Introduction to statistics
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The document discusses modeling unknown distributions with few observations by assuming they come from a canonical distribution family with unknown parameters. It explains that canonical distributions like the normal, binomial, and Poisson distributions can describe data using just a few parameters. The document then discusses how to estimate these unknown parameters, like the mean and variance, from the limited observations. Specifically, it addresses estimating the mean using the sample mean as an unbiased estimate, and estimating the variance, noting the need for an unbiased estimate that becomes more precise as the number of observations increases.
Introduction to statistics
- 2. The Quest What can we say about this black box? E.g., What is the probability that it generates a number 5 12 3 9 28 bigger than 5? Observations
- 3. Distributions What if we had many many observations? Value Frequency -1 0.3 0 0.2 Sum of 1 0.1 frequencies This table is the 2 0.1 is 1 distribution 3 0.1 associated with 4 0.1 this black box 5 0.1
- 4. Distributions Graphically -1 0 1 2 3 4 5 Area under the curve is 1
- 5. The Challenge We do not have many many observations So we cannot infer the distribution from the observations What can we do then?
- 6. What can we do with few observations? Assume distribution is known E.g., Normal, Binomial (from prior knowledge or etc other means) I.e., model approximately using a canonical distribution But the parameters are not known Can these parameters be determined from the observations?
- 7. Why Canonical Distributions Value Frequency -1 0.3 0 0.2 1 0.1 Too verbose a description for the 2 0.1 distribution 3 0.1 4 0.1 5 0.1 Can the entire distribution be described (even approximately) by just a few parameters, while modeling the data accurately
- 8. Example: Binomial Distribution A coin that yields 1 with Observations probability p and 0 with probability 1- p, tossed n 1 0 1 1 1 …. times, independently Value Frequency Number of 1’s? 0 1 Distribution, 2 μ=np,σ2=np(1-p) n-1 Can one determine p from the (few) n observations?
- 9. Other Canonical Distributions Normal μ, σ2 Poisson μ =r,σ2=r Negative Binomial μ =rp/(1-p), σ2= rp/(1-p)2 What are these? Later talk Gamma μ=kθ, σ2 =kθ2
- 10. Back to the Quest We have few observations Assume these are from a known distribution family But with unknown parameters How do we determine the parameters? How do we determine μ, σ2?
- 11. Estimating Mean μ, σ2 Estimate for the mean; a good estimate??
- 12. μ, σ2 What is the mean and variance Normal!! For of this distribution? modest n.
- 13. μ, σ2 Unbiased Tight as n grows larger
- 14. Estimating Variance μ, σ2 Estimate for the variance; a good estimate??
- 15. μ, σ2 μ, σ2 Bias
- 16. Estimating Variance Correctly μ, σ2 Unbiased!!
- 17. A Mind Reading Game • Your friend chooses a number (one of 1,3,5) in his/her mind – Call this i • He/She then rolls a 6-faced die 30 times, privately – For each roll, he/she declares Heads if the number on the die is <=i, and Tails otherwise • Your goal is to guess i solely from this sequence of n Heads and Tails. • Can you read your friend’s mind?
- 18. Thank You