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W7-slides2

# W7-slides2 - Lecture 20 Outline and Examples Distribution...

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Lecture 20- Outline and Examples Distribution of a Function of a random variables (Ross § 5.7) Joint distributions (Ross § 6.1)

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Function of random variable- discrete case Example. Let X be uniform on the 19 integers {- 9 , - 8 , . . . , 8 , 9 } (i.e. P ( X = k ) = 1 / 19 for - 9 k 9.) Problem 1 . Find P ( X 2 5).
Function of random variable- discrete case Example. Let X be uniform on the 19 integers {- 9 , - 8 , . . . , 8 , 9 } (i.e. P ( X = k ) = 1 / 19 for - 9 k 9.) Problem 1 . Find P ( X 2 5). Solution . P ( X 2 5) = P ( - 5 X 5) = P ( X ∈ {- 2 , - 1 , 0 , 1 , 2 } ) = 5 / 19.

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Function of random variable- discrete case Example. Let X be uniform on the 19 integers {- 9 , - 8 , . . . , 8 , 9 } (i.e. P ( X = k ) = 1 / 19 for - 9 k 9.) Problem 1 . Find P ( X 2 5). Solution . P ( X 2 5) = P ( - 5 X 5) = P ( X ∈ {- 2 , - 1 , 0 , 1 , 2 } ) = 5 / 19. Problem 2 . Find P ( | X - 2 | ≤ 5).
Function of random variable- discrete case Example. Let X be uniform on the 19 integers {- 9 , - 8 , . . . , 8 , 9 } (i.e. P ( X = k ) = 1 / 19 for - 9 k 9.) Problem 1 . Find P ( X 2 5). Solution . P ( X 2 5) = P ( - 5 X 5) = P ( X ∈ {- 2 , - 1 , 0 , 1 , 2 } ) = 5 / 19. Problem 2 . Find P ( | X - 2 | ≤ 5). Solution. P ( | X - 2 | ≤ 5) = P ( - 5 X - 2 5) = P ( - 3 X 7) = 11 / 19.

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Function of random variable - continuous case Continuous r.v.s. - Question: If X has density f X and Y = g ( X ), then what is the density function of Y ? Method : Use Distribution functions. Step 1. Determine values of Y . Step 2. Write cdf of Y : F Y ( y ) = P ( Y y ) = P ( X some interval A y ) = R A y f ( x ) dx ; calculate. Step 3. Take derivative: f Y ( y ) = F 0 Y ( y )
Square of a r.v. X Example. Square of a r.v. X with density f X . Let Y = X 2 . Find the density of Y . Solution.

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Square of a r.v. X Example. Square of a r.v. X with density f X . Let Y = X 2 . Find the density of Y .
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• Spring '10
• RohiniKumar
• Probability distribution, Probability theory, probability density function, Cumulative distribution function, Marginal distribution

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W7-slides2 - Lecture 20 Outline and Examples Distribution...

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