Linear combinations of random variables
DOWNLOAD
FREE
|
Linear Combinations of independent Normal distributions
If X and Y have independent Normal distributions, then aX + bY has a Normal distribution This also means that [Equation goes here ; ; download the original to see it.] Hence V Example A certain manufacturing process depends on the insertion of a metal rivet into a metal gasket.[Diagram goes here ; ; download the original to see it.] In order to create a high seal the gasket is heated. The initial diameter of the gasket, in mm, is [Equation goes here ; ; download the original to see it.] The external diameter of the rivet, in mm, i [Equation goes here ; ; download the original to see it.] a. Find the probability that a randomly selected rivet would fit inside a randomly selected gasket without heating. b. The gasket is heated so that it expands by 1%. Find to sig. fig. the proportion of rivets that fit the gasket. c. What percentage of rivets and gaskets randomly chosen will produce a high seal first time. Solution (a) We consider the distribution [Equation goes here ; ; download the original to see it.] [Equation goes here ; ; download the original to see it.] We want [Equation goes here ; ; download the original to see it.] [Diagram goes here ; ; download the original to see it.] The corresponding Z value is given by [Equation goes here ; ; download the original to see it.] (b) The heating corresponds to scale factor 1.01. Let ; 9;X ; 9; be the diameter of the gasket after heating [Equation goes here ; ; download the original to see it.] and [Equation goes here ; ; download the original to see it.] [Equation goes here ; ; download the original to see it.] [Equation goes here ; ; download the original to see it.] [Diagram goes here ; ; download the original to see it.] [Equation goes here ; ; download the original to see it.] (c)The proportion of good first time seals will be 1 ; ;(proportion not sealing when cold + proportion not filling when hot) [Equation goes here ; ; download the original to see it.]
|
Contents of
Linear combinations of random variables
1 Linear combinations of random variables
2 Independence
3 Scaling and translation of random variable X
4 Proof, for any discrete random probability distribution in general
5 Proof for any continuous random probability distribution
6 Expectation and variance of the linear combination of random variables
7 Form of the linear combination of random variables
8 Scaling and translation of a Normal distribution
9 Linear Combinations of independent Normal distributions
10 Linear combination of independent Poisson distributions.
|
