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