Random Variable Intro
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INTRODUCTION TO RANDOM VARIABLES A random variable is a function that associates a real number with each element in the sample space. Discrete Random Variable Example 1: The sample space giving a detailed description of each possible outcome when 3 components are tested. D=defective, N=nondefective S={ NNN, NND, NDN, DNN, NDD, DND, DDN, DDD} = 8 n S = 2x2x2 = 8 possible outcomes. x= the number of defective components among the 3 selected components. x= {0,1,2,3} P(x=0)=P(all components selected are nondefective)= 1/8 x=0; {NNN} P(x=1)=P(1 of 3 components is defective) = 3/8 x=1; {NND,NDN,DNN} P(x=2)=P(2 of 3 components selected are defective)=3/8 x=2; {NDD,DND,DDN} P(x=3)=P(all defective)=1/8 x=3; {DDD}
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Random Variable Intro
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INTRODUCTION TO RANDOM VARIABLESA random variable is a function that associates a real number with each element in the sample space.Discrete Random VariableExample 1: The sample space giving a detailed description of each possible outcome when 3 components are tested.D=defective, N=nondefectiveS={ NNN, NND, NDN, DNN, NDD, DND, DDN, DDD} = 8nS= 2x2x2 = 8 possible outcomes.x= the number of defective components among the 3 selected components. x= {0,1,2,3}P(x=0)=P(all components selected are nondefective)= 1/8x=0; {NNN}P(x=1)=P(1 of 3 components is defective) = 3/8x=1; {NND,NDN,DNN}P(x=2)=P(2 of 3 components selected are defective)=3/8x=2; {NDD,DND,DDN}P(x=3)=P(all defective)=1/8x=3; {DDD}
Example 3.1, p78 WalpoleURN={ R=4, B=3} two balls are drawn in succession w/o replacement.Let y = number of red balls drawn = {0,1,2}Sample Space : nS = 2x2 = 4 S={RR (y=2), RB (y=1), BR (y=1), BB (y=0)}P(y=0)=1/4, P(y=1) =2/4 = , P (y=2) = .
DISCRETE PROBABILITY DISTRIBUTIONThe set of all ordered pairs (x,f(x)) is a probability distribution, probability function or probability mass function if the following conditions are satisfied.1. 2. 3. P(x=a)= f(a).
Example 3: In Example 1, for all x, .. Thus, f(x) is a probability distribution.
Example 3.8, p80: Find the probability distribution of f(x).Solution:8 microcomputers (D=3, N=5) to purchase 2 computers.FIND: the probability distribution for the number of defective computers.nS= 8C2 = 28.x= number of defectives = 0,1,2 2 x = nondefective
The probability of getting two nondefective (D=0, N=2) is:
The probability of getting 1 defective computer (D=1, N=1) is:
The probability of getting 2 defective computers (D=2, N=0) is:
Tabulated probability distribution.X012
f(x)10/2815/283/28
Note: Sum of f(x)=10/28 + 15/28 + 3/28 = 1.
The cumulative distribution function of a discrete random variable is
EXAMPLE 5: Find the cumulative distribution of x (discrete) in example 1.Probability distribution: f(0) = 1/8, f(1) = 3/8, f(2) = 3/8, f(3) = 1/8,x=0,1 , 2 ,3
Thus, the cumulative distribution of x is: Note: x should be defined over the entire real line, .Example 3.10 p.82: READ!!!
