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Probability Basic Probability Concepts Probability Distributions Sampling Distributions

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- Slide 1
- Probability Basic Probability Concepts Probability Distributions Sampling Distributions
- Slide 2
- Probability Basic Probability Concepts
- Slide 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
- Slide 4
- Basic Probability Concepts The Classical Interpretation of Probability: P(event) = # of outcomes in the event # of outcomes in sample space
- Slide 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
- Slide 6
- Probability Random Variables and Probability Distributions
- Slide 7
- Random Variable A variable that varies in value by chance
- Slide 8
- Random Variables Discrete variable - takes on a finite, countable # of values Continuous variable - takes on an infinite # of values
- Slide 9
- Probability Distribution A listing of all possible values of the random variable, together with their associated probabilities.
- Slide 10
- Notation: Let X = defined random variable of interest x = possible values of X P(X=x) = probability that takes the value x
- Slide 11
- Example: Experiment: Toss a coin 2 times. Of interest: # of heads that show
- Slide 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
- Slide 13
- Methods for Establishing Probabilities Empirical Method Subjective Method Theoretical Method
- Slide 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
- Slide 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
- Slide 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
- Slide 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
- Slide 18
- Example: The probability distribution in tabular form: xP(X=x) 0.25 1.50 2.25 1.00
- Slide 19
- Example: The probability distribution in graphical form: P(X=x)1.00.75.50.25 012 x
- Slide 20
- Probability distribution, numerical summary form: Measure of Central Tendency: mean = expected value Measures of Dispersion: variance standard deviation Numerical Summary Measures
- Slide 21
- Expected Value Let = E(X) = mean = expected value then = E(X) = x P(X=x)
- Slide 22
- Example: xP(X=x) 0.25 1.50 2.25 1.00 = E(X) = 0(.25) + 1(.50) + 2(.25) = 1
- Slide 23
- Variance Let = variance then = ( x - ) P(X=x)
- Slide 24
- Standard Deviation Let = standard deviation then =
- Slide 25
- Example: xP(X=x) 0.25 1.50 2.25 1.00 = (0-1)(.25) + (1-1)(.50) + (2-1)(.25) =.5 = .5 =.707
- Slide 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.

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