1c-533d-434a-b118-6a61cc9abb3a)) #flashcard
id:: 648c5ae4-9e42-4c55-86d0-b0a19dfcb4dc
collapsed:: true
- ((648c5b30-e75f-4f0c-bb7d-985442728fbe))
- LATER Moments of random variable
collapsed:: true
- Definition :-> ((648c5c3a-160a-41ff-9cc7-83e750e3e8fa))
id:: 648c5c30-6a99-41f4-be4c-e5e22387647a
- Application of Expectation and Variation
collapsed:: true
- *Markov's inequality*: when **only mean** is known #flashcard
id:: 648c5c5c-623c-43de-b750-b997db48e6dd
collapsed:: true
- ((648c5c6f-d6b4-4c65-b9a4-9f51f93f86ad))
- *Chebyshev's inequality*: when **mean an variance are** known #flashcard
id:: 648c5c71-3ec9-47db-8433-e1325b3a8c20
collapsed:: true
- ((648c5c81-d93b-4859-9ee0-37cceb8fd100))
- **Usage** of Markov's inequality and Chebyshev's inequality
collapsed:: true
- ((648c5cf5-239a-4d47-a57a-d65c53c215c4))
- **Example**: ((648c5de0-d650-443a-8a67-768066dce154))
collapsed:: true
-
- ((648c5e0c-b0b1-4129-b172-3ff95ff4a41c)) and ((648c5e26-8a90-4db2-919f-356a164e5a2b)) ![section 2.4.pdf](../assets/section_2.4_1686920705980_0.pdf)
- Binomial Distribution
collapsed:: true
- Definition: #flashcard
id:: 648c5ef0-fd5e-4f17-9e2d-fde0c67e53c6
collapsed:: true
- ((648c5fc2-c838-473b-9b29-f238ae3a96ca))
- ((648c5f9a-646f-4eea-9d03-597d8598f752))
- ((648c5fa8-25a4-47ba-a285-e9399f719543))
- Properties:
collapsed:: true
- mean, variance :-> ((648c6007-c181-4e85-9a88-046841e6739d))
id:: 648c6001-da22-465c-95f1-51a7e5292cb3
- Examples:
collapsed:: true
- ((648c62d4-454b-43ab-88a8-d35a5a867523))
- ((648c63b8-21ff-42d4-ad87-70c706124ba1))
- LATER Poisson limit theorem #flashcard (when n is infinity, this become Poisson distribution)
id:: 648c63bc-2465-42e5-833f-abd152201631
collapsed:: true
- ((648c63da-deea-42ab-8f72-f3219d4fecbc))
- Poisson distribution
id:: 648c650a-e188-49a6-9f89-9a3407057ddf
collapsed:: true
- Definition: :-> ((648c653d-ed40-4938-9124-cf4c2c296867))
id:: 648c6510-49a0-4673-bf16-4cf9f47efa14
- Use cases:
collapsed:: true
- ((648c65ba-0f8f-4371-a52e-e78e8b3ae512))
- Properties:
collapsed:: true
- Variance, mean :-> ((648c65dd-356b-4209-85ae-e881033fcea8))
id:: 648c65d1-7592-43f7-a12c-d96d9e76a492
- DONE Examples
- Distribution of continuous random variables ![section 2.5.pdf](../assets/section_2.5_1686968005870_0.pdf)
- Uniform Distribution
collapsed:: true
- Definition, pdf and df :-> ((648d1aea-71f3-473b-8e72-d1ee4e629a91))
id:: 648d1aec-b256-4d1b-91d7-9a22cc30657e
- ((648d1b0d-fdbf-4373-84a8-a072590881a1))
- Standard distribution ((648d1f39-bc89-4efe-98b3-1c654716cf65))
- Properties
collapsed:: true
- mean and variance :-> ((648d1bfd-fcfb-43ed-801e-0de972c24cb1))
id:: 648d1c05-1299-42a9-80e0-d877610aa19d
- Exponential Distribution similar to -> ((648c650a-e188-49a6-9f89-9a3407057ddf))
collapsed:: true
- Definition: #flashcard
id:: 648d2593-394c-4b31-a1a4-4eef625eacac
collapsed:: true
- ((648d25a0-12d9-47c5-b9ca-f7cafdf15d89))
- Properties
collapsed:: true
- Mean, variance :-> ((648d25a0-12d9-47c5-b9ca-f7cafdf15d89))
id:: 648d25a3-9cae-4c50-a073-749b5e364bcb
- Memory-less #flashcard
id:: 648d26a1-d047-40e6-bcc5-9fab241c9a37
collapsed:: true
- ((648d26c8-0292-4c02-8d6e-2b4f5cb10385))
- LATER Example: connection between Poisson and Exponential distribution
collapsed:: true
- ((648d27e3-1886-4b09-a18d-8a89d33e745a))
- Normal Distribution
collapsed:: true
- Definition #flashcard
id:: 648d288a-4a44-443d-8490-500ddd24bed4
collapsed:: true
- ((648d2884-c239-4d00-9b6f-291bf4280dab))
- Properties:
collapsed:: true
- ((648d2931-ae90-414a-b12c-e19aa1c534fc))
- ((648d294d-7bb2-4b6e-8aca-70dc4af6005c))
- Standard normal distribution :-> ((648d2960-d588-4610-a643-e292ef5b793f))
@ -2862,11 +2821,9 @@
- (Some sort of ) linearity :-> ((648d2d62-f5e5-491f-ae12-e9638ee75f7c))
id:: 648d2d49-79da-40fc-a205-3aa0f52eb770
- Example
collapsed:: true
- in specified interval ((648d2c35-041a-4575-b39d-c42c1a6f532f))
- without knowing $$\mu$$ and $$\sigma$$: ((648d2cfd-2cef-48d2-b791-e5ac01e43fd8)) #flashcard
id:: 648d2c39-55dd-4cfd-9a39-1c17d90ff3e3
collapsed:: true
- ((648d2cfd-2cef-48d2-b791-e5ac01e43fd8))
-
- ==Summary== #flashcard Bernoulli, Binomial, Geometric, Poisson, Uniform, Exponential, Normal
@ -2960,6 +2917,7 @@
- ((648d6e79-1d02-42c0-bc45-73dca7753118))
- Continuous case
- $X + Y$
collapsed:: true
- ((648d718a-6ea6-45b9-96f7-2f89ecbf54f3))
- Can be written as: #flashcard
id:: 648d71a6-3a5b-4ef8-b461-ab67f9b65332
@ -2980,9 +2938,11 @@
collapsed:: true
- ((648d769d-a3ea-4e6c-bc93-564d88d6ed38))
- LATER $max(X, Y)$
collapsed:: true
- ((648d7726-4d29-42a5-9044-62bd69968ba1)) ... and more
- if $$X$$ and $$Y$$ are independent: ((648d77b1-1d4a-4c69-b309-e398fccab797))
- $min(X,Y)$
collapsed:: true
- similar to max
- Example
- ((648d7abf-b5e7-4d91-b909-20d45c354049))