Chapter5 - Chapter 5 lecture notes Math 431 Spring 2011...

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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 counter-intuitive 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.

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Chapter5 - Chapter 5 lecture notes Math 431 Spring 2011...

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