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Unformatted text preview: Chapter 5 lecture notes Math 431, Spring 2011 Instructor: David F. Anderson Chapter 5: Continuous Random Variables Not all random variables are discrete. Think about 1. Waiting times for anything (train, arrival of customer, production of mRNA molecule from gene, etc). 2. Distance a ball is thrown. 3. Size of an antennae on bug. The general idea is that now the set of possible values is uncountable . Probability mass functions and summation no longer works. Example : train arrives between 2am and 4 am. Function of time in each small window [ t i ,t i + Δ). Call function f . Then, P { a ≤ X ≤ b } ≈ n X i =1 f ( t i )Δ ≈ Z b a f ( t ) dt. Definition 1. We say that X is a continuous random variable if there exists a nonnegative function f , defined for all x ∈ (∞ , ∞ ), having the property that for any set B ⊂ R , P { X ∈ B } = Z B f ( x ) dx. The function f is called the probability density function of the random variable X , and is (sort of) the analogue of the probability mass function in the discrete case. So probabilities are now found by integration, rather than summation. Note: we must have that 1 = P {∞ < X < ∞} = Z ∞∞ f ( x ) dx. Also , taking B = [ a,b ] for a < b we have P { a ≤ X ≤ b } = Z b a f ( x ) dx, note that taking a = b yields the moderately counterintuitive P { X = a } = Z a a f ( x ) dx = 0 . 1 Example 1. The amount of time you must wait, in minutes, for the appearance of an mRNA molecule is a continuous random variable with density f ( t ) = λe 3 t , t ≥ , t < . . What is the probability that 1. You will have to wait between 1 and 2 minutes? 2. You will have to wait longer than 1/2 minutes? Solution : First, I haven’t given λ . We need 1 = Z ∞∞ f ( t ) dt = Z ∞ λe 3 t dt = λ 3 e 3 t ∞ t =0 = λ 3 . Therefore, λ = 3, and f ( t ) = 3 e 3 t for t ≥ 0. Thus, P { 1 < X < 2 } = Z 2 1 3 e 3 t dt = e 3 * 1 e 3 * 2 ≈ . 0473 P { X > 1 / 2 } = Z ∞ 1 / 2 3 e 3 t dt = e 3 / 2 ≈ . 2231 . Recall: for discrete random variables the probability mass function can be reconstructed from the distribution function and vice versa, and changes in the distribution function cor responded with finding where the “mass” of the probability was. We now have that F ( t ) = P { X ≤ t } = Z t∞ f ( x ) dx, and so F ( t ) = f ( t ) , agreeing with the previous interpretation. Also, P ( X ∈ ( a,b )) = P ( X ∈ [ a,b ]) = etc. = Z b a f ( t ) dt = F ( B ) F ( a ) . 2 The density function does not represent a probability . However, its integral gives prob ability of being in certain regions of R . Also, f ( a ) gives a measure of likelihood of being around a . That is, P { a / 2 < X < a + / 2 } = Z a + / 2 a / 2 f ( t ) dt ≈ f ( a ) , when is small and f is continuous at a . Thus, the probability that X will be in an interval around a of size is approximately f ( a )....
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This note was uploaded on 04/17/2011 for the course MATH 431 taught by Professor Balazs during the Spring '05 term at University of Wisconsin.
 Spring '05
 BALAZS
 Math, Probability

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