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Probability Basic Probability Concepts Probability Distributions Sampling Distributions

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Probability Basic Probability Concepts

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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

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

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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

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Probability Random Variables and Probability Distributions

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Random Variable A variable that varies in value by chance

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

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

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

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

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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

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Methods for Establishing Probabilities Empirical Method Subjective Method Theoretical Method

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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

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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

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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

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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

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

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

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

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Expected Value Let = E(X) = mean = expected value then = E(X) = x P(X=x)

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

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Variance Let ² = variance then ² = ( x - ) ² P(X=x)

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Standard Deviation Let = standard deviation then = ²

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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

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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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