Continuous RV, Random Walk, Covariance

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

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
(CONTINUOUS) RANDOM VARIABLES Continuous RVs Normal Sampling distributions and Central Limit Theorem Random Walks Covariance and Correlation
Background image of page 2
Continuous RVs Normal Sampling distributions and Central Limit Theorem Random Walks Covariance and Correlation
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
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 ]
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 6
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
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 8
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
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/07/2010 for the course ECON 0001 taught by Professor Kitsikopoulos during the Spring '08 term at NYU.

Page1 / 112

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

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online