Probability & Statistics

1. Probability & Statistics • Outline Probability and Statistics • 2 types of probability • Rules of probability • Statistical Independence • Expected Value • Normal Distributions

2. 2 Types of Probability • Subjective • Probability estimate based on what a person believes or experiences • “I think there is a 60% chance of rain tomorrow.” • Objective • Probabilities that can be stated before or a priori the occurrence of an event • Roll of a fair dice • Flip of a fair coin

3. Fundamentals and Rules of Probability • Rules of probability 1. 0 < P(A) < 1 2. SPi = 1 3. P(A or B) = P(A) + P(B), for mutually exclusive A & B

4. Mutual Exclusivity • Only one event can occur at a time A B • Addition rule • P(A) + P(B) = P(A OR B)

5. Probability Types • Marginal • Probability of a single event • e.g., P(A) = 0.1 • Joint • Probability of more than one event • e.g., P(A and B) = 0.2

6. Example of a Joint Probability • Probability of two non-mutually exclusive events occurring A B • Shaded area is a joint probability • General addition rule • P(A or B) = P(A) + P(B) - P(A and B)

7. Independence • Successive events that do not affect one another • e.g., flipping a coin • P(H and {then} T) = P(H)*P(T) • In the case of dependent events • P(A and {then} B) = P(B)*P(A|B) • General rule of multiplication

8. Conditional Probability • Probability that an event will occur given that another event has already occurred • e.g., weather forecasts, given thunder what is the probability of rain • P(A|B) • Reads probability of A given B

9. Independent vs. Dependent Events • Independent Events • P(A|B) = P(A) • P(A and B) = P(A)*P(B) • Dependent Events • P(A|B) = P(A and B) / P(B) • P(A and B) = P(B)*P(A|B)

10. Bayes’ Theorem • Famous statistician/mathematician • Created relationship for dependent events P(A|B) = P(A and B) / P(B) • Total probability law P(B) = P(B|A)*P(A) + P(B|not A)*P(not A) • Used to update probabilistic forecasts, e.g., weather forecasts

11. Example of Bayes’ Theorem • Given, A and B are dependent events P(A and B) = 0.2, P(B) = 0.4 Calculate P(A|B): P(A|B) = P(A and B) / P(B) = 0.2 / 0.4 = 0.5 • Given, A and B are independent events Calculate P(A|B): P(A|B) = P(A)*P(B) / P(B) = P(A)

12. Expected Value • Mean of the probability distribution of a random variable (RV) • E(x) = Sx*P(x) e.g., x P(x) x*P(x) 0 0.1 0.0 1 0.2 0.2 2 0.3 0.6 3 0.4 1.2 Sx*P(x) = E(x) = 2.0

13. Normal Distribution • Common probability distribution • e.g., height, weight, age, sum of two dice rolled 1,000 times, etc.

14. Normal Distribution

15. Mean and Standard Deviation • Most common statistics used • Mean or expected value E(x) = SxiP(xi) • Standard deviation s(x) = [S [xi - E(x)]2P(xi)]0.5 s(x) = [S [xi - m]2/n-1]0.5

16. Z-Scores • Standard Z-score • Measures the number of standard deviations away from the mean • Calculated as such: • Look up Z value in table to find probability