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continuous r.v.s

continuous r.v.s - Continuous Random Variable A continuous...

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Continuous Random Variable: A continuous random variable typically involves measurement. A random variable X is called a continuous random variable if P(X=x)=0 for all x in (this is the symbol for real numbers). Note (0,1) signifies {x: 0 < x < 1} but [0,1] means {x: 0 ≤ x ≤ 1} Let X be a random variable. Then the cdf of X, denoted Cumulative Distribution Functions ) ( x F X is the real-valued function defined on by ) ( x F X = P(X ≤ x) such that x is in . The CDF applies to any type of random variable (but we typically only use it with respect to continuous random variables). The CDF completely determines the probability distribution. Let us find the CDF of a coin tossing example (n=4, p=.3, X = # of heads). What happens to the CDF if we change p to .5, ceteris peribus? Let X denote a number selected at random from the interval (0,1), what is the CDF of X? What about if we change the interval from (0,1) to (0,10)? 1. It is nondecreasing Properties of a CDF 2. It is everywhere right-continuous 3. It has a value of 0 for x= - 4. It has a value of 1 for x= +
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Useful Identities: 1) P(a < X ≤ b) = ) ( ) ( a F b F X X 2) P(a ≤ X ≤ b) = ) ( ) ( a F b F X X 3) P(a < X < b) = ) ( ) ( a F b F X X 4) P(a ≤ X < b) = ) ( ) ( a F b F X X Most of the above are really important when we have a CDF that has a “jump”. However, the idea of the
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  • Spring '08
  • MARTIN
  • Probability theory, probability density function, Cumulative distribution function, CDF, continuous random variable

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