Ch4 Special Distributions

Ch4 Special Distributions - Stat1801 Probability and...

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Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 p.82 Chapter IV Special Distributions In applications of probability, certain families of distributions arise quite frequently and it is important to have a thorough understanding of these frequently occurring distributions and their properties. § 4.1 Uniform Distribution Definition For an interval () b a , , let X be the point randomly drawn from this interval. If the pdf of X is a constant function on ( ) b a ,, i . e . < < = otherwise , 0 , 1 b x a a b x f , then X is said to have a uniform distribution and is denoted as b a U X , ~. f ( x ) x 0 a b Roughly speaking, it is a point randomly selected in such a way that it has no preference to be located near any particular region. Distribution function < < = b x b x a a b a x a x x F , 1 , , 0 Mean 2 b a X E + = = μ (midpoint of interval) Variance ( ) 12 2 2 2 2 a b X E = = σ
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Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 p.83 Example 4.1 A needle drops freely onto a horizontal plane. Let θ be the angle from the north direction to the needle tip: π 2 0 < . Then ( ) 2 , 0 ~ U . μ = + = 2 2 0 , () 2 2 2 3 1 12 0 2 σ = = 2 0 2 0 = = F for 2 0 < = 2 4 E and NE between direction towards pointing P P = 4 2 ππ F F 125 . 0 4 2 2 1 = = Property Let 1 , 0 ~ U X , d cX Y + = , then d c d U Y + , ~ i f c is positive; d d c U Y , ~ + if c is negative. Proof () ( ) x x x X P x F X = = = 0 1 0 for 1 0 x . If 0 > c , then the cdf of Y is given by () ( ) ( ) y d cX P y Y P y F Y + = = d d c d y c d y c d y X P + = = = for d c y c + Compare with the cdf of a uniform random variable, we have ( ) d c c U Y + , ~. Similarly if 0 < c , then ( ) c d c U Y , ~ + .
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Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 p.84 Example 4.2 If () 1 , 0 ~ U X , then 1 , 1 ~ 1 2 U X . If 6 , 2 ~ U X , then 1 , 0 ~ 8 2 U X + . § 4.2 Bernoulli Trials and the Binomial Distribution Bernoulli experiment Possible outcome X Probability Success 1 p Fail 0 p 1 e.g. Trial Success Failure Tossing a coin Head Tail Birth of a child Boy Girl Pure guess in multiple choice Correct Wrong Randomly choose a voter Support Not support Randomly select a product Non defective Defective Insurance policy holder Claim Not claim pmf of X : () ( ) x x p p x p = 1 1, 1 , 0 = x We call the distribution of X “Bernoulli distribution”. The outcome of each experiment is called the “Bernoulli trial”. p = μ , ( ) p X E = 2 , ( ) p p p p = = 1 2 2 σ . Binomial Distribution Let X be the random variable denoting the number of successes in n Bernoulli trials. If these n Bernoulli trials are: (i) having the same success probability p , and (ii) independent , i.e. the success probability of any trial is not affected by the outcome of other trials; then X is said to have a binomial distribution with n trials and success probability p . It is denoted as p n b X , ~.
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Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 p.85 Example 4.3 Let X be the number of boys in a family with four children.
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This note was uploaded on 02/01/2012 for the course STAT 1801 taught by Professor Mrchung during the Fall '10 term at HKU.

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Ch4 Special Distributions - Stat1801 Probability and...

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