uniform and exponential

uniform and exponential - 0. Also, the parameter is a...

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Section 8.4 Uniform and Exponential Random Variables Random Variables whose PDFs are constant on their ranges characterize uniform random variables. Uniform R.V. The Uniform Distribution has 2 parameters; a and b. Both a and b are real numbers, and a<b. A continuous r.v. X is called a Uniform R.V. if, for some finite interval (a,b) of real numbers, its value is equally likely to lie anywhere in that interval. Equivalently its PDF is constant on that interval and 0 elsewhere. We write X~U(a,b) when X is Uniformly distributed on the interval (a,b). PDF: a b x f X = 1 ) ( for the interval a ≤ x ≤b, and 0 elsewhere. CDF: a b a x x F X = ) ( for the interval a ≤ x ≤b. = ) ( x F X 0 for x < a. = ) ( x F X 1 for x>b. Mean is 2 b a + . Variance is 12 ) ( 2 a b . Exponential R.V. Defn: A continuous R.V. is called an exponential R.V. if its PDF is of the form: It can be thought of as the continuous analog of the geometric distribution. x X e x f λ = ) ( for any x such that x is > 0 0 ) ( = x f X for x such that x is
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Unformatted text preview: 0. Also, the parameter is a positive real number. The exponential r.v. is often used as the distribution for the time required to complete a certain task or for the elapsed time between successive occurrences of a specified event. CDF: = ) ( x F X 0 for x 0. x X e x F = 1 ) ( for x &gt; 0. Mean: 1 Variance: 2 1 Special Property 1: Memorylessness: If X is exponentially distributed, then its conditional probability obeys the following property: s t X P + &gt; ( X &gt; s) = P(X &gt; t) for all s, t 0. Example numbers: P(X&gt;50 X&gt;35) = P(X &gt; 15). Special Property 2: Tail Probability Function: If X is exponentially distributed, then we have a specific formula for X &gt; x. x x X e e x F CDF x X P = = = = &gt; ) 1 ( 1 ) ( 1 1 ) ( . Say X is exponentially distributed with lambda equal to 10. Then, P(X&gt;50 X&gt;35) is 150 e ....
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uniform and exponential - 0. Also, the parameter is a...

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