# chap4 - Continuous random variables Chapter 4 excluding...

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Continuous random variables Chapter 4 excluding 4-9,4-10 1

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Discrete random variables z Bernoulli, Binomial, Geometric, Poisson z Binomial: # of successes out of n trials z Geometric: # of trials needed until first success z Poisson: # of events occur in an interval/region z Probability mass function z Mean and variance z Mean: probability weighted average z Variance: probability weighted squared distance 2
Continuous random variables z What is the counterpart of probability mass function for a continuous random variable X ? z What about f(x)=P(X=x) for all real x ? z For a continuous random variable, P(X=x)=0 for any possible value x 3

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z Drop a needlepoint randomly to line segment [0,1]. Assume the needlepoint lands on any part equally likely z The probability that the needlepoint falls into a subinterval [a,b] is b-a (0 a<b 1) z The probability that you will hit a predetermined point is zero because the length of a point is zero 0 1 [ ] a b 4
z When you drop the needlepoint, it will definitely land at some point in [0,1] z The probability of landing at any specific point is zero z Contradiction? ± impossible event zero probability ± Zero probability does not necessarily mean that an event is impossible; however, it is extremely improbable 5

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z How to describe the probability distribution of a continuous r.v. when pmf does not work z An analogy - a long thin beam where weight is continuously distributed over the beam z The density function is known as f(x) g/cm z The weight at any point is zero; the weight of any interval [a,b] is b a dx x f ) ( 6
Probability density function z For a continuous r.v., the probability that X falls into an interval [a, b] can be represented by z f(x) is called probability density function (pdf) z P(a X b) is the area under f(x) between a and b ) ( ) ( = b a dx x f b X a P 7

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z Shape of the pdf determines how probabilities are distributed z Small f(x) corresponds to small amount of probability distribution around x, vice versa b a f(x)dx 8
Basic properties z f(x) 0 because probabilities are always non- negative z because P( −∞ X )=1 z Because the probability of any point is zero, This is not necessarily true for discrete r.v.’s 1 ) ( = +∞ dx x f = < < = < = < = b a dx x f b X a P b X a P b X a P b X a P ) ( ) ( ) ( ) ( ) ( 9

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Uniform distribution z What should be the shape of the pdf when all points in [0,1] are equally likely (landing point X of the needlepoint) z The probability density function f(x)=1, 0 x 1 f(x)=0, otherwise z For 0 a<b 1, a b dx dx x f b X a P b a b a = = = 1 ) ( ) ( 10