Chapter 3 Conditional Probability and Independence1
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W say that X is a continuous random variable if there
Continuous random variable
Example
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A random variable is (standard) uniformly distributed if its density function f (x) follows
A random variable X~unif(α,β) has the following p.d.f. and c.d.f.
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So,
and,
So,
A variation of exponential distribution is the Laplace distribution, which has the density function
. So,
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The meaning of the ‘hazard rate’ is suggested by the following:
The hazard rate function uniquely determines the distribution function F. Why ?
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Suppose events are occurring randomly and in accordance with the three axioms for deriving Poisson distribution in sec.4.7.
As a result,
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Thus
Since
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Example
Sol.
thus
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The Cauchy distribution is an example of a distribution which has no mean, variance, or higher moments defined.
However,
Furthermore,
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where
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So,
So,
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So,
Proof
Example
Sol.