How would you prove this COS 424SML 302 Probability and Statistics Review

How would you prove this cos 424sml 302 probability

This preview shows page 31 - 39 out of 70 pages.

How would you prove this? COS 424/SML 302 Probability and Statistics Review February 6, 2019 30 / 69
Image of page 31

Subscribe to view the full document.

Continuous random variables Random variables can be continuous. We need a density p ( x ), which integrates to one. If x ∈ < then Z -∞ p ( x ) dx = 1 Probabilities are integrals over smaller intervals. E.g., P ( X ( - 2 . 4 , 6 . 5)) = Z 6 . 5 - 2 . 4 p ( x ) dx COS 424/SML 302 Probability and Statistics Review February 6, 2019 31 / 69
Image of page 32
Example: Gaussian distribution Continuous distribution: Gaussian The Gaussian (or Normal) distribution is a continuous distribution, meaning that its support is on continuous numbers. p ( x | μ, σ ) = 1 2 πσ exp - ( x - μ ) 2 2 σ 2 The density of a point x is proportional to the negative exponentiated half distance to μ scaled by σ 2 . Parameters: μ is called the mean ; σ 2 is called the variance . COS 424/SML 302 Probability and Statistics Review February 6, 2019 32 / 69
Image of page 33

Subscribe to view the full document.

Notation for random variables Discrete RVs p denotes the probability mass function , which is the same as the probability of atoms P . We can use P and p interchangeably for discrete distributions. Continuous RVs p is the probability density function the probability of any single value is always zero: P ( X = x ) = 0. We cannot use P and p interchangeably for continuous distributions. This is an unpleasant detail, but mathematically important. COS 424/SML 302 Probability and Statistics Review February 6, 2019 33 / 69
Image of page 34
Expectation Consider a function f ( · ) of a discrete random variable X . (Note: f ( X ) is also a random variable.) The expectation is a weighted average of f ( · ), where the weight of each atom is p ( x ), E [ f ( X )] = X x Ω p ( x ) f ( x ) In the continuous case, the expectation is an integral E [ f ( X )] = Z Ω p ( x ) f ( x ) dx COS 424/SML 302 Probability and Statistics Review February 6, 2019 34 / 69
Image of page 35

Subscribe to view the full document.

Conditional expectation The conditional expectation is defined similarly E [ f ( X ) | Y = y ] = X x Ω p ( x | y ) f ( x ) Examples: conditional expectations Given someone’s height Y , what is their expected shoe size X ? Given someone’s zip code, what is the expected number of dollars they will spend on your website? E [ f ( X ) | Y ] is a (function of) random variable Y . COS 424/SML 302 Probability and Statistics Review February 6, 2019 35 / 69
Image of page 36
Probability models Probability distributions are simple models of data that we observe. Assume that data are generated from a specific distribution. Infer the parameters of that distribution from the observed data. Interpret those properties in terms of properties of the underlying observed data. COS 424/SML 302 Probability and Statistics Review February 6, 2019 36 / 69
Image of page 37

Subscribe to view the full document.

Probability models, examples Examples of inferences we can make about specific data sets using a simple distribution: Inference Observation the bias of a coin 1000 coin flips the average height of a student 190 students the chance that a politician will win a primary 1200 people in Iowa the proportion of gold in a mountain 20 cubic meters of the mountain the number of bacteria species in our body 4 samples of the gut the evolutionary rate at which DNA mutates 34 completely sequenced mammals What distribution should be used in order to make each inference?
Image of page 38
Image of page 39
  • Spring '09
  • Probability theory

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes