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Week 6 and 7 Lecture Notes (1).pdf

Week 6 and 7 Lecture Notes (1).pdf - Econ 102A Introduction...

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Econ 102A Introduction to Statistical Methods for Social Scientists Stanford University Course Materials for Week 6 and Week 7 Professor Scott M. McKeon Autumn Quarter, 2017 - 18 © Scott M. McKeon All Rights Reserved
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Weeks 6 and 7 Goals: 1. Learning how to standardize a random variable. 2. Learning to compute covariance and correlation coefficients. 3. Understanding the role of covariance in portfolio optimization. 4. Becoming familiar with normal distributions. 5. Learning to use the normal distribution probability table. 6. Learning to interpret what the standardized value of a random variable represents. 7. Learning to recognize experiments with independent and identically distributed random variables. 8. Understanding the Central Limit Theorem of Sums verbally, visually and mathematically.
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Econ 102A Statistical Methods for Social Scientists Weeks 6 and 7 Definitions 1. A standardized probability distribution is a distribution having an expected value equal to 0 and a variance equal to 1. Notice that, if the variance of the distribution is 1, the standard deviation of the distribution is also 1. To create a standardized distribution, we standardize the random variable values, thereby creating new random variable values. Specifically, suppose x 0 is a random variable value. Then, we standardize x 0 according to the formula: x x 0 stan x σ μ x x z where x stan is the standardized value of x 0 , x μ is the expected value of x and x is the standard deviation of x. The standardizing process does nothing to the probability of x 0 occurring. So, when graphing the standardized distribution of x, we will use the new (standardized) random variable values along the x-axis, but the original probabilities remain unchanged. Upon transforming all the random variables into their standardized counterparts and graphing the distribution, we find that the new (standardized) distribution always has an expected value equal to 0 and a variance equal to 1. 2. The covariance between random variables X and Y (denoted Cov(X, Y)) measures the degree to which the random variables move in similar or opposite directions. Movement in the same direction (i.e., when one is high the other is high, or when one is low the other is low) implies a positive covariance while movement in opposite directions (i.e., when one is high the other is low) implies a negative covariance. 3. The correlation coefficient between random variables X and Y (denoted (X, Y)) is the covariance of the standardized values of X and Y. Handout #38 Page 1 of 1
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Econ 102A Statistical Methods for Social Scientists Correlation Between Stock Performance and Job Openings Recall the probability tree for the joint distribution example (notice, in particular, the new column on the far right): Handout #39 Page 1 of 2 Poor Fair Good .65 .20 .15 .60 .30 Aggressive Modest None .10 .2145 .1073 .0357 .48 .25 Aggressive Modest None .27 .0528 .0275 .0297 .38 .27 Aggressive Modest None .35 .0313 .0223 .0289 Poor Fair Good .28 .46 .26 .52 .34 Aggressive Modest None .14 .0655 .0428 .0177 .40 .30 Aggressive Modest None .30 .0828 .0621 .0621 .15 .27 Aggressive Modest None .58 .0175 .0316 .0679 Favorable Unfavorable Election Result?
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