1. Probability Basic Probability Concepts Probability Distributions Sampling Distributions

2. Probability Basic Probability Concepts

3. Basic Probability Concepts Probability refers to the relative chance that an event will occur. It represents a means to measure and quantify uncertainty. 0  probability  1

4. Basic Probability Concepts The Classical Interpretation of Probability: P(event) = # of outcomes in the event # of outcomes in sample space

5. Example: P(selecting a red card from deck of cards) ? Sample Space, S = all cards Event, E = red card then P(E) = # outcomes in E = 26 = 1 # outcomes in S 52 2

6. Probability Random Variables and Probability Distributions

7. Random Variable A variable that varies in value by chance

8. Random Variables Discrete variable - takes on a finite, countable # of values Continuous variable - takes on an infinite # of values

9. Probability Distribution A listing of all possible values of the random variable, together with their associated probabilities.

10. Notation: Let X = defined random variable of interest x = possible values of X P(X=x) = probability that takes the value x

11. Example: Experiment: Toss a coin 2 times. Of interest: # of heads that show

12. Example: Let X = # of heads in 2 tosses of a coin (discrete) The probability distribution of X, presented in tabular form, is: xP(X=x) 0.25 1.50 2.25 1.00

13. Methods for Establishing Probabilities Empirical Method Subjective Method Theoretical Method

14. Example: Toss 1 Toss 2 T T There are 4 possible T H outcomes in the H T sample space in this H H experiment

15. Example: Toss 1 Toss 2 T T P(X=0) = ? T H Let E = 0 heads in 2 tosses H T P(E) = # outcomes in E H H # outcomes in S = 1/4

16. Example: Toss 1 Toss 2 T T P(X=1) = ? T H Let E = 1 head in 2 tosses H T P(E) = # outcomes in E H H # outcomes in S = 2/4

17. Example: Toss 1 Toss 2 T T P(X=2) = ? T H Let E = 2 heads in 2 tosses H T P(E) = # outcomes in E H H # outcomes in S = 1/4

18. Example: The probability distribution in tabular form: xP(X=x) 0.25 1.50 2.25 1.00

19. Example: The probability distribution in graphical form: P(X=x)1.00.75.50.25 012 x

20. Probability distribution, numerical summary form: Measure of Central Tendency: mean = expected value Measures of Dispersion: variance standard deviation Numerical Summary Measures

21. Expected Value Let  = E(X) = mean = expected value then  = E(X) =  x P(X=x)

22. Example: xP(X=x) 0.25 1.50 2.25 1.00  = E(X) = 0(.25) + 1(.50) + 2(.25) = 1

23. Variance Let  ² = variance then  ² =  ( x -  ) ² P(X=x)

24. Standard Deviation Let  = standard deviation then  =   ²

25. Example: xP(X=x) 0.25 1.50 2.25 1.00  ² = (0-1)²(.25) + (1-1)²(.50) + (2-1)²(.25) =.5  = .5 =.707

26. Practical Application Risk Assessment: Investment AInvestment B E(X) E(X)  Choice of investment – the investment that yields the highest expected return and the lowest risk.