oh2 - LECTURE 2: STATISTICS REVIEW RANDOM VARIABLES X is a...

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LECTURE 2: STATISTICS REVIEW RANDOM VARIABLE S X is a random variable if it takes di f erent values according to some probability distribution Types of Random Variables: Discrete Random Variable Takes on a f nite or countably in f nite number of values Example: outcome of a coin toss Continuous Random Variable Takesonanyva lueinarea linterva l Each speci f c value has zero probability Example: height of an individual at UCLA A probability distribution is best described by the corre- sponding probability density function and the cumulative
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PROBABILITY DISTRIBUTION FUNCTION (PDF) Summarizes the information concerning the possible out- comes of X and the corresponding probabilities The PDF of a discrete random variable X that takes on values, say x 1 ,x 2 ,...,x p , is de f ned as: f ( x j )= ½ Pr( X = x j ) for j =1 , ..., p 0 for X 6 = x j The PDF of a continuous random variable is similar to that of a discrete random variable except we now mea- sure the probability the random variable is in a certain range or interval. It is de f ned as the derivative of the cumulative distribution function (CDF), de f ned next. Graph of PDF here 2
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FEATURES OF PROBABILITY DISTRIBUTIONS A. MOMENTS : Summary statistics of the distribution 1. Expected Value or Expectation or Mean μ X def = E ( X )= ½ P p j =1 x j f ( x j ) for X discrete R xf ( x ) dx for X continuous Measure of central tendency Weighted average of the possible values of X with the probability f ( x ) serving as weights Population mean value 2. Variance σ 2 X def = Var ( X )= E ¡ ( X μ X ) 2 ¢ = ½ P p j =1 ( x j μ X ) 2 f ( x j ) for X discrete R ( x μ X ) 2 f ( x ) dx for X continuous Measure of dispersion: how the values of X are spread around its mean Always positive The square root of the variance is called the stan- dard deviation: σ X def = p Var ( X ) 3
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PROPERTIES OF EXPECTATION AND VARIANCE 4
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RELATIONSHIP BETWEEN RANDOM VARIABLES In economics we are usually interested in phenomena that involve more than one random variable. Thus we have to study joint (multivariate) distributions. JOINT PROBABILITY DENSITY FUNCTION (PDF) The joint probability density function of two discrete ran- dom variables X and Y is the function f ( x, y ) such that, for any point ( x, y ) in the X Y plane, f X,Y ( x, y )=Pr( X = x, Y = y ) Properties of Joint Probability Density Func- tion: f ( x, y )= P ( X = x and Y = y ) 0 for all ( x, y ) pairs X all x X all y f ( x, y )=1 For the continuous case: Z x Z y f ( x, y )=1 5
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MARGINAL AND CONDITIONAL PDF’S From the joint PDF we can derive the marginal PDF for each of the random variables : Discrete Case: f X ( x )=P r ( X = x )= X all y f X,Y ( x, y ) f Y ( y )=P r ( Y = y )= X all x f X,Y ( x, y ) Continuous Case: f X ( x )= Z f X,Y ( x, y ) dy f Y ( y )= Z f X,Y ( x, y )
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oh2 - LECTURE 2: STATISTICS REVIEW RANDOM VARIABLES X is a...

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