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Continuous RV, Random Walk, Covariance

# Continuous RV, Random Walk, Covariance - C22.0103...

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C22.0103: Statistics for Business Control: Regression and Forecasting Hong Luo Section 003, Spring 2009 Tue/Thu/Fri, 11:00 - 12:15pm, Tisch 200 Stern School of Business New York University

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(CONTINUOUS) RANDOM VARIABLES Continuous RVs Normal Sampling distributions and Central Limit Theorem Random Walks Covariance and Correlation
Continuous RVs Normal Sampling distributions and Central Limit Theorem Random Walks Covariance and Correlation

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Continuous RVs A random variable is continuous if it can take any real value in some interval. For example: I height, weight, distance, time, and volume I prices, sales, income, stock returns Remember what is a discrete variable? counts are discrete. I sheep, pizzas, homeruns I the number of occurrences of some future event is discrete
Continuous RVs I Continuous distributions are described by smooth curves called probability density function (pdf) f ( x ). I f ( x ) is a continuous function of x , and f ( x ) 0 Discrete Uniform Distribution Continuous Uniform Distribution I Discrete uniform distribution: each value has an equal probability. e.g. die roll. I Continuous uniform distribution: all values have equal probability density. e.g. a random student picks a number randomly between [ a , b ]

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Probability of a Continuous RV I Key property of probability density: Probability = Area Under Curve I CDF F ( x ) is the area under the curve f ( · ) in the interval ( -∞ , x ]. F ( x ) = P ( X x ) = Z x -∞ f ( x ) dx
Probability of a Continuous RV I Probability that X is between a and b is P ( a < X b ) = R b a f ( x ) dx = R b f ( x ) dx - R a f ( x ) dx = P ( X b ) - P ( X a ) I Thus the total area under f ( x ) is 1: P ( -∞ < X < ) = Z -∞ f ( x ) dx = 1 I e.g. for the uniform distribution, what should be the height of the line? 1 b - a

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Probability of a Continuous RV I What is the probability of X = x ? I 0! Why? I There is no area under the curve between x and x I Since there are an inﬁnite number of values for a continuous RV, if each single value has a positive probability, then, the total probability would add up to ! I Therefore, P ( X = x ) = 0 for any speciﬁc x , and f ( x ) is now interpreted as a relative intensity. I For continuous RV, it only makes sense to talk about probability of an interval, not the probability of a particular value.
EX: Light Bulb Lifetimes A box of light bulbs states “average life 2000 hours”. I What is the probability a light bulb fails at exactly 2000 hours? Let the life of the light bulb be represented by the RV T , and let’s model T using the exponential distribution f ( t ) = 1 μ e - t μ where I μ is the mean (we’ll assume it’s 2000). I

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Continuous RV, Random Walk, Covariance - C22.0103...

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