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### Lecture15

Course: ECON 220, Spring 2011
School: University of Toronto
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Lecture ECO220Y 15 Continuous Distribution Part 1 Part Migiwa Tanaka Reading: 8.1 1 Outline Introduction Discrete vs. Continuous Discrete Continuous Continuous Distribution Examples of Continuous Distribution Uniform Distribution Triangle Distribution Normal Distribution Student t Distribution F Distribution 2 Continuous Random Variable It can take any values within a interval --- infinite...

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Lecture ECO220Y 15 Continuous Distribution Part 1 Part Migiwa Tanaka Reading: 8.1 1 Outline Introduction Discrete vs. Continuous Discrete Continuous Continuous Distribution Examples of Continuous Distribution Uniform Distribution Triangle Distribution Normal Distribution Student t Distribution F Distribution 2 Continuous Random Variable It can take any values within a interval --- infinite possibilities. Many variables in the real world are continuous. th Height, Weight, Temperature, Speed We may only see finite set of values in data but it is because 3 Sample size is finite and Our ability to measure is limited. Discrete Discrete vs. Continuous Continuous Infinitely many possible values Finite possible values. Probability Distribution Probability Distribution P(X=x)=p(x) = the probability f(X=x)=f(x)=density of r.v. X at value value x. of of r.v. X taking the value x. Discrete Continuous In distribution diagram, height of of distribution =probability. Probability can be specified only only for a interval of values. Probability that r.v. X takes a value between x1 and x2 is area is below f(x) line for the interval. Single value does not have any probability why? 4 Continuous Distribution Di Properties of Continuous Distribution: 1. 2. f(x) 0 for all x. Area below f(x) across all x sum up to 1. f( f(x) P(x1Xx2) Area=1 0 5 x1 x2 x Are they probability density? f(x) f(x) 1 0 x 0 5x 6 Uniform Distribution (1) Di (1) Uniform Distribution is defined by following density function: What are the parameters of this distribution? Any values between a and b are equally likely to appear. There is no possibility that X can take value below a or above b. 7 0 density 1 f ( x) for a x b. ba 1 ba a b Uniform Distribution (2) Di (2) U[a, b] denotes uniform random variable X distributed between a and b. Mean ba E[ X ] 2 Variance P ( x1 X x2 ) 1 ba (b a ) 2 E [( X ) 2 ] 2 12 12 0 a x1 x2 b Probability that X takes values between x1 and x2 if a<x1<x2<b: 8 x2 P( x1 x1 X x2 ) ba = Height x Width Wid Application: Per customer waiting time at a fast food restaurant is uniformly distributed between 1 minute to 6 minutes. What is the probability that a randomly chosen customer waits between 2 to 4 minutes? Density of this distribution= 1/(6-1)=1/5 1/(6 P(2<X<4)= 1/5*(4-2)=2/5=0.4 What is the probability that a randomly chosen customer waits less than 4 minutes? P(X<3)=1/5*(4-1)=3/5=.6 What is the probability that a randomly chosen customer waits waits more than 5 minutes? P(X>5)=1-P(X5)=1-1/5*(5-1)=1/5=.2 9 Triangle Distribution (1) Di (1) Suppose X1~U[a, b], X2~U[a, b], b] X1 and X2 are independent. What does the distribution of X1+X2 look like? 2b] => Triangle distribution X1+X2~T[2a, 2b]. 10 Triangle Distribution (2) Di (2) 11 0 .2 .4 density .6 .8 1 1.2 0 .2 X1 .4 .6 0 .2 density .4 .6 .8 1 1.2 0 .5 1 X1+X2 1.5 2 X2 .8 1 0 .2 density .4 .6 .8 1 1.2 0 .2 .4 .6 .8 1 Intuition? Triangle Distribution (3) Di (3) For For X ~T[2a,2b], Mean? Remember that E[X1+X2] X X=X1+X2 X1~U[a,b] X2~U[a,b] X1 and X2 are independent Variance? and mean and variance of X1 and X2 are V[X1+X2] =V[X1]+V[X2]+2COV[X1+X2] ab E[ X 1 ] E[ X 2 ] , 2 (b a ) 2 V [ X1] V [ X 2 ] 12 12 Triangle Distribution - Probabilit Di bility 1.2 A. 1.2 C. 1 .8 .6 .4 .2 0 0 .5 1 1.5 2 0 .2 .4 .6 .8 1 0 .5 1 1.5 2 1.2 B. What is the shaded area of each diagram? Area of triangle= (Base x Height)/2 0 .2 .4 .6 .8 1 0 .5 1 1.5 2 .25 13 Summary Summary Uniform Distribution Notation: X~U[a, b] Shape: Triangle Distribution Notation: X ~ T[2a, 2b] Shape: Symmetric, Rectangle Even Density Bounded support: [a, b] Parameters: a and b Probability Symmetric Triangle More density around mean Bounded support: [2a, 2b] Parameter: 2a and 2b Probability =area=base x height Mean: Variance: 14 ab 2 (b a ) 2 2 12 =area=(base x height)/2 use symmetry ab Mean: (b a ) 2 Variance: 2 6
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