Understanding Point Estimation and Maximum Likelihood in Statistics

This collection of images and text delves into various topics in statistics essential for engineers, such as point estimation, unbiased estimators, maximum likelihood, and estimating parameters from different probability distributions. Concepts like estimating from Uniform samples, choosing between sample mean and sample max, minimum variance principle, and maximum likelihood for exponential distributions are also explored.

• Statistics
• Estimation
• Engineers
• Maximum Likelihood
• Probability Distributions

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1. ENGG 2780A / ESTR 2020: Statistics for Engineers Spring 2021 4. Point Estimation Andrej Bogdanov

2. Poisson() Normal( , ) Alice Bob Charlie PASS PASS FAIL Binomial(150, p)

3. Estimators X = (X1, , Xn) independent samples ^ Unbiased: E[ n] = ^ Consistent: n converges to in probability

4. How to estimate from Uniform(0, ) samples?

5. How to estimate E[Uniform(0, )] from 3 samples? Option 1: sample mean Option 2: via sample max

6. sample mean PDF = 1/2 = 1/6 sample max PDF = 1/2 1/8

7. Minimum variance principle Among unbiased estimators, it is preferable* to choose the one with smaller variance.

8. Maximum likelihood maximize fX(x | ) recall Bayesian MAP estimate: maximize f |X( |x) fX| (x | ) f ( )

9. What is the ML for from Uniform(0, ) samples?

10. Maximum likelihood for Indicator() k heads, n k tails. ML bias estimate?

11. Within the first 3 seconds, raindrops arrive at times 1.2, 1.9, and 2.5. What is the estimated rate? 3 0

12. The first 3 raindrops arrive at 1.2, 1.9, and 2.5 sec. What is the estimated rate? 0

13. Maximum likelihood for Exponential()

14. A Normal(, ) RV takes values 2.9, 3.3. What is the ML estimate for ?

15. Maximum likelihood for Normal(, ) (X1, , Xn) independent Normal( , ) X1 + + Xn n ML estimate of : X= Xis the Normal( , ) estimator of smallest possible variance.

16. A Normal(, ) RV takes values 2.9, 3.3. What is the ML estimate for v = ?