random_variables

# random_variables -

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± Discrete random variables ± Continuous random variables ± Cumulative distribution functions ± Simulating random variables ± Expected value and variance
d ib l dl f th t i ± Random variables are models for stochastic phenomena that result in a real number value ± Some applications with similar probabilities occur so frequently that general models have been developed to handle specific questions ormally a random variable X is a function from a ± Formally, a random variable, X, is a function from a sample space W to the real numbers X 0 W

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± A random variable, X, is discrete if it takes a finite or at most a countably infinite number of values uppose X takes the values x ± Suppose X takes the values x 1 , x 2 , … ± The probability mass function , p X ( Ä ), for X is given by ) ( ) ( i i X x X P x p = = ± Note that 1 ) ( = i X x p i
uppose you bet 10o 000 Dong on red ± Suppose you bet 10o,000 Dong on red coming up on one spin of a roulette wheel et X denote the net amount won ± Let X denote the net amount won 19 ) 100 ( , 18 ) 100 ( = = K p K p X X ± Suppose now X is net winnings after betting n red on two spins of the roulette wheel 37 37 on red on two spins of the roulette wheel 9 9 8 8 2 2 24 . 37 19 ) 200 ( .5, 37 19 37 18 2 ) 0 ( .26, 37 18 ) 200 ( = = = = = = K p p K p X X X

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he mulative distribution function f X is defined to be ± The cumulative distribution function of X is defined to be < < = x x X P x F for , ) ( ) ( ± CDFs for discrete random variables is a step function with X step height equal to mass given to value at step 1 .3 .5
is net winnings after betting on red on two ± X is net winnings after betting on red on two spins of the roulette wheel DF for X is ± CDF for X is < < x K K x 0 200 26 . 200 0 < = K x x F X 00 200 0 76 . ) ( < x K 200 1

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± Often used as an “indicator” random variable ± PDF q p p p p X X = = = ) 1 ( ) 0 ( , ) 1 ( ± Example ± X equals 1 if an individual reproduces, otherwise X = 0

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± sed to model the number of successes in n Used to model the number of successes in n independent repetitions of an experiment ± wo parameter class of distributions: ( ,p Two parameter class of distributions: (n,p) n k - n k ( ) ( ) n k p k k p X K , 1 , 0 , p - 1 ) ( = = B(10,.5)
± uppose both parents are carriers for a single Suppose both parents are carriers for a single locus genetic disease the couple has four children what is the ± If the couple has four children, what is the probability mass function for the number of children that have the disease? 4 1 4 0 , 4 3 4 1 1 4 ) 1 ( , 4 3 4 1 0 4 ) 0 ( = = X X p p 1 3 2 2 4 3 4 1 3 4 ) 3 ( , 4 3 4 1 2 4 ) 2 ( = = X X p p 0 4 4 3 4 1 4 4 ) 4 ( = X p

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random_variables -

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